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**HIS GLOSSARY** comprises some of the terms that
geometers use to describe objects in higher-dimensional spaces.
We need a specialized vocabulary because most of the
multidimensional figures we discuss either have no common names
at all or have very cumbersome, notational names that are awkward
to use. Which is more convenient: the word “glome,” or
the phrases “hypersphere in four-space” or
“four-dimensional hypersphere”?

Many of these terms were recently created by Jonathan
Bowers, Norman Johnson, or me as part of our Uniform Polychora
Project and do not yet appear in standard geometry texts. Part of
the purpose of the Glossary for Hyperspace is to acquaint readers
with these terms. These are *working* definitions; I make no
particular claims of rigor or formality or abstractness here!
Many of these definitions have extensions and generalizations
that are not particularly relevant to the kind of geometry
we’re doing, so do not expect the Glossary to cover all
aspects of the concepts being described. I’m trying to be
thorough here, but not to the point of absolute formality
or completeness.

The Glossary is (or should be) self-contained. If one finds an unfamiliar term in a Glossary entry, it has (or should have) an entry of its own in the Glossary. Mathematical terms that everyone should be familiar with, however, do not necessarily appear in the Glossary.

Links to latest terms/revisions in Glossary:

Alteration

Alternation

Antiprism

Apiculation

Faceting

Glome

Hecatonicosachoron,
regular

Hexacosichoron,
regular

Hexadecachoron,
regular

Hypersphere

Icositetrachoron,
regular

Measure polytope

Monogon

Pentachoron, regular

Surcell

Tessellation

Tesseract

Tesseract army

Transitivity

This Web page is perpetually under construction, and new terms will be added to the Glossary from time to time as needed. Revisions, corrections (nothing is perfect the first time out!), and other updates will also appear whenever required. I am particularly grateful to Norman Johnson and Jonathan Bowers for their reviews of the entries. I intend eventually to add some desperately needed illustrations to the Glossary, and if it grows too large I will have to make separate Web pages for some of the longer entries. Readers who wish to comment on or correct the contents should email me directly at Polycell.

Text ©2001 George Olshevsky. Use of the terminology as set forth herein for mathematical discourse and publication is, of course, strongly encouraged!

**Logo polyhedra**: On the left is
the vertex figure of a
uniform star-polychoron
from the *gadtaxady* regiment. Jonathan Bowers calls
this star-polychoron a *gard tapady*, for *grand
retroditetrahedronary prismatodishecatonicosachoron*. Its cells are 120 great stellated
dodecahedra, 120 great rhombidodekahedra, and 720 pentagrammatic
prisms; its faces are 3600
squares, 1440 pentagrams, and 720 decagrams. On the right is the
vertex figure of its conjugate, from the
*sidtaxhi* regiment. Jonathan calls this star-polychoron a
*sidtapady*, for *small ditetrahedronary
prismatodishecatonicosachoron*. Its cells are 120 dodecahedra,
120 small rhombidodekahedra, and 720 pentagonal prisms; its faces
are 3600 squares, 1440 pentagons, and 720 decagons. Both
star-polychora have 3600 edges,
and the 600 vertices of a hecatonicosachoron.
These vertex figures show how the cells come together at each
corner. Both vertex figures are toroidal polyhedra, which is not
quite so obvious in the right figure as in the left; the holes
opening into the center of the right figure are smaller. A
hypothetical four-dimensional model-maker could pass a
four-dimensional thread through the hole under a vertex and
thereby hang either star-polychoron from the ceiling. No
three-dimensional uniform polytope has this property.

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*See*Apiculation.

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- Stellation of a core polychoron
in four-space. In aggrandizement, the cells of*P*expand in their realms, passing through other such realms if necessary, to meet the expansions of other facets of the polychoron. This is usually done symmetrically, so that identical cells of the core polychoron expand exactly the same way, to produce a star-polychoron with the same symmetry group as the core polychoron. A synonym for aggrandizement is*P**cell-stellation*.

- A more restrictive definition of aggrandizement requires
that the newly expanded cells be similar to the original cells.
Then, given a regular polychoron
, the*R**grand*has the same kinds of cells as*R*, only larger and more interpenetrating.*R**See also*Hecatonicosachoron, regular; Hexacosichoron, regular; Regular polytope; Stellation; Symmetry.

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- Respectively, the product and process of producing a new
polytope by joining or blending stellachunks of a convex central
polytope with one another. Aggregates are simple polytopes, since
their facets have no interior regions. Aggregation is the kind of
stellation described for the regular icosahedron by H. S. M.
Coxeter, P. Du Val, H. T. Flather & J. F. Petrie in their 1938
publication
*The Fifty-Nine Icosahedra*(reprinted in 1999 by Tarquin Publications). Calling their operation*stellation*, they enumerated all the externally different irreducible aggregates of the icosahedron that preserve either the 60 rotational symmetries or the full 120 symmetries of the regular icosahedron.

- It is useful to examine the distinction between the kind
of stellation described in
*The Fifty-Nine Icosahedra*and the kind of stellation known as*greatening*, which is what produces, for example, the great icosahedron from the icosahedron. In*The Fifty-Nine Icosahedra*, the great icosahedron appears as figure G—except that it doesn’t! The great icosahedron is a polyhedron whose faces are 20 interpenetrating equilateral triangles that lie in the face planes of a core icosahedron. The polyhedron illustrated as figure G*looks like*the great icosahedron, but because of the way it is aggregated from the icosahedron’s stellachunks, its faces are 180 rather long and slender triangles: 120 scalene and 60 isosceles. Six of the former and three of the latter lie coplanar in each of 20 face-planes. The reason figure G looks like a great icosahedron is that these 180 triangles are precisely the external parts of the triangular faces of the great icosahedron. In other words, figure G is the*surhedron*of the great icosahedron: the simple polyhedron whose faces, all nonintersecting, are the external parts of the great icosahedron’s faces.

- So aggregation produces only “hollow shell”
polytopes: aggregates, such as surtopes. The other kinds of
stellations, such as edge-stellation, greatening, and
aggrandizement, produce polytopes with internal structure (which
would be readily apparent to a hypothetical being in a
higher-dimensional space).
*See also*Aggrandizement; Greatening; Join; Stellation; Surtope.

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- A kind of faceting that creates a new polytope or compound
polytope a
from a polytope*P*simply by cyclically reordering the vertices of the faces. The usual reordering is to put the vertices in alternating order, so that a face whose vertices are numbered {1,2,3,...} becomes a face (or pair of coplanar faces) with the same vertices in order {1,3,5,...,2,4,6,...}. For polytopes of dimension greater than two, new faces and higher-dimensional elements must almost always be constructed to take up newly created free edges, and old faces and higher-dimensional elements may be removed whose edges are no longer elements of a*P*. Alteration splits an even-sided polygonal face into a compound of two faces (or, if the face is a quadrilateral, into a pair of edges, the diagonals of the quadrilateral), which is why a*P*may become a compound of two polytopes. In this case, either polytope will be an alternated*P*, that is, h*P*. Among symmetric polytopes, alteration frequently produces a new uniform polytope or a compound of two new uniform polytopes from another uniform or isogonal polytope.*P*

- For example, an altered triangle a
is simply the same triangle*T*with its vertices in reverse order; an altered regular pentagon a{5} is a regular pentagram {*T*^{5}/_{2}}; an altered regular pentagram a{^{5}/_{2}} is a regular pentagon {5} (the vertices of aa{5} are the vertices of {5} backward); and so on. Extending this construction to polyhedra shows that an altered dodecahedron a{5,3} is the uniform star-polyhedron known as a*small ditrigonary icosidodecahedron*, and an altered great stellated dodecahedron a{^{5}/_{2},3} is the uniform star-polyhedron known as a*great ditrigonary icosidodecahedron*. An altered pentagonal prism is a pentagrammatic antiprism. An altered cube a(4,3) is the stella octangula compound of two regular tetrahedra, either of which is an alternated cube h{4,3} = {3,3}. The altered tesseract a{4,3,3} is the regular compound of two hexadecachora {4,3,3}*2*[{3,3,4}], either of which is an alternated tesseract.*See also*Alternation; Faceting.

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- A kind of partial truncation in which exactly half of the
vertices of a symmetric polytope
are removed and replaced by stump facets. The truncating hyperplanes are situated so as to pass through the vertices adjacent to the vertex being removed. The resulting polytope h*P*retains half the vertices (and half the symmetries) of the original polytope*P*, and its facets are (1) the facets of*P*alternated (if these exist as facets), and (2) the vertex figures of*P*where the truncated vertices once were. For example, an alternated regular polygon of 2*P**p*sides, h{2*p*}, is a regular polygon of*p*sides, {*p*}, and an alternated cube, h{4,3}, is a regular tetrahedron, {3,3}. An alternated tesseract h{4,3,3} is a regular hexadecachoron {3,3,4}, and in general an alternated measure polytope is a half measure polytope or demihypercube.*See also*Alteration; Half measure polytope; Truncation.

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- A polyhedron whose faces consist of two congruent, parallel
*p*-gons and a belt of 2*p*triangles positioned so that each triangle shares one edge with a base and the opposite vertex with the other base, and its other two edges with a triangle on either side of it. When the triangles are all congruent and equilateral and the bases are convex regular*p*-gons, the antiprism is Archimedean. (If*p*=3, the Archimedean antiprism is a regular octahedron.) If instead the bases are regular star-polygons, the antiprism is Colohimian.

- Analogues of antiprisms exist in spaces of dimension
greater than three, but uniform analogues are quite rare. The
regular hexadecachoron is a uniform antiprism whose bases are two
parallel regular tetrahedra oriented so that the corners of
either are situated opposite the faces of the other. The bases
are connected by 14 more regular tetrahedra defined by either a
face of one base and the corresponding vertex of the other, or by
a pair of corresponding edges of the bases.

- The general four-dimensional antiprism is
constructed with a particular polyhedron as a base and its dual
polyhedron as the other base. Only then will a vertex of one base
have a corresponding face in the other, allowing construction of
a pyramid as a lateral cell; or an edge in one base have a
corresponding edge in the other, allowing construction of a
disphenoid-like tetrahedron as a lateral cell. Such polychora
have no hope of even being isogonal, let alone uniform, unless
their bases are self-dual.

- If we allow higher-dimensional antiprisms to have
antiprisms as well as simplexes among their lateral facets
joining two base polytopes, then three such uniform antiprisms
are known in four-space and two more in five-space. These were
discovered by Norman Johnson in the 1960s. All are
star-antiprisms.

- In three dimensions, any two
*p*-gons, regular or not, convex or not, congruent or not, parallel or not, can serve as bases for a generalized antiprism, as long as no vertex of either base lies in the plane of the other. Simply join the sides of one*p*-gon in cyclic order to the vertices of the other*p*-gon with triangles, and vice versa, making sure that each vertex common to two edges of one*p*-gon is in the triangle whose other two vertices are in the two triangles formed from those edges. Because any three noncollinear points determine a triangle, this construction always yields a polyhedron. If the*p*-gons are wildly irregular, the resulting antiprism will be, too.*See also*Hexadecachoron, regular; Johnson antiprisms; Prismatic polytopes; Retroprism.

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- An Archimedean prism in four-space based on an Archimedean or
Colohimian antiprism. There is one (Archimedean) for every convex
regular polygon and one or two (Colohimian: antiprismatic or
retroprismatic) for every regular star-polygon. If the polygon is
an equilateral triangle, the antiprismatic prism is also the
*octahedral prism*. To name an Archimedean antiprismatic prism it is sufficient to specify the base polygon of the base and whether the base is an antiprism or a retroprism. Thus the*square antiprismatic prism*is the Archimedean antiprismatic prism whose base is a square antiprism; and the*pentagrammatic retroprismatic prism*is the Archimedean antiprismatic prism whose base is a pentagrammatic retroprism.*See also*Antiprism; Archimedean polytopes; Colohimian polyhedra; Retroprism.

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- The tiling of a straight line by line segments. If the line
segments are of equal length, the apeirogon is regular. This is
(trivially) the only regular tiling of the line.
*See also*Honeycomb; Infinite polytopes; Tiling.

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- The operation of constructing a new polytope
v
from a polytope*P*by joining pyramids to the facets of*P*. Each pyramid must, of course, have as its base an exact copy of the facet to which it is to be joined and have a single vertex as its apex. This operation increases the number of vertices of*P*by one for each apiculated facet, decreases the number of facets of*P*by one for each apiculated facet, and adds to the elements of*P*all the lateral elements, from edges to facets, of the apiculated pyramid. If*P*is convex, it is always possible to apiculate pyramids onto all of its facets that are low enough that v*P*remains convex. On the other hand, apiculating a tall enough pyramid will convert a polytope into a star-polytope. Apiculation may also be done by joining the pyramids*P**inwardly*to the facets, which results in a different kind of star-polytope. Inward apiculation is better known as*dimpling*, so we may reserve the term apiculation to mean*outward*apiculation. The dual operation to apiculation is truncation or quasitruncation, depending on the details of how the pyramids are apiculated (short or tall, inward or outward).

- Also, since we are primarily interested in symmetric
polytopes, what we usually mean by apiculation and dimpling is
the
*simultaneous*joining of congruent pyramids in the same way to the set of*all the facets*that the symmetry group of the polytope is transitive on, with the apices of the pyramids symmetrically positioned relative to their bases, so that the pyramids have the symmetry of the facets they are joined to. Since all the pyramids are congruent, they all have the same altitude, which can then be taken as the*altitude*of the apiculation itself. In dimpling, the altitude is negative.

- For example, a unit regular dodecahedron apiculated at an
altitude of approximately +0.2515 has twelve low identical
regular-pentagonal pyramids joined outwardly to all twelve
dodecahedral faces; it is the
*pentakis dodecahedron*, the Catalan dual of the truncated icosahedron. And a*dimpled dodecahedron*with altitude approximately –0.5257 is the aggregate of the regular icosahedron that has twelve identical pentagonal pyramids joined inwardly to all twelve faces; its 60 faces are all identical equilateral triangles that lie by threes in the face planes of the icosahedron.

- Other kinds of apiculations, such as joining only one or
two pyramids to one or two out of a larger number of faces, may
then be called
*asymmetric*,*subsymmetric*, or*partial*apiculations. For example, joining identical pentagonal pyramids of altitude approximately +1.3764 to two opposite faces of a unit regular dodecahedron produces a pentagonal antibipyramid, whose symmetry group is a subgroup of the symmetry group of the dodecahedron. It is a partial subsymmetric apiculation of the dodecahedron.

- The apiculation operation is sometimes called
“stellation,” but this is incorrect. Sticking pyramids
onto the facets of a core polytope may produce a pretty starlike
figure, but it is not stellation as defined in this Glossary. I
once called apiculation
*akisation*, because the names of the Catalan duals of the truncated Archimedean polyhedra contain the Greek root*-akis*, which means “times” (as in multiplication); but for some reason Norman Johnson dissuaded me. He had originally called this operation*accretion*, but has since renamed it apiculation because it describes the addition of new apices to a polytope.

- Here are a few more examples of symmetric apiculations
(with unspecified altitudes):

v{1}, where {1} denotes a dyad, is an isosceles triangle;

v{2}, where {2} denotes a digon, is a rhombus;

v{*p*,1}, where*p*>2;, is a*p*-gonal pyramid;

v{*p*,2}, where*p*>2, is a*p*-gonal bipyramid;

v{*p*,*q*}, where {*p*,*q*} is the Schläfli symbol of a polyhedron, spherical tiling, plane tiling, or hyperbolic tiling, is the Catalan*p*-akis {*p*,*q*} polyhedron, spherical tiling, plane tiling, or hyperbolic tiling.

- In the case of a tessellation, the apiculated pyramids
usually (though not necessarily) lie in the
*n*-space of the tessellation, and thus have altitude 0. When they do not, the apiculation is*skew*. For example, a skew apiculation of an apeirogon produces a regular*zigzag apeirogon*.*See also*Apeirogon; Catalan polytopes; Duality; Dyad; Join; Stellation; Truncation.

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- The semiregular convex uniform polytopes, so called because
Greek mathematician Archimedes of Syracuse (287–212
BCE) is said to have proposed and solved the
problem of finding the 13 nonprismatic convex uniform polyhedra
whose faces are two or more kinds of regular polygons. Together
with the five Platonic solids, or convex regular polyhedra, and
the infinite families of Archimedean convex prisms and
antiprisms, they constitute the convex uniform polyhedra.

- The concept of an Archimedean polyhedron may be naturally
extended to spaces of two or more dimensions: It is either
a convex regular polygon (in two dimensions) or (in three or more
dimensions) a convex polytope that is not regular, whose facets
are regular and/or Archimedean polytopes, and whose symmetry
group is transitive on its vertices(that is, its corners are
“all alike”). The Archimedean polytopes of four-space,
or
*Archimedean polychora*, were first completely enumerated via computer search by John Horton Conway and Michael Guy in the mid-1960s, although many had been discovered by Alicia Boole Stott and Thorold Gosset in the early years of the 20^{th}century. There are two infinite families (the duoprisms and the antiprismatic prisms) and 58 others. All, along with the six convex regular polychora, are tabulated at the compiler’s website Uniform Convex Polychora.

- The Archimedean polytopes in spaces of dimension greater
than four are not yet completely enumerated, although many are
known or can be derived straightforwardly using Wythoff’s
construction.
*See also*Catalan polytopes; Regular polytopes; Uniform polytopes.

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- The two-dimensional content of a closed plane figure, such as
a polygon. More generally, the content of a two-dimensional
manifold embedded in a metric space.
*See also*Content of a figure.

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- The set of all uniform polytopes (and, if need be, uniform
compound polytopes) that share the same set of vertices; a
0-regiment. For example, in three-space, the 20 vertices of a
regular dodecahedron are shared by an army that includes the
dodecahedron itself, the three ditrigonary polyhedra (small and
great ditrigonary icosidodecahedra and the ditrigonary
dodecadodecahedron), the regular compound of five cubes, the
regular compounds of five and ten tetrahedra, and the great
stellated dodecahedron. There are also three exotic uniform
polyhedra in the army: the small and great complex
rhombicosidodecahedra and the complex rhombidodecadodecahedron.
In four-space, armies may include hundreds and even thousands of
uniform polychora and uniform compound polychora. The uniform
compound polychora in many of the armies are not yet very well
known.

- The
*general*of an army is the uniform polytope or, sometimes, uniform compound polytope whose vertex figure contains the vertex figures of all the other polytopes in the army. Armies are subdivided into*regiments*, which are polytopes that share the same edges as well as the same vertices. For example, the dodecahedral army includes five regiments: [1] the dodecahedron itself; [2] the ditrigonary polyhedra and the compound of five cubes; [3] the compound of five tetrahedra; [4] the compound of ten tetrahedra, and [5] the great stellated dodecahedron. The three exotics are in regiment [2].

- Armies can be denoted by the number of vertices they
contain together with the dimension of the space in which the
vertices are embedded. For example, the army whose vertices are
the 720 corners of a rectified hexacosichoron (or
*rox*, as Jonathan Bowers calls it) is denoted 4/720/1; the 4/ can be omitted if the dimension of the space is understood. The /1 identifies the 720 vertices as belonging specifically to the rox and not to one of the numerous prismatic 720-vertex armies; we could also denote it 720/rox, or even simply /rox, for example. The rox is the general of this army, so the army can be called the*rox army*.*See also*Regiment; Exotic polytopes; Mix-and-match notation.

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- The pivot of a rotation in three-space. More generally, if
the set of all the points in
*n*-space that are invariant under a particular isometry is a line, that line is the*axis*of the isometry. Also, any invariant line of a translation in Euclidean*n*-space. In a translation, the points of*n*-space move along parallel lines that do not move; these are the invariant lines of the translation.*See also*Pivot.

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- The interior of a sphere together with its surface. If the
surface is excluded, the ball is
*open*. A ball is the three-dimensional hyperball. Sometimes a disk, which is a two-dimensional hyperball, is referred to as a two-dimensional ball; and sometimes an*n*-dimensional hyperball is referred to as an*n*-dimensional ball.*See also*Hyperball.

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- A polytope b
in*B**n*-space derived from a base polytopein (*B**n*–1)-space in the following way: The hyperplane containingdivides*B**n*-space into two regions. Choose two points,*a1*and*a2*, called*apices*(singular:*apex*), one in each region, and construct bby (1) discarding the body of*B*, and (2) joining*B**a1*and*a2*to every remaining element ofto make the elements of b*B*. So each vertex of*B*makes an edge of b*B*with*B**a1*and with*a2*; each edge ofmakes a triangle with*B**a1*and*a2*; each face ofmakes a three-dimensional pyramid with*B**a1*and*a2*; and so on.

- While any choice of apices and base will make a
bipyramid, one typically wants some kind of symmetry in the
resulting polytope, so the usual choice of apices is on a line
perpendicular to the base hyperplane that passes through the
center of symmetry of the base (if it has one; centroid if it
doesn’t). Furthermore, the distance from either apex to the
center should be the same, so that either apex is a reflection of
the other in the base hyperplane; this becomes the
*height*or*altitude*of the bipyramid. This more symmetric kind of figure is what is most often referred to as a bipyramid. The dual of a right prism based on a polytopeis this kind of symmetric bipyramid, based on the dual of*P*.*P*

- If the base polytope
is isotopic, any symmetric bipyramid sb*B*having that base will also be isotopic. Its facets are two identical pyramids for each facet of*B*, one with each apex. If the base*B*is convex, sb*B*will also be convex (but a less constrained bipyramid need not be convex). The*B**n*-dimensional cross polytope is a symmetric bipyramid whose base is an (*n*–1)-dimensional cross polytope and whose altitude is adjusted so that its lateral edges are all the same length as the base edges—just as its dual, the*n*-dimensional measure polytope, is a right prism based on an (*n*–1)-dimensional measure polytope, whose altitude is adjusted to equal the edge length of its base. The symmetry group of a symmetric bipyramid is (or includes as a subgroup) the symmetry group of the base adjoined to the reflection in the base hyperplane.*See also*Cross polytope; Duality; Isotope; Prismatic polytopes; Pyramid.

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*See under*Truncation.

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- A polytope constructed by superimposing one or more polytopes
and discarding coincident facets. This usually forms a polytope
because the ridges of the discarded facets are shared by the
remaining facets of both polytopes and become ridges of the new
polytope, but sometimes blends can be degenerate, so each blend
has to be examined individually to see whether or not it is a
proper polytope. If two polytopes share only one facet that is
discarded, the blend is called a
*join*, and the two polytopes are merely*joined together*. When two polytopes blend to make a third, then blending the third with either of the first two produces, as a blend, the other of the first two. Such a set of polytopes is called a*blend multiplet*(*e.g.*, a blend triplet, a blend quadruplet, and so forth).*See also*Join.

- A different kind of blend occurs when some of the
coinciding facets do not cancel completely but themselves merely
blend. This is called an
*incomplete*blend at those facets, as opposed to a*complete*blend, wherein the coinciding facets cancel completely. New uniform polytopes are often constructed as blends, both complete and incomplete, of other uniform polytopes. The vertex figure of a blended uniform polytope is always a blend or join of the vertex figures of the original uniform polytopes.

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- The unique
*n*-dimensional element of an*n*-dimensional polytope. The body of a nullitope is the empty set; the body of a monad is a point; the body of a dyad is a line segment; the body of a polygon is the interior of the polygon bounded by its sides; the body of a polyhedron is the interior of the polyhedron bounded by its faces; and so on.

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- The four-dimensional content of a closed four-dimensional
figure, such as a polychoron. More generally, the content of a
four-dimensional manifold embedded in a metric space.
*See also*Content of a figure; Tesseract.

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- The (usually uniform) polytope that contains itself and all
the other members of a company.
*See also*Company.

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- A means of creating a polytope in a higher-dimensional space
from two or more polytopes in lower-dimensional spaces. Let
and*p*be points in*q**j*-space and*k*-space, respectively. Then the Cartesian product ofand*p*is a point in (*q**j*+*k*)-space whose coordinates are given by stringing together the*j*coordinates ofand the*p**k*coordinates of. It’s as simple as that.*q*

- To obtain the Cartesian product of two polytopes, simply
form
the Cartesian product of all the points—interior and
boundary—of the first with all the points—interior
and boundary—of the second. Then the vertices of the
Cartesian product polytope will be the Cartesian products of all
the vertices of the first polytope with all the vertices of the
second polytope; the edges of the Cartesian product will be the
Cartesian products of the edges of the first with the vertices of
the second, together with the Cartesian products of the vertices
of the first with the edges of the second; the faces of the
Cartesian product will be the Cartesian products of the faces of
the first with the vertices of the second, together with the
Cartesian products of the vertices of the first with the faces of
the second, together with the rectangles that are the Cartesian
products of the edges of the first and the edges of the second;
and so on.

- Polytopes that are constructed as Cartesian products of
lower-dimensional polytopes are called
*prismatic*polytopes. The Cartesian product of a set of polytopes will be a uniform polytope if and only if the polytopes themselves are uniform and have the same edge length. For example, the square whose corners are at {0,0}, {0,1}, {1,0}, and {1,1} in the plane is the Cartesian product of two line segments whose ends are at {0} and {1} in the real line. If one of the line segments had length two instead of one, the Cartesian product would be a 1 x 2 rectangle. The Cartesian product of any polytope with a line segment is called a*prism*. For example, the uniform hexagonal prism is the Cartesian product of a unit hexagon with a unit line segment. In higher-dimensional spaces,*n*>10 or so, uniform prismatic polytopes probably represent the vast majority of uniform polytopes, because of the huge number of possible Cartesian products.

- Cartesian products can easily be extended to all kinds of
figures. A
*cylinder*is the Cartesian product of any polytope or other figure with a circle or an ellipse. If both figures are circles, the cylinder becomes a*spherical torus*. The Cartesian product of a helix in three-space and a Euclidean plane is a kind of helically spiraling 3-manifold embedded in five-space. The combinations and possibilities far outstrip our ability to name them all.

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- The smallest convex polytope that contains a nonconvex
polytope or other kind of figure in
*n*-space.

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- The duals of nonprismatic Archimedean polytopes. In
three-space, these are the 13 Catalan solids, described by
Eugène Charles Catalan in 1865. Here is a table of the
Catalan solids:

Triakis tetrahedron: dual of truncated tetrahedron

Triakis octahedron: dual of truncated cube

Tetrakis hexahedron: dual of truncated octahedron

Triakis icosahedron: dual of truncated dodecahedron

Pentakis dodecahedron: dual of truncated icosahedron

Rhombic dodecahedron: dual of cuboctahedron

Rhombic triacontahedron: dual of icosidodecahedron

Hexakis octahedron: dual of truncated cuboctahedron

Hexakis icosahedron: dual of truncated icosidodecahedron

Strombic icositetrahedron: dual of rhombicuboctahedron

Strombic hexecontahedron: dual of rhombicosidodecahedron

Pentagonal icositetrahedron: dual of snub cuboctahedron

Pentagonal hexecontahedron: dual of snub icosidodecahedron

- In four-space, each of the 58 Archimedean polychora has a
dual Catalan polychoron. A Catalan polychoron has a number of
identical irregular cells, each cell being the dual of the vertex
figure of the corresponding Archimedean polychoron. The Catalan
polychora have one to four kinds of vertex figures, each vertex
figure being the dual of a cell of the corresponding Archimedean
polychoron. For example, the dual of the great
diprismatohexacosihecatonicosachoron is a polychoron with 14,400
cells. These are identical scalene tetrahedra, 7200 being
left-handed and the other 7200 being right-handed, the duals of
the
vertex-figure scalene tetrahedra of the great
diprismatohexacosihecatonicosachoron. One hundred twenty (60
left, 60 right) tetrahedra come together at their narrowest ends
at each of 120 vertices, in a hexakis-icosahedral vertex figure;
24 (12 left, 12 right) tetrahedra come together at each of 600
vertices in a tetrakis-hexahedral vertex figure; 20 (ten left,
ten right) tetrahedra come together at each of 720 vertices in a
decagonal-bipyramidal vertex figure; and 12 (six left, six right)
tetrahedra come together at each of 1200 vertices in a
hexagonal-bipyramidal vertex figure. This is the most complicated
Catalan polychoron. Having 14,400 tetrahedral cells makes it a
**scalene-tetrahedral myriatetrakischiliatetracosichoron**.*See also*Archimedean polytopes; Duality.

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- A facet of a polychoron; a three-dimensional element of a
polytope. Sometimes used in place of facet for an
*n*-dimensional polytope.*See also*Facet.

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*See*Aggrandizement .

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- A point that remains unmoved by any symmetry operation in a
symmetry group, if there is a single such point. In a uniform
polytope, the center of symmetry is the center of the
hypersphere that contains all its vertices.

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- Figures whose symmetry groups comprise only direct
isometries, so that they come in left-handed (
*laevo*) and right-handed (*dextro*) forms. Such figures are also called*enantiomorphic*. A left-handed figure in*n*-space can be reflected in a mirror hyperplane to become the right-handed version, and*vice versa*. It can also be rotated 180° in (*n*+1)-space with the mirror hyperplane as pivot to become a right-handed figure in*n*-space.*See also*Mirror; Reflection; Rotation.

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*See*Line .

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- The 53 nonprismatic semiregular uniform star-polyhedra, first
completely enumerated by H. S. M. Coxeter, M. S. Longuet-Higgins,
and J. C. P. Miller in their 1954 monograph “Uniform
Polyhedra,”
*Philosophical Transactions of the Royal Society of London*(Series A),*246*: 401–450. Colohimi is an acronym of the surnames of the three authors of the monograph, hence the term Colohimian.*See also*Uniform polyhedra.

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- The (usually uniform, not necessarily convex) polytope that
contains itself and all the other members of a regiment. The
vertex figure of a colonel is frequently but not necessarily a
convex polytope.
*See also*Regiment.

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- The set of all uniform polytopes (and, if need be, uniform
compound polytopes) that share the same set of faces; a
2-regiment. In three-space, companies generally include only one
polyhedron each, because polyhedra that have the same faces (and
thus edges and vertices as well) are equivalent. (It is possible
for two star-polyhedra to have the same faces but not the same
interiors, however, so three-space companies with more than one
polyhedron can be contrived.) But in four-space, uniform
polychora may have the same faces but not the same cells. For
example, the hexacosichoron and the icosahedral
hecatonicosachoron have the same 1200 triangles, but in the
former they are shared by 600 tetrahedra and in the latter they
are shared by 120 icosahedra. These two polychora form the
*hexacosichoric company*, of which the hexacosichoron is the*captain*(being the polychoron that contains the other). Likewise, the grand hexacosichoron and the great icosahedral hecatonicosachoron form the conjugate company of two, the*great icosahedral hexacosichoric company*. Here the captain is the great icosahedral hecatonicosachoron.

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- A figure comprising two or more polytopes, its
*components*, usually but not necessarily of the same dimensionality, usually but not necessarily interpenetrating. A compound polytope is*uniform*if its symmetry group is transitive on its vertices, that is, its corners are “all alike,” irrespective of which component of the compound a vertex might belong to. From this it follows that the components of a uniform polytope must themselves be congruent uniform polytopes.

- Tessellations can also be compounded. For example, the
checkerboard tiling contains the vertices of larger checkerboard
tilings in infinitely many different ways, so there are
infinitely many uniform compounds of two or more larger
checkerboard tilings within a unit checkerboard tiling. Some of
these compounds are chiral.
*See also*Polytope; Uniform polytope; Tessellation; Transitivity.

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- Different polytopes that share the same abstract description
(not to be confused with duals, whose abstract descriptions are
the reverse of each other), sometimes called
*isomorphic*polytopes. Typically, one polytope may have pentagonal faces and the other may instead have pentagrammatic faces, or decagonal faces versus decagrammatic faces, or octagonal faces versus octagrammatic faces. Conjugate polytopes usually occur in pairs, but there also exist stand-alone polytopes that are*self-conjugate*. As a rule, the facets of a conjugate polytope are conjugates of the facets of the first polytope, and the vertex figures of a conjugate polytope are conjugates of the vertex figures of the first polytope, whenever such conjugacies are apparent. Among uniform polychora, if conjugates occur in different regiments, then each polychoron in one regiment will have a conjugate in the other. But if conjugates occur in the same regiment, then each polychoron in the regiment will have a conjugate in the same regiment, or will be self-conjugate.

- Each regular polytope with pentagonal symmetry has a
regular conjugate; for example, the dodecahedron and great
stellated dodecahedron are conjugates, the icosahedron and great
icosahedron are conjugates, and the small stellated dodecahedron
and great dodecahedron are conjugates. To obtain the
Schläfli symbol of the conjugate of a regular polytope,
exchange the 5’s for
^{5}/_{2}’s and*vice versa*. Among uniform polytopes, conjugates are sometimes distinguished by names using the adjectives*small*and*great*, with great usually applied to the more intricate conjugate polytope.*See also*Regiment; Regular polytopes; Uniform polytopes.

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- In Euclidean
*n*-space, the number of unit measure polytopes and fractions thereof that would exactly fill the interior of a polytope or other closed figure. The content of a point is conventionally taken to be 0, since a point has no interior. The content of a line segment is its*length*; the content of a polygon or other closed figure in two-space is its*area*; the content of a polyhedron or other closed figure in three-space is its*volume*; the content of a polychoron or other closed figure in four-space is its*bulk*. In*n*-space with*n*>4, we use numeric Greek roots for their names:*pentabulk*,*hexabulk*,*heptabulk*,*octabulk*,*enneabulk*, and so on.

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- A polytope with the property that, given any two points in
its interior, the line segment joining the two points also lies
entirely in the interior. In addition, the facets of the polytope
may not overlap or have any parts in the interior.

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*See*Plane.

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- Star-polytopes in
*n*-space that share the same surtope but differ internally. A hypothetical*n*-dimensional observer outside the copycats would be unable to distinguish them, but a hypothetical (*n*+1)-dimensional observer would see their internal differences. For example, the small stellated dodecahedron is different from its surhedron, which is a nonconvex hexecontahedron whose faces are sixty acute golden triangles. Only an observer inside the polyhedra, or a four-dimensional observer looking from outside three-space, would see that the small stellated dodecahedron includes the portions of the faces that comprise an internal dodecahedron, whereas the hexecontahedron is entirely hollow.*See also*Polytope; Surtope.

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*See*Realm.

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- A vertex of a polytope.
*See also*Vertex.

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- Also known as an
*orthoplex*. In Euclidean*n*-space, the analogue of the*line segment*in one-space, the*square*in two-space, the*octahedron*in three-space, and the*hexadecachoron*in four-space. One of the three kinds of regular polytopes that exist in spaces of dimension greater than four. So called because its construction in*n*-space involves locating two points symmetrically across from each other on either side of an (*n*–1)-dimensional cross polytope embedded in the*n*-space. The*n*-dimensional cross polytope, a special kind of bipyramid, is the dual of the*n*-dimensional measure polytope. The vertex figure of an*n*-dimensional cross polytope of edge 1 is an (*n*–1)-dimensional cross polytope of edge 1. The vertex figure of an*n*-dimensional measure polytope honeycomb is an*n*-dimensional cross polytope of edge*sqrt*(2).

- For
*n*>4, we use Greek numerical roots to name the cross polytopes:*pentacross*for*n*=5,*hexacross*for*n*=6,*heptacross*for*n*=7,*octacross*for*n*=8,*enneacross*for*n*=9,*decacross*for*n*=10,*hendecacross*for*n*=11, and so on. The facets of an*n*-dimensional cross polytope are 2^{n}(*n*–1)-dimensional simplexes, and an*n*-dimensional cross polytope has 2*n*vertices (corners).

- The coordinates of the vertices of an
*n*-dimensional cross polytope centered on the origin are simply all possible strings of*n*–1 zeros and a +1 or a –1. These coordinates place two vertices of the cross polytope on each of the*n*coordinate axes, on opposite sides of the origin; the edge length of the cross polytope is then*sqrt*(2). Note also that the section of the cross polytope by any coordinate*k*-flat, 0<*k*<*n*, is a*k*-dimensional cross polytope. From this, it follows that the number of*k*-dimensional elements of an*n*-dimensional cross polytope is given by the coefficient of*a*^{k+1}in the binomial expansion of (2*a*+1)^{n}. For example, the binomial expansion of (2*a*+1)^{7}is

128*a*^{7}+ 448*a*^{6}+ 672*a*^{5}+ 560*a*^{4}+ 280*a*^{3}+ 84*a*^{2}+ 14*a*+ 1.

From this, a seven-dimensional cross polytope, or heptacross, has

128 hexons (heptapenta),

448 pentons (hexatetra),

672 tetrons (pentachora),

560 cells (tetrahedra),

280 faces (triangles),

84 edges,

14 vertices, and

1 nullitope (element of dimension –1).

There is also the heptacross itself (element of dimension 7), which is left out of the binomial expansion.*See also*Bipyramid; Duality; Measure polytope; Regular polytope; Simplex; Vertex figure.

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- A uniform polytope in
*n*-space,*n*>2, constructed from the cross polytope by discarding as facets 2^{n–1}alternate (*n*–1)-simplexes and filling in the free ridges with*n*mutually perpendicular (*n*–1)-dimensional cross polytopes that all pass through the polytope’s center. If*n*=2, the resulting non-uniform polygon is a*square bowtie*; if*n*=3, the resulting uniform polyhedron is the*tetrahemihexahedron*; if*n*=4, the resulting uniform polychoron is the*octahemioctachoron*; if*n*=5, the resulting polytetron is the*hexadecahemidecatetron*; and so on. For*n*>2, the demicross polytope is not orientable;*e. g.*, in three-space, the tetrahemihexahedron has topologically the same surface as a cross-cap.

- The vertex figure of a demicross polytope is the
demicross
polytope of one fewer dimensions, so for
*n*>3, the*rectified demicross polytope*and*truncated demicross polytope*are both uniform as well. In four-space, the rectified demicross polytope belongs to the ico regiment, that is, the regiment whose colonel is the regular icositetrachoron. Jonathan Bowers calls it a*ratho*; its formal Greekish name is*disoctatetrachoron*. The facets of a rectified demicross polytope are 2*n*demicross polytopes,*n*rectified cross polytopes, and 2^{n–1}rectified simplexes, of*n*–1 dimensions. The facets of a truncated demicross polytope are 2*n*demicross polytopes,*n*truncated cross polytopes, and 2^{n–1}truncated simplexes, of*n*–1 dimensions.*See also*Ico regiment.

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*See*Half measure polytope.

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- Loosely speaking, the least number of facets that a path must
cross to reach infinity from an interior region of an
*n*-dimensional polytope in*n*-space is that region’s density. If the path must cross a facet region of density*d*>1, or if the path must pass through a retrograde region of density*d*<0, that counts as a crossing of*d*facets. The density of the polytope as a whole is the density of its densest interior region. For example, the sides of a regular pentagram circle the center twice, so its interior pentagonal region suffers two overlaps; a path from this region of the pentagram must cross two sides of the pentagram before it exits. The density of the pentagram as a whole is therefore two, although the density of each of its five triangular points is one. The density of any convex polytope is one.

- In general, the density of a regular polygon
{
*p*/*q*} is*q*. The density of a small stellated dodecahedron or a great dodecahedron is 3, and the density of a great stellated dodecahedron or a great icosahedron is 7. In four-space, the densities of the regular star-polychora are:

Stellated hecatonicosachoron and icosahedral hecatonicosachoron: 4

Great hecatonicosachoron: 6

Great stellated hecatonicosachoron and grand hecatonicosachoron: 20

Grand stellated hecatonicosachoron: 66

Great grand hecatonicosachoron and great icosahedral hecatonicosachoron: 76

Great grand stellated hecatonicosachoron and grand hexacosichoron: 191

- The densities of all the regular star-polychora were
originally calculated by Ludwig Schläfli and Edmund Hess in
the mid-1800s and were tabulated by H. S. M. Coxeter in
*Regular Polytopes*(3rd edition: Dover Books, 1973).*See also*Hecatonicosachoron, regular; Hexacosichoron, regular.

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- The angle between two intersecting (
*n*–1)-flats (hyperplanes) in*n*-space. Two (*n*–1)-flats intersect in an (*n*–2)-flat. To measure the dihedral angle, choose any point in the (*n*–2)-flat and construct two rays perpendicular to that (*n*–2)-flat that lie in each of the two (*n*–1)-flats. The dihedral angle between the (*n*–1)-flats is then the angle between the two rays. Depending on the orientations of the rays, the dihedral angle will be acute or obtuse.

- In a polytope in
*n*-space, the dihedral angle between two adjoining facets is the dihedral angle between the hyperplanes that contain them. In a skew polytope, the dihedral angle between adjoining*k*-dimensional facets is the dihedral angle between the facets in the (*k*+1)-dimensional space that contains them.

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- The number of parameters (called
*coordinates*) required to locate a point uniquely in a space relative to another point in that space.

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- Inward apiculation.
*See also*Apiculation.

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- A company of three star-polychora within the
*sishi*/*gaghi*regiment. As members of this regiment, they share the 120 vertices of a hexacosichoron and the 1200 edges of both a stellated hecatonicosachoron (sishi: the colonel of the regiment) and a great grand hecatonicosachoron (gaghi). Being a company, they also share their 2400 triangular faces. They have just one kind of face to share, which is rare for a company. The triangles also happen to belong to the regular compound of 25 icositetrachora in a hexacosichoron, but the star-polychora do not share the compound’s 600 octahedral cells. Each ditrigonary star-polychoron has two kinds of cells, drawn from cell sets of three different kinds: 600 tetrahedra, 120 icosahedra, or 120 great icosahedra.

- The star-polychoron from this company whose cells are 600
tetrahedra and 120 great icosahedra is constructed by replacing
the 120 small stellated dodecahedral cells of a sishi with the
120 great icosahedra that have the same vertices and edges. This
leaves all 2400 (that is, 120 times 20) new triangular faces
free. They fall into 600 edge-connected sets of four that can be
replaced by the cell set of 600 regular tetrahedra to make a
closed polychoron: the
*small ditrigonary hexacosihecatonicosachoron*, or*sidtixhi*as Jonathan Bowers calls it.

- The next star-polychoron from this company is the
conjugate
of the preceding, known as the
*great ditrigonary hexacosihecatonicosachoron*, or as Jonathan calls it, a*gidtixhi*. Its cells are the same 600 tetrahedra as above, but instead of the 120 great icosahedral cells it has 120 ordinary (or small) icosahedral cells. Each icosahedral cell has the vertices and edges of the great dodecahedral cells of a gaghi, the conjugate of the sishi.*See also*Hecatonicosachoron.

- The third star-polychoron in this company is constructed
by
blending the first two, that is, by positioning a sidtixhi and a
gidtixhi so that their vertices coincide, then knocking out the
two overlapping sets of 600 tetrahedra. The 120 icosahedra and
120 great icosahedra adjoin along their triangles after the
tetrahedra are discarded. This star-polychoron is the
*ditrigonary dyakishecatonicosachoron*(the “dyakis” means “two times,” since it has two times 120 cells), or*dittady*as Jonathan Bowers calls it.

- These three star-polychora were discovered by Jonathan
Bowers
and, independently and somewhat later, by this Glossary’s
compiler. They are closely related to their three
three-dimensional counterparts, the
*ditrigonary star-polyhedra:*the small and great ditrigonary icosidodecahedra and the ditrigonary dodecadodecahedron. These uniform polyhedra have the same sets of vertices and edges, the vertices belonging to a regular dodecahedron and the edges being the diagonals of the dodecahedron’s twelve faces. Their three different kinds of faces are twelve pentagrams, 20 triangles, and/or twelve pentagons; the three star-polyhedra have two out of three of these face sets. Each ditrigonary polyhedron is the vertex figure of its corresponding ditrigonary polychoron. Just as the triangles of the ditrigonary star-polychora belong to the compound of 25 icositetrachora in a hexacosichoron, so do the edges of the ditrigonary polyhedra also belong to the regular compound of five cubes, which is, naturally, the vertex figure of the compound of 25 icositetrachora. Pictures of the three ditrigonary polychora appear among the uniform dodecahedron facetings; just click and scroll down.

- The ditrigonary star-polyhedra have nonuniform polygonal
counterparts in two-space: three
*ditrigons*, that is, three equiangular hexagons inscribed in a circle, with alternate sides of two different lengths (hence the name “ditrigonary”). One is convex; the second uses the three long sides of the first with the three long diagonals; and the third uses the three short sides of the first with the three long diagonals. When the long and short sides are of length 1 and 1/*tau*(or [*sqrt*{5}–1]/2), the long diagonal is of length*tau*(or [*sqrt*{5}+1]/2), and these three particular ditrigons become the vertex figures of the ditrigonary star-polyhedra. The six short diagonals of the outer hexagon form a compound of two triangles of edge*sqrt*(2), which is the vertex figure of the compound of five cubes. In two-, three-, and four-space, the ditrigonary polytopes form blending triplets. Unfortunately, this cute ditrigonary sequence runs just from two- to four-space.

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- The dual d
of a polytope*P*is a figure, usually but not necessarily a polytope under some definitions of the term, whose elements are the elements of*P*“backward.” That is, to every facet of*P*there corresponds a vertex of d*P*; to every ridge of*P*there corresponds an edge of d*P*; to every peak of*P*there corresponds a face of d*P*; ...; to every face of*P*there corresponds a peak of d*P*; to every edge of*P*there corresponds a ridge of d*P*; and to every vertex of*P*there corresponds a facet of d*P*.*P*

- In projective spaces, duality is truly exact, but in
Euclidean spaces, which lack points at infinity, duality breaks
down when the facets of a polytope pass through its center. Such
facets dualize to points at infinity, which are not permitted for
Euclidean polytopes. Also, cohyperplanar facets, that is, facets
that lie in the same hyperplane, dualize to coincident vertices,
and these may not be permitted under certain definitions of a
polytope.

- Otherwise, pretty much any polytope has a dual. In
particular, the dual of a regular polytope with Schläfli
symbol
{
*a*,*b*,*c*,...,*x*,*y*,*z*} is the regular polytope with Schläfli symbol {*z*,*y*,*x*,...,*c*,*b*,*a*}. The dual of an antiprism in three-space is an antibipyramid, and the dual of a polytopal prism is a bipyramid based on the dual of the base of the prism. In the plane, the dual of a regular polygon {*n*} is another {*n*}. The duals of the 13 Archimedean polyhedra are the 13 Catalan polyhedra. The duals of the Colohimian polyhedra were described by Magnus Wenninger in his book*Dual Models*(Cambridge University Press, 1983).

- The dual operation to truncation is apiculation. That is,
the dual of a polytope whose vertices are truncated away may be
constructed by apiculating pyramids onto the facets of the dual
that correspond to the truncated vertices. Likewise, the dual
operation to stellation is faceting. The concept of duality,
including the details of how to construct the dual of a
particular polytope, has too wide a scope and too many
interesting ramifications to be covered adequately in a short
Glossary entry such as this.

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- Also called a
*double prism*. The four-dimensional polytope that is the Cartesian product of two plane polygons, its*base polygons*. If the two polygons are convex and regular and have sides of the same length, the resulting duoprism is Archimedean (that is, convex uniform). If the two polygons are congruent squares, the resulting duoprism is a tesseract (that is, regular). If either polygon is a regular star-polygon and both have the same edge length, the resulting figure is a uniform (nonconvex)*star-duoprism*. The cells of a duoprism based on a*p*-gon and a*q*-gon are*p**q*-gonal rectangular prisms and*q**p*-gonal rectangular prisms, and it has*pq*vertices.

- Duoprisms may be defined in spaces of dimension greater
than four as Cartesian products of any two
*j*-dimensional and*k*-dimensional polytopes,*j*+*k*>4, not just polygons (for which*j*=*k*=2).*Triaprisms*,*quadriprisms*, and other kinds of*multiprisms*can be defined as Cartesian products of more than two kinds of polytopes.*See also*Cartesian product; Prismatic polytopes.

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- In four-space, a polytope whose vertices form polygons in two
absolutely perpendicular planes. The edges of a duopyramid are
the edges of the two base polygons together with the line
segments that connect each vertex of one base polygon to each
vertex of the other base polygon. The faces of a duopyramid are
the triangles formed by any edge of a base polygon and any vertex
of the other base polygon. And the cells of a duopyramid are the
tetrahedra defined by any edge of one base polygon together with
any edge of the other base polygon. If both base polygons are
regular, have equal edges, and are centered on the same point
(the point of intersection of their absolutely perpendicular
planes), the resulting
*p*-gonal*q*-gonal duopyramid is the dual of the uniform*p*-gonal*q*-gonal duoprism. In particular, the square duopyramid is the regular hexadecachoron, dual of the square duoprism, which is the tesseract. This definition generalizes to pyramids in spaces of any number of dimensions.*See also*Duality; Duoprism; Hexadecachoron, regular; Polytope; Prismatic polytopes; Pyramid.

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- A one-dimensional polytope. Its elements include the empty
set, two points (its ends or vertices), and the line segment
bounded by the two points (the body of the dyad).
*See also*Line segment.

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- A facet of a polygon, also called a
*side*; a one-dimensional element of a polytope.*See also*Facet.

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- Producing a new polytope from a core polytope by extending
the edges of the core polytope until they meet extended other
edges. The quasitruncated cube is an edge-stellation of the
truncated cube, for example: The edges of the truncated cube are
extended until the regular octagons {8} become regular octagrams
{
^{8}/_{3}}. Then the small “corner” triangles of the truncated cube are replaced by larger, parallel triangles formed by some of the extended edges.*See also*Stellation.

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- A polytope that belongs to another polytope. The elements of
a polytope are the nullitope, which has dimension –1; the
*vertices*or*corners*, which have dimension 0; the*edges*, which have dimension 1; the*faces*, which have dimension 2; the*cells*, which have dimension 3; the*tetrons*, which have dimension 4; the*pentons*, which have dimension 5; and so on up to and including the polytope itself, which has dimension*n*. The nullitopes of all the elements of the polytope are shared as the nullitope of the entire polytope.

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- The set that contains no elements.
*See also*Empty space, the; Nullitope, the.

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- The space devoid of points; the empty set considered as a
space. By convention, the empty space has –1 dimensions.
*See also*Nullitope, the.

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*See*Chiral figures.

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- A facet or vertex of a dyad.
*See also*Dyad; Facet; Point.

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- A space of
*n*dimensions with the Euclidean metric: the distance between two points= {*a**a*_{1},*a*_{2},*a*_{3},...,*a*_{n}} and= {*b**b*_{1},*b*_{2},*b*_{3},...,*b*_{n}}

is given by the general Pythagorean rule

D[,*a*] =*b**sqrt*[(*a*_{1}–*b*_{1})^{2}+(*a*_{2}–*b*_{2})^{2}+(*a*_{3}–*b*_{3})^{2}+...+(*a*_{n}–*b*_{n})^{2}].

In Euclidean*n*-space, given a hyperplane*H*and a pointnot in that hyperplane, there is exactly one hyperplane that passes through*p*parallel to*p**H*.

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- Polytopes that have one or more coincident elements, also
called
*exopolytopes*or simply*exotopes*. Two-dimensional exopolytopes are*exopolygons*; three-dimensional exopolytopes as*exopolyhedra*; four-dimensional exopolytopes are*exopolychora*; and so on.*Coincident*elements are distinct elements that occupy exactly the same position in the polytope. Some definitions of a proper polytope prohibit all exopolytopes because coincident facets render the connectivity of a figure ambiguous. For example, in an exopolygon with a pair of coincident vertices, there are three ways that the four sides incident at the coincident-vertex pair may adjoin. Because these cannot be distinguished by simply examining a diagram of the sides of the exopolygon, an*adjunct rule*is required to specify which sides adjoin to which at coincident vertices, and this complicates the description of the figure. (Polytopes cannot by definition admit more than two facets at a ridge; coincident ridges are thus a bookkeeping trick to insure that this dyadic rule is not violated.) Also, allowing facets to coincide opens a Pandora’s box of endless numbers of apparently identical exopolytopes, such as Riemannian polygons and polyhedra whose vertex figures are Riemannian polygons, that are merely multiple overlapping copies of one or more underlying proper polytopes. And it may produce*k*-dimensional*membranes*and*whiskers*(membranes of dimension 1), 0<*k*<*n*: thin elements that hang out into*n*-space without support.

- Many symmetric and uniform exopolytopes are interesting
figures even if they are not proper polytopes. One compromise
that allows certain exopolytopes to be proper polytopes and
avoids the problem of infinitely many permissible multiple-copy
exopolytopes is to permit proper
*n*-dimensional polytopes to have (*n*–1)-dimensional exopolytopes as facets, provided that these exopolytopes do not themselves have coincident facets. Ridges and other lower-dimensional elements of facets are already shared among three or more facets in a polytope, so it makes little difference to its structure whether some of these elements already coincide in a facet.

- So, for example, if an exopolygon has coincident vertices
but no coincident sides, it may appear as a face of a proper
polyhedron even though it is not a proper polygon itself; if an
exopolyhedron has coincident edges but no coincident faces, it
may appear as a cell of a proper polychoron even though it is not
a proper polyhedron itself; and so on. The inherent connectivity
ambiguity at coincident elements may be dealt with by
appropriately structuring their interiors or by adjunct rules, or
simply ignored (so that, for example, exopolytopes that differ
solely in their adjunct rules would not be considered different:
a slippery slope, since different adjunct rules often generate
substantially different duals from otherwise identical
polytopes). Note that when the vertex figure of a uniform
exopolytope is a compound, its components must be symmetric
images of each other; and the number of vertices coincident at a
corner equals the number of components of the compound vertex
figure. A uniform exopolytope must, of course, have the same
vertex-figure adjunct rules as well as the same vertex figure at
every vertex. Under this definition, several of the figures
described as
*degenerate*by H. S. M. Coxeter, M. S. Longuet-Higgins, and J. C. P. Miller in their 1954 monograph “Uniform Polyhedra,”*Philosophical Transactions of the Royal Society of London*(Series A),*246*: 401–450 may be redescribed as uniform exopolyhedra and used as cells of proper uniform polychora.

- Exopolytopes may occur as duals of stellations that have
distinct cohyperplanar facets and are perfectly well-behaved,
proper polytopes. For example, the surtope of a regular pentagram
is a proper star-shaped decagon called a
*hollow pentagram*; its vertices are the five vertices of the pentagram alternating with the five points of intersection of the pentagram’s sides, and its sides form the pentagram’s periphery. These ten sides lie in collinear pairs in the five lines of the pentagram’s sides, so the dual of the hollow pentagram is a*complete pentagon:*an exodecagon whose ten vertices coincide pairwise at the corners of a regular pentagon and whose sides are all of the pentagon’s sides and diagonals. If the edges of the hollow pentagram are slightly tilted symmetrically out of collinearity (by, for example, moving the outer five vertices slightly inward), the corners of the dual complete pentagon will no longer coincide, and it becomes a proper star-decagon, not an exodecagon.

- The two different adjunct rules, applied
uniformly to all the vertex-pairs, create two different isogonal
(but not uniform) complete pentagons, which may be distinguished
by organizing their interiors. The (circular) vertex figure of
the exodecagon with the more acute 36° dihedral angles is a
pair of nonoverlapping 36° circular arcs separated by a
36° arc; the vertex figure of the one with the less acute
72° dihedral angles is a pair of 72° arcs overlapping in
a 36° arc. It is the former that is the dual of the hollow
pentagon. (Exercise: Describe the proper star-decagon that is the
dual of the other complete pentagon.) The nonoverlapping arcs of
the former complete pentagon indicate that a portion of the
exterior has been “trapped” beneath each vertex, so
that the interior has five exterior triangular “holes”
inside it, and no overlapping regions. In the latter complete
pentagon, these holes are covered up by parts of the interior
that overlap. Having no overlapping interior regions means the
*density*(which equals the greatest number of overlaps of an interior region of a polytope or exopolytope) of the former is 1. The density of the latter complete pentagon, however, is 3, the number of times the complete pentagon covers the central pentagonal region. Note that in the former complete pentagon, the short sides are retrograde with respect to the long sides, and in calculating its density a short-side crossing is subtracted from the total.

- These two complete pentagons are the two-dimensional
members of a small transdimensional family of mainly uniform
exopolytopes. The three-dimensional members are the
*small complex icosidodecahedron*and*great complex icosidodecahedron*, each also coming in two density-different versions. Both exopolyhedra were listed among the degenerate uniform figures by Coxeter, Longuet-Higgins, and Miller, since they arise naturally as uniform figures via Wythoff’s construction. The adjective “complex” signals their exotic nature. Just as a complete pentagon resembles a regular pentagon circumscribed about a regular pentagram, so does the small complex icosidodecahedron resemble a regular icosahedron circumscribed about a great dodecahedron that has the same vertices and edges, and the great complex icosidodecahedron resembles a small stellated dodecahedron circumscribed about a great icosahedron that has the same vertices and edges. The former exopolyhedron has 20 triangles and twelve pentagons as faces, the latter has 20 triangles and twelve pentagrams. They are each other’s conjugates. The vertex figure of either of these exopolyhedra is a complete pentagon: one of short side 1 for the small complex icosidodecahedron and one of short side (*sqrt*[5]–1)/2 for the great complex icosidodecahedron.

- Just as the two different isogonal complete pentagons may
be distinguished by their adjunct rules, so may two different
small and great complex icosidodecahedra be distinguished by
which of the complete pentagons their vertex figures are. Each
complex icosidodecahedron has ten faces at each vertex, five
triangles alternating with five pentagons or pentagrams; the
adjunct rules at the edges correspond to the adjunct rules at the
vertices of the complete pentagon vertex figure. To an external
three-dimensional observer, the small complex icosidodecahedron
looks like a regular icosahedron and the great complex
icosidodecahedron looks like a small stellated dodecahedron, but
a hypothetical four-dimensional observer would instantly
distinguish the exopolyhedra from their surtopes by their
internal structures, which would be open and evident to him or
her. The small complex icosidodecahedron with density-1 vertex
figures has internal holes (30 of them, like Swiss cheese), and
its density is 2; the small complex icosidodecahedron with
density-3 vertex figures has density 4; the great complex
icosidodecahedron with density-1 vertex figures has internal
holes (80 of them), and its density is 4; and the great complex
icosidodecahedron with density-3 vertex figures has density 10.
Quite a few uniform polychora are known with these exopolyhedra
as cells.

- Continuing into four-space, there are eight uniform
four-dimensional members of this family, two for each of the four
different complex exopolyhedra above, which are their vertex
figures, just as complete pentagons are the vertex figures of the
complex exopolyhedra. They are constructed from the regular
polychora and star-polychora by combining those that share the
same vertices, edges, and faces but have different cells. All
have the 120 vertices of a hexacosichoron and the 720 edges of
either a hexacosichoron or a grand hexacosichoron. The first two
pairs are each other’s conjugates, as are the latter two
pairs. Here is a table of these exopolychora:

- Components: Hexacosichoron around icosahedral hecatonicosachoron
- Faces: 2400 triangles, in 1200 coincident pairs
- Cells: 600 tetrahedra, 120 icosahedra
- Density: 3
- Vertex figure: Small complex icosidodecahedron density 2,
edge 1

- Density: 5
- Vertex figure: Small complex icosidodecahedron density 4,
edge 1

**Small complex hexacosihecatonicosachoron**- Components: Great icosahedral hecatonicosachoron around grand hexacosichoron
- Faces: 2400 triangles, in 1200 coincident pairs
- Cells: 600 tetrahedra, 120 great icosahedra
- Density: 115
- Vertex figure: Great complex icosidodecahedron density 4,
edge 1

- Density: 267
- Vertex figure: Great complex icosidodecahedron density 10,
edge 1

**Grand complex hexacosihecatonicosachoron**- Components: Great hecatonicosachoron around grand hecatonicosachoron
- Faces: 1440 pentagons, in 720 coincident pairs
- Cells: 120 great dodecahedra, 120 dodecahedra
- Density: 14
- Vertex figure: Great complex icosidodecahedron density 4,
edge (
*sqrt*[5]+1)/2

- Density: 26
- Vertex figure: Great complex icosidodecahedron density 10,
edge (
*sqrt*[5]+1)/2

**Great complex dishecatonicosachoron**- Components: Great stellated hecatonicosachoron around grand stellated hecatonicosachoron
- Faces: 1440 pentagrams, in 720 coincident pairs
- Cells: 120 great stellated dodecahedra, 120 small stellated dodecahedra
- Density: 46
- Vertex figure: Small complex icosidodecahedron density 2,
edge (
*sqrt*[5]–1)/2

- Density: 86
- Vertex figure: Small complex icosidodecahedron density 4,
edge (
*sqrt*[5]–1)/2

**Stellated complex dishecatonicosachoron** - The density of any complex polytope in this family is
found by either adding together or subtracting smaller from
larger the densities of its inner and outer components. This
transdimensional family stops with dimension 4, since there are
no pentagonal polytopes in spaces of dimension greater than 4.
Although the eight complex exopolychora are uniform, they are not
proper polychora (having coincident faces) and are not counted
among the uniform polychora, but they may potentially be used as
tetrons in proper uniform polytetra.

- Three other uniform exopolyhedra usable as cells turn up
via Wythoff’s construction besides the complex
icosidodecahedra mentioned above. They may be described as each
of the three ditrigonary polyhedra surrounding the compound of
five cubes that has the same vertices and edges. Unlike the
complex icosidodecahedra, all but one of these are externally
different from their components. Twelve faces meet at each of
their 20 vertices, but the vertex figures are compounds of three
isosceles trapezoids or isosceles crossed trapezoids, so they are
essentially uniform exopolyhedra with four faces at each of 60
vertices that have collapsed by threes into the 20 vertices of a
regular dodecahedron. Their names are
*small complex rhombicosidodecahedron*,*great complex rhombicosidodecahedron*, and*complex dodecadodecahedron*. The first two are conjugates; the third is self-conjugate. Quite a few uniform exopolychora are known with these exopolyhedra as cells. Pictures of these exopolyhedra are among the uniform dodecahedron facetings; simply click and scroll.

- In order not to burden the already huge table of known
uniform polychora unnecessarily, Jonathan Bowers and I have
adopted the convention of not counting polychora that differ from
other polychora in the table solely in the densities of their
cells. Furthermore, in displaying uniform polychora with
exopolyhedral cells, the interiors of the cells are
*checkerboarded*. That is, inner volumes of even density are rendered as holes, and inner volumes of odd density are rendered solid. This gives the sections of such cells a more interesting appearance and readily signals their exotic nature. (Exercise: Count the holes and solid volumes inside a checkerboarded great complex icosidodecahedron!)

- It is useful to provide an example of a uniform
polychoron with exopolyhedral cells. In the icosahedral
hecatonicosachoron, replace the icosahedra with small complex
icosidodecahedra that have the same triangular faces. These cells
adjoin one another at their triangular faces. What about the
pentagons? The cells adjoin one another at their pentagonal
faces, too. The vertex figure of the icosahedral
hecatonicosachoron is a great dodecahedron of edge 1, whose
pentagons are the vertex figures of the twelve icosahedra that
meet at each vertex. After the substitution, the vertex figure of
the new uniform polychoron becomes the great dodecahedron with
its pentagons replaced by complete pentagons. These adjoin one
another along their short and long sides. Internally, and by our
convention, this vertex figure differs from the great
dodecahedron in having twelve internal holes, each a pentagonal
pyramid atop the dodecahedron deep inside the figure formed by
the internal pentagons of the complete pentagons. This uniform
polychoron can be called a
*small complete-icosidodecahedral hecatonicosachoron*(Jonathan Bowers calls it a*sachi*, for*small complexhecatonicosachoron*). Its conjugate is constructed by replacing the small stellated dodecahedra of the grand stellated hecatonicosachoron with great complex icosidodecahedra. These adjoin along their external pentagrams and their internal triangles. The vertex figure of this*great complete-icosidodecahedral hecatonicosachoron*(which Jonathan Bowers calls a*gachi*, for*great complexhecatonicosachoron*) is a smaller “holey great dodecahedron” of edge (*sqrt*[5]–1)/2. Both polychora are not only uniform but*isochoric*; that is, they are uniform four-dimensional isotopes: Their symmetry groups are transitive on all their cells, so the cells are “all alike.” Such polychora are scarce; fewer than three dozen uniform isochora are known, among which are the obvious 16 regular polychora.*See also*Colohimian polyhedra; Density; Hecatonicosachoron, regular; Hexacosichoron, regular; Isotope; Polytope; Riemannian polygon; Surtope; Thin; Transdimensional polytope families; Vertex figure; Wythoff’s construction.

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- A facet of a polyhedron; a two-dimensional element of a
polytope.
*See also*Facet.

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*See*Greatening.

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- An (
*n*–1)-dimensional element of an*n*-dimensional polytope. The facets of a line segment are its*end points*or*ends*; the facets of a polygon are its*edges*or*sides*; the facets of a polyhedron are its*faces*; the facets of a polychoron are its*cells*; the facets of a polytetron are its*tetrons*; and so on. - For
*n*<5, facet names are non-numeric, but for*n*>4, we use numeric Greek roots for their names:*tetron*,*penton*,*hexon*,*hepton*,*octon*,*enneon*,*decon*, and so on. Plurals of these are formed by adding -s.

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- The general operation of constructing a polytope or a
compound polytope that has exactly the same vertices as another
polytope. For example, in three-space a great stellated
dodecahedron is a faceting of a dodecahedron, since both
polyhedra have the same set of vertices. Instead of using the
pentagons of the dodecahedron, the great stellated dodecahedron
uses pentagrams that lie inside the dodecahedron. Pictures of all
the uniform facetings of a regular dodecahedron may be viewed at this Web
page; simply click and scroll. There are many more facetings
of the regular dodecahedron, a total of 44 of which (36 fully
symmetric and 8 subsymmetric) serve as vertex figures of uniform
star-polychora in the sishi/gaghi regiment and in the swirlprism
group. Many more—
*covertical compounds*of these—serve as vertex figures of uniform exopolychora that are not counted as proper uniform polychora.

- All uniform facetings of a uniform polytope belong to the
same army. If a faceting has not only the same vertices but also
the same edges as a polytope, each is an
*edge-faceting*of the other, and if they are uniform they both belong to the same regiment. For example, in three-space the great dodecahedron is an edge-faceting of the regular icosahedron.

- Faceting is, generally speaking, the dual operation to
stellation. That is, the dual of a stellation of a polytope is a
faceting of the dual of the polytope. For example, the great
icosahedron is a stellation of the icosahedron, and the dual of
the great icosahedron is the great stellated dodecahedron, a
faceting of the dodecahedron, the dual of the icosahedron.

- A
*symmetric*faceting is one that preserves the full symmetry group of the polytope; other kinds of facetings are*subsymmetric*. When the word faceting stands alone without a modifier, then it denotes symmetric faceting. A symmetric faceting of a uniform polyhedron is always at least isogonal. Although strict faceting requires using all the vertices of the polytope to be faceted, it is sometimes interesting to consider*partial*faceting, wherein only a subset of the vertices of the original polytope is used. For example, a regular tetrahedron may be obtained as a partial faceting of a cube. So alternation can be a kind of partial faceting as well as a truncation.

- All the symmetric facetings are known for all the
Platonic solids, although a systematic description of the
facetings of the regular dodecahedron has yet to be published.
When the number of vertices of a polyhedron exceeds twelve, the
number of facetings grows quickly (much like the number of
stellations of a polyhedron with more than twelve faces). The
convex uniform polyhedra with 60 and 120 vertices have very many
symmetric facetings, only a few of which are themselves uniform
polyhedra.

- The faceting operation is best illustrated using a
polyhedron that has not too many vertices and symmetries, so that
the number of symmetric facetings remains reasonably manageable,
but has enough vertices and symmetries to keep the operation from
becoming trivial. A good choice is the Archimedean pentagonal
prism. It has 20 symmetries, ten vertices, 15 edges, and seven
faces. Furthermore, it is the vertex figure of the rectified
hexacosichoron, or
*rox*, also called an*icosahedral hexacosihecatonicosachoron*. So any facetings of the prism will likely be vertex figures of uniform star-polychora in the rox regiment. The rox has 720 vertices, 3600 edges, 3600 triangular faces, and 120 icosahedral and 600 octahedral cells. In the prism, the two pentagonal bases are the vertex figures of the two icosahedra that meet at each corner, and the five squares are the vertex figures of the five octahedra that meet at each corner. The icosahedra form a*corner-connected set*of cells; they share no elements except their vertices.

- To begin faceting, we first identify all the different
sets of symmetrically equivalent line segments that connect all
possible pairs of the prism’s vertices. The prism has ten
vertices, so the total number of such line segments must be
10*9/2 = 45. Here are the symmetric sets:

A: The five edges that join the corners of the base pentagons

B: The ten edges of the base pentagons

C: The ten diagonals of the lateral squares

D: The ten diagonals of the base pentagons

E: The ten long diagonals that connect a vertex to the farthest vertices in the other base

- If the edge of the prism has length 1, then the edges in
sets A and B have length 1; the edges in set C have length
*sqrt*(2); the edges in set D have length (*sqrt*[5]+1)/2; and the edges in set E have length*sqrt*[(5+*sqrt*[5])/2] = 2**cos*(18°). These sets exhaust the 45 possible line segments connecting ten points. Experienced geometers will instantly recognize these edges as the vertex figures of triangles, squares, regular pentagons, and regular decagons. All the polychora in the rox regiment will have only these polygons as their faces.

- The next step in faceting is to find all the different
face planes defined by the prism’s vertices. The most
obvious ones are, of course, the two planes of the base pentagons
(defined by the A edges) and the five planes of the lateral
squares (defined by two A edges alternating with two B edges).
These are the external face planes. Internally there are four
more kinds, which may be found by systematic inspection: the five
planes of the rectangles defined by two A edges and two D edges;
the ten planes defined by wide isosceles triangles having two C
edges and a D edge; the ten planes defined by narrow isosceles
triangles having an A edge and two E edges; and the ten planes of
the trapezoids defined by an A edge, two C edges, and a D edge.
This is quite a menu of planes from which to construct
polyhedra.

- The third step in faceting is to find all the different
kinds of faces that can be constructed in the face planes from
the sets of edges. Because we are searching just for fully
symmetric facetings, we need consider only the symmetric
polygons. Were we searching for subsymmetric facetings, we would
need to consider all kinds of faces, symmetric and asymmetric,
that can be constructed from the sets of edges. Here is an
example of a subsymmetric faceting of the pentagonal prism. Its
symmetry group includes only the rotations of the prism, not the
reflections, so it is a chiral polyhedron. Its faces are the two
pentagonal bases of the prism and ten lateral scalene triangles,
each formed by an A edge, a C edge, and an E edge:

- The scalene triangles lie in the planes of
the ten trapezoids, each being one of a pair of congruent
triangles. This figure uses only the left-handed triangles of
each pair. There is another pair of congruent scalene triangles
in those planes (having a C edge, a D edge, and an E edge), and
they may be used with the diagonals of the pentagons, that is,
the base pentagrams, to form a different chiral polyhedron:

- Other subsymmetric facetings of the
pentagonal prism might have two different polygons for their
bases, or asymmetric pentagons for their bases, and so on. There
are quite a few possibilities up to and including completely
asymmetric polyhedra; no one has exhibited them all or even
enumerated them. It would likely require a pretty deep computer
search to exhaust all the possibilities. Here is a pretty
subsymmetric faceting of the pentagonal prism whose symmetry
group lacks the reflection in the mirror parallel to and halfway
between the bases:

- This is one reason we limit ourselves to
the fully symmetric facetings: to keep the size of the face sets
manageable. A fully symmetric faceting will either use all the
faces in a set or none. The other reason is that the vertex
figures of the rox regiment polychora will be found only among
the fully symmetric facetings. The symmetry group of the rox
itself forces this constraint.

- Here is a table of all the different groups of symmetric
faces that can be used to make fully symmetric facetings of the
pentagonal prism. The number in brackets designates their
mix-and-match notation when used to enumerate the star-polychora
in
the rox regiment. Free edges in a face set can be used to join
the faces to another face set that has the same free edges; this
is how the faces are mixed and matched to construct new
polyhedra.

[1] 2 base pentagons (all A edges, free)

[2] 5 lateral squares (all A edges, free; all B edges)

[3] 2 base pentagrams (all D edges, free)

[4] 5 rectangles (all D edges, free; all B edges)

[5] 10 wide isosceles triangles (all D edges, free; all C edges)

[6] 10 narrow isosceles triangles (all A edges, free; all E edges)

[7] 10 trapezoids (all A edges, free; all D edges, free; all C edges)

[8] 10 neckties (all A edges, free; all D edges, free; all E edges)

[9] 10 butterflies (all C edges; all E edges)

[10] 5 square bowties (all A edges, free; all C edges, free)

[11] 5 square bowties (all B edges; all C edges, free)

[12] 5 short bowties (all B edges; all E edges, free)

[13] 5 long bowties (all D edges, free; all E edges, free)

[14] 2 complete pentagons, density 1 (all A edges, free; all D edges, free)

[15] 2 complete pentagons, density 3 (all A edges, free; all D edges, free)

- Here a
*necktie*is a crossed quadrilateral whose sides are the two parallel bases and two long diagonals of a trapezoid; a*butterfly*is a crossed quadrilateral whose sides are the lateral sides and two long diagonals of a trapezoid; and a*bowtie*is a crossed quadrilateral whose sides are two parallel edges and the two diagonals of a rectangle or a square. A*short bowtie*uses the short sides of a rectangle, and a*long bowtie*uses the long sides of a rectangle, along with the diagonals of the rectangle.

- A quick inventory of the faces and their edges shows that
the bowties cannot be mixed and matched to any other kinds of
faces in the list. It also shows that the ten butterfly faces
form a closed face set: there are no free edges by which they can
join to other faces. All by themselves, they form a toroidal
isogonal isohedron called a
*stephanoid*(from Greek*stephanos*, meaning “crown” or “garland”). It corresponds to a uniform isochoric polychoron in the rox regiment whose cells are 120 identical small rhombidodekahedra (in a small rhombidodekahedron, the faces are the 30 squares of a rhombicosidodecahedron together with twelve decagons; its vertex figure is the butterfly quadrilateral).

- The faces in sets [14] and [15] are the exopolygons
called complete pentagons, of density 1 and 3 respectively. They
make externally different facetings of the prism, and as vertex
figures of uniform polychora they represent the small complex
icosidodecahedra of density 2 and 4, respectively. But in
counting members of the rox regiment, we ignore cells that differ
just in their densities. So these facetings correspond to the 22
different uniform rox regiment polychora.

- To make a long story short, all 25 facetings of the
pentagonal prism are depicted in the associated diagram.
Simply click and scroll through it to see how the various
combinations of faces fit together to make the facetings. For
completeness, here is a table of the cells in the rox regiment
that correspond to the face sets of the pentagonal prism:

[1] 120 icosahedra

[2] 600 octahedra

[3] 120 great dodecahedra

[4] 120 icosidodecahedra

[5] 720 pentagonal prisms

[6] 120 truncated dodecahedra

[7] 120 [small] rhombicosidodecahedra

[8] 120 small dodekicosidodecahedra

[9] 120 small rhombidodekahedra

[10] 600 tetrahemihexahedra (not used as cells)

[11] 600 more tetrahemihexahedra (not used as cells)

[12] 120 small icosihemidodecahedra (not used as cells)

[13] 120 small dodecahemidodecahedra (not used as cells)

[14] 120 small complex icosidodecahedra, density 2

[15] 120 small complex icosidodecahedra, density 4 - The faceting of the Archimedean pentagonal prism
extends trivially to a pentagonal prism of any height, not just
unit height. That is, any pentagonal prism has the same 25 kinds
of symmetric facetings, just stretched or flattened. In
particular, there are four major regiments in the 720-vertex rox
army (the rox is the general of its army as well as the colonel
of its regiment), all of whose vertex figures are facetings of
regular-pentagonal prisms of various height-to-base-edge ratios.
Thus, each regiment includes 22 different uniform polychora, for
a total of 88.
*See also*Alteration; Alternation; Army; Colonel; Density; Exotic polytopes; General; Mix-and-match notation; Platonic solids; Regiment; Stellation; Truncation.

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*See under*Regular polytope.

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- A
*k*-dimensional space, –1<*k*<*n*, embedded in an*n*-dimensional space. If*k*=*n*–1, the flat is a*hyperplane*. In Euclidean*n*-space, a 0-flat is a point, a 1-flat is a line, a 2-flat is a plane, a 3-flat is a*realm*, a 4-flat is a*tetrealm*, a 5-flat is a*pentrealm*, a 6-flat is a*hexrealm*, a 7-flat is a*heptrealm*, an 8-flat is an*octrealm*, a 9-flat is an*ennearealm*, a 10-flat is a*decrealm*, and so on, prefixing Greek numerical roots to the word*realm*for*k*>3. Such subspaces are called flat because they have no intrinsic curvature; they are as flat as possible in their embedding spaces.

- As an adjective,
*flat*can also describe an (*n*–1)-dimensional figure embedded in*n*-dimensional space. A finite set of at least*n*+1 flat figures embedded in*n*-space can be positioned so as to isolate a region of*n*-space, but no countable set of thin figures can.*See also*Thick; Thin.

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- A facet of a potential polytope that has at least one ridge
not common to another facet of the polytope. Adjoining a new
facet to the potential polytope at such a ridge is necessary
(though by no means sufficient) to convert the potential polytope
into a true polytope.

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- The (usually uniform, quite often convex) polytope that
contains itself and all the other members of its army.
*See also*Army.

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- An opposite isometry in
*n*-space,*n*>1 that combines a translation with a reflection in a mirror positioned parallel to the translation. No finite polytope can have a glide reflection as a symmetry, because no point of*n*-space remains fixed, but honeycombs, infinite polytopes, and tilings can.*See also*Honeycomb; Infinite polytope; Symmetry; Tiling.

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- A four-dimensional hypersphere: the locus of all the points
of any metrical four-space at a particular distance, called the
*radius*, from a given point, called the*center*. From the Latin*glomus*, meaning “ball”; possessive*glomeric*. Although a glome lies in Euclidean four-space, it is actually a curved three-dimensional manifold, the surcell of a gongyl.

- The volume of a glome is its three-dimensional content,
which is given in terms of its radius
*r*by the formula 2*pi*^{2}*r*^{3}. The bulk of a glome is the four-dimensional content of the glome’s gongyl, and it is given in terms of the radius*r*by the formula^{1}/_{2}*pi*^{2}*r*^{4}.

- A glome may be regularly tessellated in 16 ways,
corresponding to the 16 regular polcyhora.
*See also*Content of a figure; Gongyl; Hypersphere; Regular polytope; Surcell.

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- A glome together with its finite interior; a four-dimensional
hyperball. From the Greek
*gongylos*, meaning “ball”; pronounced “GON-jil”. The interior of a gongyl by itself is called an*open*gongyl. The boundary of a gongyl is a glome.*See also*Glome.

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*See*Aggrandizement.

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- Stellation of a polyhedron
in three-space. In greatening, the faces of*P*expand in their planes, passing through other such planes if necessary, to meet the expansions of other faces of the polyhedron. Usually this is done symmetrically, so that identical faces of the polyhedron expand exactly the same way to produce a symmetric star-polyhedron. A synonym for greatening is*P**face-stellation*.

- A more restrictive definition of greatening requires that
the expanded faces be similar to the original faces. Then, given
a regular polyhedron
, the*R**great*has the same kinds of faces as*R*, only larger and more interpenetrating. This definition extends to regular polychora*R*, so that the cells of a great*R*are greatened cells of*R*.*R**See also*Regular polytope; Stellation.

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- Also known as a
*demihypercube*in*n*-space,*n*>2. The uniform polytope constructed by completely truncating the alternate vertices of a measure polytope, that is, truncating half its vertices by hyperplanes that pass through a truncated vertex’s edge-neighboring vertices. If*n*=2, this truncation of a square leaves only the square’s diagonal, which is not a polygon. But when*n*=3, this truncation of a cube produces a regular tetrahedron, and when*n*=4, it produces a regular hexadecachoron from a tesseract. For*n*>4, demihypercubes are no longer regular, only uniform, with two different kinds of facets: 2*n*(*n*–1)-dimensional demihypercubes and 2^{n–1}(*n*–1)-dimensional simplexes. The names of the demihypercubes are constructed by prefixing demi- to the name of a measure polytope:*demipenteract*,*demihexeract*,*demihepteract*, and so forth. As noted above, a*demicube*is a regular tetrahedron, and a*demitesseract*is a regular hexadecachoron.

- The vertex figure of a half measure polytope
in*H**n*-space is a rectified (*n*–1)-dimensional simplex (the simplex has*n*vertices and*n*facets). The facets of the rectified simplex are (1)*n*(*n*–2)-dimensional simplexes, which are the vertex figures of the (*n*–1)-dimensional simplex facets of, and (2)*H**n*rectified (*n*–2)-dimensional simplexes, which are the vertex figures of the (*n*–1)-dimensional demihypercube facets of. Thus, for example, the vertex figure of the demicube is a rectified triangle, which is a smaller triangle, the vertex figure of a tetrahedron (the demicube). The vertex figure of the demitesseract is a rectified tetrahedron, which is an octahedron, the vertex figure of a regular hexadecachoron (the demitesseract). And so on.*H*

- Euclidean
*n*-space can always be uniformly honeycombed by half measure polytopes and cross polytopes: In the regular honeycomb of*n*-space by measure polytopes, remove alternate vertices, thereby transforming each measure polytope into a half measure polytope, and fill the gaps with*n*-dimensional cross polytopes (the vertex figures of the measure polytope honeycomb). In the plane, this changes the checkerboard tiling into another checkerboard tiling rotated 45° to the original; in 3-space this produces the uniform honeycomb of regular tetrahedra and octahedra; and in 4-space this produces the regular honeycomb of hexadecachora.*See also*Alteration; Alternation; Rectification; Simplex.

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- The convex regular polychoron with 120 regular-dodecahedral
cells, also called a
*120-cell*and a*hi*(by Jonathan Bowers). The name is from the Greek*hecaton*, meaning 100, and*icosa*, meaning 20. It has 600 vertices, 1200 edges, and 720 pentagonal faces as its non-trivial elements besides its cells. It was discovered by Ludwig Schläfli in the mid-1850s; its Schläfli symbol is {5,3,3}. The 120 dodecahedra lie in rings of ten that girdle the polychoron; each dodecahedron belongs to six such rings. This means the dihedral angle of the hecatonicosachoron is exactly 144°.

- Of the ten regular star-polychora, nine are also various
kinds of hecatonicosachora, each having 120 cells constructed by
extending the cells (a three-dimensional “stellation”)
of the convex regular hecatonicosachoron until they meet the like
extensions of other cells. Four of the ten regular star-polychora
were discovered by Ludwig Schläfli (who also discovered the
six convex regular polychora), and the other six by Edmund Hess
(who also independently rediscovered the four of Schläfli).
The nine regular star-hecatonicosachora are constructed from the
convex regular hecatonicosachoron as follows:

- [1] Extend (stellate) each of the 120 dodecahedral cells
into a small stellated dodecahedron. The resulting polychoron is
called a
*stellated hecatonicosachoron*, and it has 120 vertices (one above each cell of the original hecatonicosachoron, 1200 edges (extensions of the edges of the hecatonicosachoron), and 720 pentagrammatic faces (stellations of the pentagonal faces of the hecatonicosachoron) in addition to its 120 small stellated dodecahedral cells. It was described by Edmund Hess in 1876; its Schläfli symbol is {^{5}/_{2},5,3}. The operation in which a pentagon’s edges are extended until they meet is an example of*stellation*; hence the name of this star-polychoron. Jonathan Bowers calls it a*sishi*(an acronym for*small*stellated hecatonicosachoron: here the “small” is to contrast with the “great” and “grand” names used below).

- [2] Replace each small stellated dodecahedral cell in the
preceding star-polychoron with the great dodecahedron that has
the same vertices. The resulting polychoron is called a
*great hecatonicosachoron*, and it has 120 vertices (the same as the vertices of the preceding star-polychoron), 720 edges, and 720 pentagonal faces (where the pentagrams of the preceding star-polychoron were located) in addition to its 120 great dodecahedral cells. This operation, wherein each pentagram is replaced by the pentagon that has the same vertices, is called*greatening*; hence the name of this star-polychoron. It was described by Edmund Hess in 1876; its Schläfli symbol is {5,^{5}/_{2},5}. Jonathan Bowers calls it a*gohi*(an acronym for great hecatonicosachoron).

- [3] Replace each great dodecahedral cell in the preceding
star-polychoron with the icosahedron that has the same vertices
and edges. The resulting polychoron is called an
*icosahedral hecatonicosachoron*, and it has 120 vertices (the same as the vertices of the preceding star-polychoron), 720 edges (the same as the edges of the preceding star-polychoron), and 1200 triangular faces in addition to its 120 icosahedral cells. Its triangles are the same as those of the regular hexacosichoron (600-cell) that has the same vertices, which is why Jonathan Bowers calls it a*fix*(an acronym for*faceted hexacosichoron*). It was described by Edmund Hess in 1876; its Schläfli symbol is {3,5,^{5}/_{2}}.

- [4] Extend (stellate) the edges of [2] above (the great
hecatonicosachoron), so that each great dodecahedral cell becomes
a great stellated dodecahedron. The resulting star-polychoron has
120 vertices, 720 edges, 720 pentagrammatic faces, and 120 great
stellated dodecahedral cells, and is called a
*great stellated hecatonicosachoron*. It was discovered by Ludwig Schläfli in the mid-1850s; its Schläfli symbol is {^{5}/_{2},3,5}. Jonathan Bowers calls it a*gishi*(an acronym for great stellated hecatonicosachoron).

- [5] Replace each great stellated dodecahedral cell of the
preceding star-polychoron with the dodecahedron that has the same
vertices. This creates a new star-polychoron whose cells are 120
dodecahedra, only they’re packed together 20 at each corner
rather than only four as in the ordinary hecatonicosachoron.
Replacing the great stellated dodecahedra by dodecahedra in this
manner is an example of
*aggrandizement*(the big outer dodecahedron is an enlarged, or aggrandized, inverted version of the little dodecahedron buried deep within the great stellated dodecahedron and bounded by its face-planes), so we call this star-polychoron the*grand hecatonicosachoron*. It has 120 vertices, 720 edges, 720 pentagonal faces, and 120 dodecahedral cells, and it has a very complicated surtope, as do all the succeeding star-polychora. It was discovered by Ludwig Schläfli in the mid-1850s; its Schläfli symbol is {5,3,^{5}/_{2}}. Jonathan Bowers calls it a*gahi*(an acronym for grand hecatonicosachoron).

- [6] Extend (stellate) each dodecahedral cell of the
preceding star-polychoron into a small stellated dodecahedron.
(Amazingly enough, this does create another regular
star-polychoron!). The resulting figure has 120 vertices, 720
edges (extensions of the edges of the preceding), 720
pentagrammatic faces (stellations of the pentagrams of the
preceding), and 120 small stellated dodecahedral cells. It is
called a
*grand stellated hecatonicosachoron*. It was described by Edmund Hess in 1876; its Schläfli symbol is {^{5}/_{2},5,^{5}/_{2}}. Jonathan Bowers’s name for it is*gashi*(an acronym for grand stellated hecatonicosachoron).

- [7] Replace the small stellated dodecahedral cells of the
preceding star-polychoron with the great icosahedra that have the
same vertices and edges. It is one of those wonderful
coincidences of polytope geometry that the face planes of this
particular great icosahedron are the same as the face planes of
the icosahedra of star-polychoron [3], so that it is a greatening
of that figure: the
*great icosahedral hecatonicosachoron*. It has the same 120 vertices and 720 edges of the grand stellated hecatonicosachoron, but its faces are 1200 triangles that are shared among its 120 great icosahedral cells. It was described by Edmund Hess in 1876; its Schläfli symbol is {3,^{5}/_{2},5}. Jonathan Bowers’s name for it is*gofix*(an acronym for*great faceted hexacosichoron*). It has the same faces as the*grand hexacosichoron*(grand 600-cell). The latter is the only regular star-polychoron whose cells are 600 tetrahedra, so it falls outside the present series of star-hecatonicosachora.*See also*Hexacosichoron, regular.

- [8] Replace the pentagrams of the grand stellated
hecatonicosachoron with pentagons, thereby greatening the small
stellated dodecahedral cells of the former into great
dodecahedra. The new figure has 120 vertices, 1200 edges, 720
pentagonal faces, and 120 great dodecahedral cells: the
*great grand hecatonicosachoron*. It was described by Edmund Hess in 1876; its Schläfli symbol is {5,^{5}/_{2},3}. Jonathan Bowers’s name for it is*gaghi*(an acronym for great grand hecatonicosachoron). Digression: You cannot make a further regular star-polychoron by replacing the great dodecahedra with icosahedra, because the icosahedra will not adjoin along their faces. But you*can*close the figure with an additional 600 tetrahedra, giving a complicated uniform star-polychoron with 600 tetrahedral and 120 icosahedral cells: the*great ditrigonary hexacosihecatonicosachoron*(which Jonathan Bowers acronymizes into*gidtixhi*). End of digression.*See also*Ditrigonary polychora.

- [9] Finally, stellate the great dodecahedral cells of
star-polychoron [8] into great stellated dodecahedra to produce
the whopping
*great grand stellated hecatonicosachoron*. Jonathan Bowers calls it a*gogishi*(an acronym for great grand stellated hexacosichoron). This formidable figure has 600 vertices, 1200 edges, 720 pentagrammatic faces, and 120 great stellated dodecahedral cells. It was discovered by Ludwig Schläfli in the mid-1850s; its Schläfli symbol is {^{5}/_{2},3,3}. If we could see it in its entirety, it would resemble a four-dimensional sea urchin.

- Unfortunately, here the process of
stellation-greatening-aggrandizement ceases to create new
star-polychora; substituting dodecahedra for the great stellated
dodecahedral cells leaves all the pentagonal faces free, and the
figure does not close. The free faces
*can*be taken up by numerous other kinds of uniform polyhedra, to create a regiment of uniform star-polychora, but that is beyond the scope of this particular Glossary entry.

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- The convex regular polychoron with 600 regular-tetrahedral
cells, also called a
*600-cell*and an*ex*(by Jonathan Bowers). The name is from the Greek*hexacosa*, meaning 600. It has 120 vertices, 720 edges, and 1200 triangular faces as its non-trivial elements besides its cells. It was discovered by Ludwig Schläfli in the mid-1850s; its Schläfli symbol is {3,3,5}. Its 720 edges lie on 72 equatorial decagons. It is the general of the 120-vertex*hexacosichoric army*and the colonel of the*hexacosichoric regiment*.

- One of the ten regular star-polychora is another
hexacosichoron, namely, the
*grand hexacosichoron*, or*gax*as Jonathan Bowers calls it. It is the conjugate of the regular convex, or small, hexacosichoron. It was discovered by Ludwig Schläfli in the mid-1850s; its Schläfli symbol is {3,3,^{5}/_{2}}. It has the same vertices, edges, and faces as the great icosahedral hecatonicosachoron (or gofix, as Jonathan Bowers calls it) and is therefore in the same company.*See also*Hecatonicosachoron, regular.

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- The convex regular polychoron with 16 regular-tetrahedral
cells, also called a
*16-cell*and a*hex*(by Jonathan Bowers); the four-dimensional regular cross polytope; the four-dimensional half measure polytope. The name is from the Greek*hexadeca*, meaning 16. It has eight vertices, 24 edges, and 32 triangular faces as its non-trivial elements besides its cells. It was discovered by Ludwig Schläfli in the mid-1850s; its Schläfli symbol is {3,3,4}. Its 24 edges lie on six equatorial squares. Four-space may be uniformly honeycombed by hexes, three sharing each face. This*hexadecachoric honeycomb*has Schläfli symbol {3,3,4,3}.*See also*Cross polytope.

- The simplest uniform star-polychoron has the vertices,
edges,
and faces of a hex. Discard an alternating set of eight
tetrahedra from the hex, and adjoin the resulting 32 free
triangular faces to the free faces of a set of four concentric
octahedra placed at the center of the hex in four mutually
perpendicular realms. The resulting polychoron is called an
*octahemioctachoron*, or a*tho*(for*tesseractihemioctachoron*) by Jonathan Bowers. It is the four-dimensional demicross polytope.*See also*Demicross polytope.

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- A homotope in two-space, also called an
*equilateral polygon*.*See also*Homotope.

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- A homotope in three-space; a polyhedron all of whose faces
are congruent. All regular polyhedra are homohedra, but not
*vice versa*.*See also*Homotope.

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- A polytope whose facets are all congruent. All line segments
are one-dimensional homotopes. In two-space, a homotope is any
polygon whose sides have the same length, and we can call such a
figure a
*homogon*or (more usually) an*equilateral polygon*. In three-space, a homotope is a polyhedron whose faces are congruent polygons, and we call such a figure a*homohedron*. In four-space a homotope is a*homochoron*, and in higher spaces we use Greek numerical suffixes:*homotetron*,*homopenton*,*homohexon*, and so on. To denote a specific kind of homotope, we insert an appropriate Greek numerical prefix in between the “homo” prefix and the suffix. For example, a tetrahedron whose faces are all congruent is a*homotetrahedron*.*See also*Isotope.

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- An infinite collection of
*n*-dimensional polytopes,*n*>2, adjoining along their facets, that fills*n*-space completely. If the polytopes are uniform and the symmetry group of the honeycomb is transitive on its vertices (that is, its corners are all “surrounded alike”), the honeycomb is*uniform*. If in addition the polytopes are regular and congruent, the honeycomb is*regular*. If the polytopes overlap, so that the honeycomb fills*n*-space more than once, the honeycomb is a*star-honeycomb*.

- Honeycombs can also be defined that fill
*n*-dimensional elliptic spaces, spherical spaces, and hyperbolic spaces. Spherical*n*-dimensional honeycombs strongly resemble Euclidean polytopes of*n*+1 dimensions, because Euclidean polytopes can be mapped very straightforwardly onto hyperspheres. Considerable mathematical literature exists on tilings and honeycombs, far beyond the scope of this Glossary entry.

- Examples of honeycombs include the regular honeycombs of
measure polytopes in Euclidean
*n*-space for all*n*>0; the uniform honeycomb of tetrahedra and octahedra in three-space; and the regular honeycombs of hexadecachora and icositetrachora in four-space.*See also*Polytope; Regular polytope; Uniform polytope; Tessellation; Tiling.

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- Any of a family of regular skew polytopes whose facets are
*k*-dimensional measure polytopes embedded in 2*k*-space. In the general hotel, the measure polytopes adjoin one another along their facets like the rooms of a hotel. Specifically, a {*p*/*q*}-gonal*k*-hotel is a skew polytope in 2*k*-space whose cells are*p*^{k}*k*-dimensional measure polytopes arranged in regular {*p*/*q*}-gonal girdles.

- When
*k*=1, the {*p*/*q*}-gonal 1-hotel is simply the edges of the {*p*/*q*}, so the hotel is not skew, just interiorless. When*k*=2, the {*p*/*q*}-gonal 2-hotel comprises the*p*^{2}square faces of a {*p*/*q*}-gonal duoprism (although when*p*=4 and*q*=1, we have to exclude eight squares). In general, the facets of a {*p*/*q*}-gonal*k*-hotel are the*p*^{k}*k*-hypercubes of the Cartesian product prism {*p*/*q*}^{k}.

- For
*k*>1, we might denote a {*p*/*q*}-gonal*k*-hotel by

{4,3,3,...|*k*–2 3’s altogether|...,3,3,4|*p*/*q*}.

- When
*k*=2, there are no threes in the notation, and it becomes Coxeter’s notation for a*p*/*q*-gonal 2-hotel:

{4,4|*p*/*q*}.

- The vertex figure of a
*k*-hotel is a regular skew*k*-orthotope in (2*k*–1)-space, whose facets are among the (*k*–1)-simplexes of an irregular (2*k*–1)-dimensional simplex, all of whose edges have length*sqrt*(2)*except**k*edges that join pairs of non-adjoining vertices. These*k*edges have length 2*sqrt*(pi**q*/*p*). For example, for*k*=2, the vertex figure of a {*p*/*q*}-gonal dihotel (we can use the Greek prefix instead of*k*- here: thus, dihotel, trihotel, tetrahotel, etc.) is the skew zigzag polygon of a disphenoid whose two (that is,*k*) opposite edges are of length 2*sqrt*(pi**q*/*p*) and whose (other four) zigzag edges are all of length*sqrt*(2).

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- The interior of a hypersphere together with its boundary. If
the boundary is excluded, the hyperball is
*open*. In Euclidean 1-space, a hyperball is called a*line segment*or a*closed interval*; in 2-space, a hyperball is called a*disk*; in 3-space, a*ball*; in 4-space, a*gongyl*; in 5-space, a*pentaball*; in 6-space, a*hexaball*; in 7-space, a*heptaball*; and so on, prefixing a Greek numerical root to the word ball in spaces of*k*dimensions,*k*>4.

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- A section of a hypersphere by a hyperplane. A
*great hypercircle*is the section of a hypersphere by a hyperplane through the center. A hypercircle has one fewer dimensions than a hypersphere. In particular, a hypercircle of a circle is a pair of points, or*point-pair*; a hypercircle of a sphere is a circle; a hypercircle of a glome is a sphere; a hypercircle of a tetraglome is a glome; a hypercircle of a pentaglome is a tetraglome; and so on. The modifier*great*is used when the sectioning hyperplane passes through the center.

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- Synonym of measure polytope in
*n*-space, but often used specifically as the name of the four-dimensional measure polytope, or tesseract.*See also*Measure polytope.

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- An (
*n*–1)-dimensional space imbedded in*n*-space, for*n*>–1. A hyperplane in a point is the empty space; a hyperplane in a line is a point; a hyperplane in a plane is a line; a hyperplane in a realm is a plane; a hyperplane in a tetrealm is a realm; and so on. A hyperplane divides*n*-space into two regions, so that a path from any point in one region to any point in the other must intersect the hyperplane.

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- The locus of all the points of any metrical
*n*-dimensional space,*n*>0, at a particular distance, called the*radius*, from a given point, called the*center*. For*n*=1, a hypersphere is a*dyad*; for*n*=2, a hypersphere is a*circle*; for*n*=3, a hypersphere is a*sphere*; for*n*=4, a hypersphere is a*glome*; for*n*=5, a hypersphere is a*tetraglome*; for*n*=6, a hypersphere is a*pentaglome*; and so on. Greek numerical prefixes are used for hyperspheres in dimensions greater than four. Note that a hypersphere in Euclidean*n*-dimensional space is an (*n*–1)-dimensional manifold, so the Greek prefix is chosen accordingly. Some authors, such as H. S. M. Coxeter in*Regular Polytopes*, use the term “sphere” not just for a 2-hypersphere in 3-space but for a general (*n*–1)-hypersphere in*n*-space.

- In 1983, F. Le Lionnais investigated the
(
*n*–1)-dimensional and*n*-dimensional contents of hyperspheres in*n*-dimensional spaces. It turns out that the surhexon hexabulk (6-dimensional content) of a unit hexaglome (*n*=7) is numerically the largest for any*n*-dimensional hypersphere, at^{16}/_{15 }*pi*^{3}, and the pentabulk (5-dimensional content) of a unit tetraglome (*n*=5) is numerically the largest for any*n*-dimensional hypersphere, at^{8}/_{15 }*pi*^{2}.*See also*Content of a figure; Glome; Gongyl; Hyperball; Hypercircle.

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- A simplex imbedded in a hypersphere of
*n*dimensions. If the hypersphere is two points (*n*=0), a hyperspherical simplex is one of those points; if the hypersphere is a circle (*n*=1), a hyperspherical simplex is an*arc*; if the hypersphere is a sphere (*n*=2), a hyperspherical simplex is a*spherical triangle*; if the hypersphere is a glome (*n*=3), a hyperspherical simplex is a*glomeric tetrahedron*; if the hypersphere is a tetraglome (*n*=4), a hyperspherical simplex is a*tetraglomeric pentachoron*; and so on.

- The elements of a hyperspherical simplex, as with a
Euclidean
simplex, are themselves hyperspherical simplexes of dimension
*j*, –2<*j*<*n*+1. The corners are*n*+1 points that do not all lie in the same great hypercircle; the edges are arcs of great circles that join any pair of corners; the faces are spherical triangles of great spheres incident with any three corners; the cells are glomeric tetrahedra of great glomes incident with any four corners; and so on.*See also*Hypersphere; Simplex.

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- The regiment of 14 uniform polychora and one regular compound
that have the same 24 vertices and 96 edges as a regular
icositetrachoron, or
*ico*. It is arbitrarily coded as regiment 24/1/1: the first regiment of the first 24-vertex army. The possible faces (face sets) for the ico regiment are the 96 triangles of the icositetrachoron, the 72 squares that are the equatorial squares of the octahedra of the icositetrachoron, and the 16 equatorial hexagons of the icositetrachoron. The possible cells (cell sets) for the ico regiment are the 24 octahedra of the icositetrahedron, the 48 tetrahemihexahedra inscribable in the 24 octahedra, the twelve equatorial cuboctahedra of the icositetrachoron, the twelve octahemioctahedra inscribable in those cuboctahedra, and the twelve hexahemioctahedra inscribable in those cuboctahedra. There are also 24 cubes, but these form the well-known regular compound of three tesseracts in an icositetrachoron and no uniform subsets of these cubes are available as cells of uniform star-polychora in the ico regiment. All the uniform polychora in the ico regiment use various combinations of faces and cells taken from the above-listed face sets and cell sets.

- Excluding the ico itself and the compound of three
tesseracts, all 13 uniform star-polychora of the ico regiment are
subsymmetric facetings of the ico: They do not have all the
ico’s 1152 symmetries. One has half the symmetries (576),
six have
^{1}/_{3}the symmetries (384), and six have^{1}/_{6}the symmetries (192). Only the ico itself uses all the available cells in a cell set (the 24 octahedra). Because each central hexagon of the ico is common to the three octahemioctahedra or hexahemioctahedra that pass through it, no ico polychoron can use all twelve of either cell that are available; a face must belong to exactly two (not three) cells in any polychoron. Also, all 13 ico star-polychora are not orientable.

- The hexadecachoric symmetry group (which is the symmetry
group of the tesseract and the hexadecachoron) has order 384, and
it is a subgroup of index three in in the icositetrachoric
symmetry group. This permits a uniform coloring of the 24
octahedra of the icositetrachoron in three colors, so that eight
octahedra receive each color. Let these colors be red, yellow,
and blue. Color one octahedron red, and then color red the six
octahedra that touch the octahedron just at its six corners, and
finally color red the octahedron opposite the first red
octahedron, which also touches the six red octahedra just at
their corners. Each red octahedron then touches another red
octahedron at its six corners. The eight octahedra together
form a
*corner-connected subset*of the 24 octahedra of the icositetrachoron. Now color an uncolored octahedron yellow, and also the seven other octahedra that form a corner-connected set with it. No red octahedron can be in the yellow set, because all its corner-neighbors are already colored red. This leaves eight octahedra uncolored, so color them blue. They touch only other blue octahedra at their corners.

- This coloring of the icositetrachoron has some
interesting properties. Each edge of the icositetrachoron is
common to three octahedra, one of each different color. The eight
octahedra of any one color lie inside the eight cubes of one of
the tesseracts of the regular compound of three tesseracts, so
the tesseracts may be correspondingly colored red, yellow, and
blue. In fact, each octahedron is situated so that its corners
are centered in the faces of one of the cubes of the tesseract.
If the colored icositetrachoron is projected vertex-first into a
rhombic dodecahedron in three-space, its three equatorial bands
of four rhombi acquire the ico’s three colors.

- The ico vertex figure is a unit cube. Its twelve edges
are the vertex figures of the ico triangles incident at a vertex,
its twelve facial diagonals are the vertex figures of twelve of
the squares in the square face set incident at a vertex, and its
four long diagonals are the vertex figures of four equatorial
hexagons incident at a vertex. The faces of the cube are the
vertex figures of the six octahedra incident at a vertex. If the
octahedra are three-colored as described above, then all three
colors will occur in the cube. Two opposite squares of the cube
will have the same color, corresponding to the two octahedra of
that color incident at each vertex. The fact that all the vertex
figures will have the same coloring is what makes the coloring
uniform.

- The vertex figures of the uniform polychora in the ico
regiment are all facetings of the cube. The accompanying diagram of
the face sets of the ico regiment shows all the different
faceting faces and their mix-and-match numberings. Because the
octahedra fall into three symmetric subsets, it is necessary to
give the squares three numbers ([1][2][3]), and this threefold
numbering carries over to all the other face sets. This little
hitch unfortunately makes the mix-and-match notation somewhat
cumbersome for this regiment, because a polychoron will not have
a unique notation. But on the other hand, the number of different
notations for a particular polychoron equals the index of its
symmetry group in the icositetrachoric symmetry group, so one may
determine the order of its symmetry group from its full
mix-and-match notation at a glance.

- Because the ico uses all three subsets of eight
octahedra, its mix-and-match notation becomes 24/1/1[1][2][3].
The regular compound of three tesseracts uses faceting cell set
[19], the 24 cubes, so its mix-and-match notation becomes
24/1/1[19]. The 13 star-polychora in the ico regiment have their
vertex figures illustrated in the accompanying chart. The
chart also provides Jonathan Bowers’s names for these
polychora, as well as formal Greekish names and their full
mix-and-match notations. In the chart, the little BF and GF
notations indicate whether it was Bowers or George (your humble
Glossary compiler) who found the particular star-polychoron. Both
of us discovered the 13th one simultaneously. One configuration
(not shown in the chart) in Jonathan’s original list was
pretty deceptive and turned out to be a compound. We now have a
rather simple proof that these figures do exhaust the possible
star-polychora in the ico regiment.

- The mix-and-match notations indicate which cell sets
compose each star-polychoron. For example, the
*disoctachoron*, or*oh*, has mix-and-match notation 24/1/1[1][13][14]=[2][14][15]=[3][13][15]. This indicates that its cells are the eight octahedra of set [1], [2], or [3], the four octahemioctahedra of set [13], [14], or [15], and four more octahemioctahedra of set [14], [15], or [13], respectively. It is called a disoctachoron because it has altogether two different sets (*dis-*) of eight cells each (*-octa-*). A hypothetical four-dimensional viewer would see this figure as having a surcell of eight corner-connected octahedra separated by 16 octahedral chasms that extend to the center of the polychoron, much like the octahemioctahedron has a surface of eight corner-connected triangles separated by six square-pyramidal cavities whose apices are all at its center. Because it has three different mix-and-match notations, it has^{1152}/_{3}= 384 symmetries, the symmetries of a tesseract or hexadecachoron.

- The other two symmetry groups represented among the
star-polychora of the ico regiment are the
*ionic diminished icositetrachoric group*of order 576, which is the symmetry group of the icositetrahemiicositetrachoron, or*ihi*, and the*ionic diminished hexadecachoric group*of order 192, which is the symmetry group of the six ico star-polychora that have six different mix-and-match notations. The former is also the symmetry group of the snub icositetrachoron, a convex uniform polychoron whose cells are 120 tetrahedra and 24 icosahedra.

- It should be clear from the preceding that the
icositetrachoron is the colonel of the ico regiment 24/1/1. It is
also the general of the ico army 24/1, and this army includes
another regiment, 24/1/2, whose only members are two uniform
compounds. Whereas the edges of the 24/1/1 regiment are the edges
of the icositetrachoron, the edges of the 24/1/2 are the shortest
diagonals of the icositetrachoron, namely, the 72 diagonals of
its octahedral cells. These are the edges of the regular compound
of three hexadecachora that is dual to the regular compound of
three tesseracts. This compound is the colonel of the 24/1/2
regiment. The vertex figure of this compound is the regular
octahedron whose corners are at the centers of the faces of the
cube that is the vertex figure of the icositetrachoron.

- The octahedron vertex figure has just one uniform
faceting, into a tetrahemihexahedron (the demicross polyhedron),
which is the vertex figure of the demicross polychoron that has
the same vertices, edges, and faces as the hexadecachoron. So the
three hexadecachora can be replaced by three demicross polychora
in the compound of three hexadecachora. This can be done in two
distinct ways, but only one of these is the uniform compound of
three demicross polychora (the other way yields a nonuniform
compound of three demicross polychora). This is the other member
of the 24/1/2 regiment.

- Twenty-four distinct points may be connected by a total
of 24*23/2 = 276 line segments. When these 24 points are the
vertices of an icositetrachoron, the line segments fall by their
lengths into four sets. The shortest are the 96 edges of the
icositetrachoron. Next come the 72 diagonals of its 24 octahedral
cells; these are the edges of regiment 24/1/2, the edges of the
regular compound of three hexadecachora. Then come the 96 long
diagonals of the cubic cells of the regular compound of three
tesseracts, and finally the twelve long diagonals of the
icositetrachoron. It can be shown that the latter two sets of
edges do not form polyhedra that can be used as cells of uniform
polychora (trivially so in the case of the long diagonals), so
the faceting of the icositetrachoron for uniform polychora and
compounds is complete with the two regiments described above.
*See also*Army; Colonel; Faceting; General; Icositetrachoron, regular; Mix-and-match notation; Regiment; Surcell; Uniform polytope; Vertex figure.

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- The convex regular polychoron with 24 regular-octahedral
cells, also called a
*24-cell*and an*ico*(by Jonathan Bowers). The name is from the Greek*icosa*, meaning 20, and*tetra*, meaning four. It has 24 vertices, 96 edges, and 96 triangular faces as its non-trivial elements besides its cells. It was discovered by Ludwig Schläfli in the mid-1850s; its Schläfli symbol is {3,4,3}. Its 96 edges lie on 16 equatorial hexagons, and its cells girdle the polychoron in rings of six, which gives it a dihedral angle of 120°. Four-space may be uniformly tiled by icos, three around each face, producng the*icositetrachoric honeycomb*, whose Schläfli symbol is {3,4,3,3}. It is the general of the 24-vertex*icositetrachoron army*and the colonel of the*ico regiment*.

- The 24 vertices of an icositetrachoron include the 16
vertices of a tesseract together with the eight vertices of a
regular hexadecachoron, oriented and sized so that each vertex is
on a line from the center of the tesseract through the center of
each cubic cell, at a distance from the center equal to the
circumradius of the tesseract. Let the vertex coordinates of
the tesseract be all 16 possible combinations of +1 and –1
in four positions. Then the coordinates of the other eight
vertices of the icositetrachoron will be the eight permutations
of +2 or –2 with three zeros. The edge length of this
icositetrachoron is 2, the same as the edge length and
circumradius of the tesseract (whose 32 edges are among the 96
edges of the icositetrachoron; it is one of the three tesseracts
of the regular compound of three tesseracts in the ico regiment).
The circumradius of the tesseract is the same as the circumradius
of the entire icositetrachoron, of course. Each of the 24 squares
of the tesseract is a diametral square of one of the 24
octahedral cells of the icositetrachoron. It is not difficult to
show that the 96 edges of an icositetrachoron form 16 regular
equatorial hexagons, four passing through each vertex. Each such
hexagon includes two opposite edges of the original tesseract.
*See also*Hexadecachoron; Ico regiment; Tesseract.

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- A countably infinite collection of facets of dimension
*k*>0 that otherwise satisfies all the criteria of the definition of a polytope. If all the facets lie in the same*k*-dimensional space, the polytope is an*apeirogon*(for*k*=1), a*tiling*(for*k*=2), or a*honeycomb*(for*k*>2), and it fills the entire space. Otherwise, it is embedded in a space of more than*k*dimensions, and it is an*infinite skew polytope*.

- <MORE TO COME>
*See also*Apeirogon; Honeycomb; Polytope; Regular polytope; Skew polytope; Tiling.

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- A polytope whose symmetry group is transitive on its
vertices, so that all its vertex figures are congruent; not to be
confused with an
*isogon*, which is an equilateral polygon whose symmetry group is transitive on its sides (and is thus the dual of an isogonal polygon). All uniform polytopes are isogonal by definition, but there also exist uncountably many nonuniform isogonal polytopes. For example, a regular octahedron can be “stretched” by moving two opposite faces apart, or “flattened” by moving two opposite faces closer together, and letting the other six faces stretch or flatten to keep the polyhedron closed. The resulting nonuniform triangular antiprisms have six irregular isosceles triangles as their lateral faces. But their vertices remain “all alike,” so they are isogonal. The opposite faces of any of these octahedra can additionally be “twisted” relative to each other, deforming the lateral faces into scalene triangles but still keeping the vertex figures identical and the polyhedra as a whole isogonal. The dual of an isogonal polytope is an isotope.

- All the vertices of an isogonal polytope must lie on the
same hypersphere, centered on the center of symmetry of the
polytope. Isogonal or
*equiangular*polygons may have sides of at most two different lengths. Therefore, if an equiangular polygon has an odd number of sides it must be regular. An equiangular polygon with an even number of sides is either regular or has alternating sides of two different lengths. An equiangular tetragon (polygon with four sides) is either a square, a rectangle, or one of three kinds of bowties. An equiangular hexagon is either regular, ditrigonary, or one of three kinds of star-hexagons.

- All regular polytopes are both isogonal and isotopic, and
in addition there exist polytopes in all spaces of dimension
greater than two that are not regular but are both isogonal and
isotopic. For example, the
*small*and*great prismosaurus*in four-space are uniform polychora with a single kind of cell, which is not a regular polyhedron (720 identical pentagrammatic antiprisms make up the small prismosaurus, 720 identical pentagonal antiprisms make up the great prismosaurus; the two prismosauri are conjugates). Their duals, whose 120 cells are two kinds of greatenings of a regular icosahedron with nine-sided stars (enneagrams) as faces, are likewise both isogonal and isotopic. Incidentally, the vertex figures of the prismosauri, as well as the cells of the prismosauri duals, are isogonal isohedra. Surprisingly, nobody has yet produced a provably complete enumeration of all the kinds of isogonal isotopes, even for three-space (where there are infinitely many, almost all having prismatic symmetries). Isogonal isohedra are noted by H. S. M. Coxeter in*Regular Polytopes*(pp. 116–117).*See also*Antiprism; Conjugate polytopes; Duality; Isotope; Transitivity; Uniform polytope; Vertex figure.

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- A transformation of any metric space that leaves the distance
between any two points of the space unchanged. The
*symmetries*of a polytope are isometries that permute the polytope’s elements. An isometry that preserves orientation is a*direct isometry*; an isometry that reverses orientation is an*opposite isometry*. Reflection is an opposite isometry; rotation is a direct isometry. Every isometry of*n*-space can be expressed as the composition of up to*n*+1 reflections. If the number of reflections is even, the isometry is direct, and if the number is odd, the isometry is opposite. For example, a rotation can always be expressed as the composition of two reflections in intersecting mirrors; the intersection of the two mirrors is the pivot of the rotation. Other kinds of isometries are translations, glide reflections, screw displacements, swirls, double rotations, central inversions, and half-turns.*See also*Axis; Glide reflection; Pivot; Reflection; Rotation; Symmetry; Translation.

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*See*Conjugate polytopes.

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- A homotope whose symmetry group is transitive on its facets.
Not to be confused with the term from nuclear physics. An isotope
in two-space is called an
*isogon*; an isotope in three-space is called an*isohedron*; an isotope in four-space is called an*isochoron*; and so forth. Naming conventions for isotopes are as for homotopes, except the prefix “iso“ is substituted for “homo.” In two-space, there is only one kind of isotrigon, the*equilateral triangle*, which is also the only kind of homotrigon; and every homotetragon and isotetragon is a*rhombus*or a*square*. The dual of an isotope is an isogonal polytope.*See also*Duality; Isogonal polytope; Homotope.

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- A small regiment of three uniform antiprism-like
star-polychora discovered by Norman Johnson in the early 1960s
and described in his 1966 doctoral dissertation. The
*small ditrigonary icosidodecahedral antiprism*(Jonathan Bowers’s name:*sidtidap*) has two small ditrigonary icosidodecahedra in parallel planes, aligned directly opposite each other with corresponding faces parallel. The lateral cells are [1] 40 regular tetrahedra defined by a vertex of either base and the triangle closest to that vertex in the other base, and [2] twelve pentagrammatic antiprisms that connect parallel pentagrams of the bases. The conjugate polychoron is the*great ditrigonary icosidodecahedral antiprism*(Jonathan Bowers’s name:*gidtidap*), in which the bases are two great ditrigonary icosidodecahedra that have the same vertices and edges as the small ditrigonary icosidodecahedra of the small ditrigonary icosidodecahedral antiprism. Instead of the [2] twelve pentagrammatic antiprisms, the lateral cells include [3] twelve pentagonal antiprisms that connect parallel opposite pentagons of the great ditrigonary icosidodecahedra (so the pentagonal antiprisms all pass through the center of the antiprism). Finally, these two polychora can be blended (because they belong to the same regiment) to produce the*ditrigonary dodecadodecahedral antiantiprism*(Jonathan Bowers’s name:*ditdidap*). Its cells are two base ditrigonary dodecadodecahedra connected by the [2] twelve pentagrammatic antiprisms of the small ditrigonary icosidodecahedral antiprism and the [3] twelve pentagonal antiprisms of the great ditrigonary icosidodecahedral antiprism. The [1] 40 tetrahedral cells are blended out.

- Interestingly, the central sections of these antiprisms
by realms parallel to and midway between the base realms are
themselves uniform polyhedra. The central section of the small
ditrigonary icosidodecahedral antiprism is an icosidodecahedron;
the central section of the great ditrigonary icosidodecahedral
antiprism is a small icosihemidodecahedron; and the central
section of the ditrigonary dodecadodecahedral antiantiprism is a
small dodecahemidodecahedron.

- Also in his dissertation, Johnson described two conjugate
antiprisms in five-space, analogues of the three antiprisms in
four-space. He did not name them, but they may be called the
*small altered-hecatonicosachoric antiprism*(Jonathan Bowers’s name:*sidtaxhiap*) and the*great altered-hecatonicosachoric antiprism*(Jonathan Bowers’s name:*gadtaxhiap*). The bases of the former are two altered hecatonicosachora, connected by 120 small ditrigonary icosidodecahedral antiprisms and 1200 regular pentachora; the bases of the latter are two altered great grand stellated hecatonicosachora, connected by 120 great ditrigonary icosidodecahedral antiprisms and 1200 regular pentachora. They belong to different 1200-vertex armies (their vertices are those of hecatonicosachoric prisms of two different altitudes) and thus cannot be blended to produce a third five-dimensional antiprism. An altered hecatonicosachoron is constructed by replacing the 120 dodecahedra of a hecatonicosachoron with small ditrigonary icosidodecahedra that have the same vertices, and taking up the free triangles with 600 interpenetrating tetrahedra; an altered great grand stellated hecatonicosachoron is constructed by likewise replacing the 120 great stellated dodecahedra of a great grand stellated hecatonicosachoron with highly interpenetrating great ditrigonary icosidodecahedra that have the same vertices, and taking up the free triangles with 600 interpenetrating tetrahedra.

- The vertex figures of the five-dimensional
Johnson antiprisms are four-dimensional cupolas, analogues of the
vertex figures of the four-dimensional Johnson antiprisms. The
vertex figure of the small altered-hecatonicosachoric
antiprism is a convex cupola whose small base is a regular
tetrahedron and whose large base is a larger regular tetrahedron,
truncated so that the stump triangles are the same size as the
faces of the tetrahedron and the hexagons are ditrigonary and
isogonal, precisely the vertex figures of small ditrigonary
icosidodecahedra. The hexagons adjoin one another along their
short edges and adjoin the stump triangles along their long
edges. This truncated tetrahedron is the vertex figure of the
altered hecatonicosachoron that is the base of the antiprism.
The truncated tetrahedron is aligned relative to the small-base
tetrahedron so that its four hexagons are parallel to the
tetrahedron’s four triangles. In this position, the two
bases are joined by lateral cells that are (1) four cupolas that
are the vertex figures of a small ditrigonary icosidodecahedral
antiprism, which adjoin one another along their trapezoidal
lateral faces, and (2) four tetrahedra, each defined by a stump
triangle and the corner of the small-base tetrahedron closest to
it. Altogether, the vertex figure has ten cells: five regular
tetrahedra, four cupolas, and one nonuniform truncated
tetrahedron.

- The vertex figure of the great
altered-hecatonicosachoric antiprism is a nonconvex cupola
that is not quite so easily described, although it also has ten
cells. The small base is again a regular tetrahedron, but the
large base is what might be described as a
“quasitruncated” tetrahedron: Extend the six edges of a
small regular tetrahedron a distance (
*sqrt*[5]+1)/2 of their length in both directions, to convert each triangle into the ditrigonary hexagram that is the vertex figure of a great ditrigonary icosidodecahedron. Cap off the free edges of the hexagrams with four unit triangles; the resulting star-polyhedron is the vertex figure of the altered great grand stellated hecatonicosachoron. Orient this base so that the four hexagrams are in planes parallel to the four face-planes of the small-base tetrahedron. Then join the hexagrams to the tetrahedral triangles by four cupolas that are the vertex figures of great ditrigonary icosidodecahedral antiprisms; these adjoin along their interpenetrating trapezoids. Add in four more tetrahedra to take up the free triangles, and the four-dimensional vertex figure is finished. Its ten cells are five tetrahedra (one base and four lateral), four star-cupolas, and the “quasitruncated” tetrahedron.*See also*Antiprism; Conjugate polytopes; Ditrigonary star-polychora; Mix-and-match notation.

Picture of Johnson antiprism vertex figures.

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- A new polytope created by adjoining two polytopes along a
common congruent facet and discarding the facet, so that
thereafter the free facets of adjoin the free facets of the
other along their shared ridges. Also, the process of creating
such a polytope. If the two joined polytopes happen to have other
common facets, these must also be discarded, and the new polytope
is a multi-faceted join known as a
*blend*. If as a result of a join two facets acquire a 180-degree dihedral angle (as, for example, in the join of a regular tetrahedron and regular octahedron), then their common ridge dissolves and the two facets join into a single facet (unless cohyperplanar facets are permitted in the context of the join).*See also*Blend.

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- The set of mirrors combined reflections in which generate all
the symmetries of a symmetry group. Three kinds of kaleidoscopes
are special: the
*complete*kaleidoscope, a*minimal*kaleidoscope, and the*fundamental*kaleidoscope. The complete kaleidoscope is the set of*all*the mirrors of the symmetry group, whereas a minimal kaleidoscope is the smallest set of mirrors that will generate all the symmetries of the symmetry group. We are most interested in the unique (up to orientation) fundamental kaleidoscope, which is the minimal kaleidoscope whose mirrors make the smallest acute dihedral angles with one another. The complete kaleidoscope is unique up to orientation of the mirrors, but there are often several minimal sets of mirrors that generate all the symmetries of the group. For example, a regular pentagon has ten symmetries. The complete kaleidoscope of the pentagon has five mirrors, all intersecting at the center of the pentagon and each passing through a vertex and the midpoint of the opposite side. A minimal kaleidoscope consists of any two of these mirrors, and the fundamental kaleidoscope consists of two mirrors that make an angle of 36° with each other. In general, a minimal kaleidoscope of a polytope in*n*-space will have at most*n*mirrors.

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- The one-dimensional content of a line segment. More
generally, the content of a curved line embedded in a metric
space.
*See also*Content of a figure.

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- The set of all points of any
*n*-dimensional projective space,*n*>0, that are collinear with two distinct points. The property of*collinearity*is undefined in Euclidean geometry, but in Euclidean*n*-dimensional space, where an arbitrary pointhas coordinates {*x**x*_{1},*x*_{2},*x*_{3},...,*x*_{n}}, a line is the set of points whose coordinates are expressed as

{*t*(*b*_{1}–*a*_{1})+*a*_{1},*t*(*b*_{2}–*a*_{2})+*a*_{2},*t*(*b*_{3}–*a*_{3})+*a*_{3},...,*t*(*b*_{n}–*a*_{n})+*a*_{n}}

where*t*ranges from –infinity to +infinity and

= {**a***a*_{1},*a*_{2},*a*_{3},...,*a*_{n}} and

= {**b***b*_{1},*b*_{2},*b*_{3},...,*b*_{n}}

are the two points that determine the line. (Simple, eh?) The single parameter*t*shows that a line is one-dimensional, because one needs only the value of*t*to locate a point on the line throughand**a**. In the above parametrization,**b***t*=0 corresponds to the pointand**a***t*=1 corresponds to the point.**b**

- The above linear equation is more compactly expressed as
a
vector expression:

*t*(–**b**)+**a****a**

where*t*is a scalar andand**a**are vectors extending from the origin to the points**b**and**a**. For any value of**b***t*the expression yields a vector that extends from the origin to a point somewhere on the line through the pointsand**a**.**b**

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- A one-dimensional element of a polytope; an interval of a
line bounded between two points, called its
*ends*or*end points*; the body of a dyad. All line segments are similar.*See also*Dyad; End [point].

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- Also known as a
*hypercube*and an*orthotope*. In Euclidean*n*-space, the analogue of the*line segment*in one-space, the*square*in two-space, and the*cube*in three-space. One of the three kinds of regular polytopes that exist in spaces of dimension greater than four. So called because Euclidean*n*-space can be honeycombed by identical unit measure polytopes, allowing us to measure the*n*-dimensional content of a figure in*n*-space; this honeycomb is called the*measure polytope honeycomb*. The*n*-dimensional measure polytope is the dual of the*n*-dimensional cross polytope. The vertex figure of an*n*-dimensional measure polytope of edge 1 is an (*n*–1)-dimensional simplex of edge*sqrt*(2).

- For
*n*>3, we use Greek numerical roots to name the measure polytopes:*tesseract*for*n*=4,*penteract*for*n*=5,*hexeract*for*n*=6,*hepteract*for*n*=7,*octeract*for*n*=8,*eneneract*for*n*=9,*dekeract*for*n*=10,*hendekeract*for*n*=11, and so on. The dihedral angle of an*n*-dimensional measure polytope is always a right angle for*n*>1. The facets of an*n*-dimensional measure polytope are 2*n*(*n*–1)-dimensional measure polytopes, and an*n*-dimensional measure polytope has 2^{n}vertices (corners).

- The coordinates of the vertices of an
*n*-dimensional measure polytope of edge length 2 centered on the origin, whose elements are all parallel to the various coordinate hyperplanes, are quite simple: all 2^{n}possible sign-change permutations of a string of*n*+1’s or –1’s. For this reason, the number of*k*-dimensional elements of an*n*-dimensional measure polytope is given by the coefficient of*a*^{k}in the binomial expansion of (*a*+2)^{n}. For example, the binomial expansion of (*a*+2)^{7}is*a*^{7}+ 14*a*^{6}+ 84*a*^{5}+ 280*a*^{4}+ 560*a*^{3}+ 672*a*^{2}+ 448*a*+ 128.

From this, a seven-dimensional measure polytope, or hepteract, has

one hepton (the hepteract itself),

14 hexons (hexeracts),

84 pentons (penteracts),

280 tetrons (tesseracts),

560 cells (cubes),

672 faces (squares),

448 edges, and

128 vertices.

There is also one nullitope (element of dimension –1), which is left out of the binomial expansion.

- From the above, it is also easy to see that the length of
the long diagonal of a unit
*n*-dimensional measure polytope is*sqrt*(*n*).*See also*Cross polytope; Duality; Regular polytope; Simplex; Vertex figure.

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- A space in which the distance between any two points can
always be assigned a unique value. Euclidean, spherical,
elliptic, and hyperbolic spaces are metric; but projective space,
for which distances are not defined, is not.

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- The set of points of
*n*-space that do not change under a reflection,*n*>0. For*n*=1, a mirror is a point; for*n*=2, a mirror is a line; for*n*=3, a mirror is a plane; for*n*=4, a mirror is a realm; and so on. In general a mirror is an (*n*–1)-flat, or hyperplane, in*n*-space.*See also*Reflection; Rotation .

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- A method of denoting uniform polytopes by army and regiment.
A mix-and-match notation is a more or less arbitrary string of
numbers separated by slashes and brackets that uniquely specifies
a uniform polytope. The small regiment of three Johnson antiprisms provides a
simple example.

- All three Johnson antiprisms have the same set of 40
vertices and 180 edges; nine edges meet at each vertex. The
vertex figure of the regiment’s colonel is a kind of cupola
with an equilateral triangle as one base and a convex ditrigonary
hexagon as the other base. The edge of the triangle has length 1,
and the edges of the ditrigonary hexagon are of length 1 and
1/
*tau*alternating. (*Tau*is the Golden Ratio, [*sqrt*{5}+1]/2. The value of 1/*tau*is [*sqrt*{5}–1]/2, less than 1) The triangle and hexagon are parallel, and the triangle is aligned so that its edges are parallel to the short edges of the hexagon. Joining the triangle to the hexagon are three more equilateral triangles and three trapezoids. The equilateral triangles are determined by the long edges of the base hexagon and the corresponding vertices of the base triangle; the trapezoids are determined by the short edges of the base hexagon and the corresponding edges of the base triangle.

- In the vertex figure, the triangles are the vertex
figures of the four tetrahedra that meet at each vertex, the
trapezoids are the vertex figures of the three pentagrammatic
antiprisms that meet at each vertex, and the base ditrigonary
hexagon is the vertex figure of one of the two base small
ditrigonary icosidodecahedra of the Johnson antiprism. This
particular vertex figure is that of the small ditrigonary
icosidodecahedral antiprism, which is the colonel of the
regiment.

- Inscribable within the colonel’s vertex figure are
two other polyhedra, which are the vertex figures of the other
two Johnson antiprisms. Replacing the base hexagon with the
hexagram that has the same vertices but the unit edges and long
diagonals (which have length
*tau*) as its edges gives the vertex figure of the great ditrigonary icosidodecahedral antiprism. Since the short edges of the hexagon are absent, so are the three trapezoids that represent the pentagrammatic antiprisms. Instead, three trapezoids whose edges are 1, 1, 1, and*tau*take up the free edges and close the vertex figure. These correspond to the three pentagonal antiprisms that meet at each vertex.

- Replacing the base hexagon with the “propeller
hexagram” that has the same vertices but the short edges and
long diagonals as its edges gives the vertex figure of the
ditrigonary dodecadodecahedral antiantiprism. Since the unit
edges of the hexagon are absent, so are the triangles that join
them to the vertices of the base triangle. The base triangle
itself is also absent, and all the free edges are taken up by the
two kinds of trapezoids.

- For the mix-and-match notation for this regiment, the six
kinds of cells are given the numbers 1–6 arbitrarily by
convenience as follows:

[1] The 40 tetrahedra

[2] The 12 pentagrammatic antiprisms

[3] The 12 pentagonal antiprisms

[4] The 2 small ditrigonary icosidodecahedra

[5] The 2 great ditrigonary icosidodecahedra

[6] The 2 ditrigonary dodecadodecahedra

- There are 40 vertices, the vertices of a dodecahedral prism
of a particular height (this is the case polychoron of the
regiment). This set of vertices is arbitrarily designated 40/J/1
(J for Johnson, 1 because it is the first [and only] regiment of
the 40/J army). The mix-and-match notations for the three
polychora then become:

40/J/1[1][2][4]

40/J/1[1][3][5]

40/J/1[2][3][6]

- This is a simple example, with a regiment of only three
polychora, of how to mix and match cell groups to yield closed
polychora. The way any two of the figures will blend to give the
third is very apparent (the blend is incomplete, since [4] and
[5] do not cancel but blend to give [6]). Mix-and-match notation
is very handy for quickly finding and designating polychora in
the larger regiments, which might have hundreds or even thousands
of members. Also, the string of numbers can specify a polychoron
uniquely to a computer program that draws projections, sections,
or vertex figures.
*See also*Army; Blend; Regiment; Uniform polytope.

Picture of Johnson antiprism vertex figures.

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- A zero-dimensional polytope. Its elements comprise the empty
set and a single point.
*See also*Point.

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- The simplest tiling or tessellation of a circle, consisting
of a single vertex and a single edge, which is the periphery of
the circle starting and ending at the vertex. The Schläfli
symbol of a monogon is {1}; it is trivially a regular
tessellation. The body of a monogon is the interior of its
circle. Although a monogon cannot be used as an element of a
Euclidean polytope, it can be used as a cellet in rudimentary
*n*-dimensional hyperspherical tessellations. For example, the*monogonal dihedron*, whose Schläfli symbol is {1,2}, is the regular tiling of the sphere by two hemispheres that share a common equatorial monogon. Its dual {2,1} is the*digonal monohedron*, a regular tiling of the sphere that has two vertices (at the poles; hence the adjective digonal) joined by a single edge. Its single face (hence the name monohedron) is the entire surface of the sphere. Having but a single face, the digonal monohedron is the simplest hosohedron. There is also an even simpler*monogonal monohedron*{1,1}, which is the simplest regular tessellation of a sphere. It consists of a single vertex (so it is monogonal), no edges, and a single face (so it is monohedral), the entire surface of the sphere except for the vertex. It is self-dual.*See also*Body; Polygon; Tessellation; Tiling.

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- A space with
*n*dimensions,*n*>–2.*n*-space can be projective, affine, Euclidean, spherical, elliptical, or hyperbolic, for example, depending on the axioms and postulates that the space obeys. If*n*=–1, the space is the empty space; if*n*=0, the space is a single point or a countable collection of points. The letter*n*is customarily replaced by a number, perhaps spelled out, when discussing a space with a particular number of dimensions: 1-space or one-space, 5-space or five-space, and so on.

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- The polytope whose only element is the empty set. By
convention, the nullitope has –1 dimensions. The nullitope
is an element of every polytope, just as, dually, every polytope
has one
*n*-dimensional element, namely, its body.*See also*Body; Empty set, the; Polytope.

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- Perpendicularity.

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*See*Cross polytope.

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*See*Measure polytope.

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- An (
*n*–3)-dimensional element of an*n*-dimensional polytope. Three or more facets of the polytope adjoin along every peak and form a cycle around it.

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- The simplest regular polychoron in four-space, with five
vertices, ten edges, ten triangular faces, and five tetrahedral
cells. Its symmetry group is the dyadic pentachoric group, of
order 120, which is the permutation group of five elements. It
was discovered by Ludwig Schläfli in the mid-1850s; its
Schläfli symbol is {3,3,3}. The regular pentachoron has no
facetings, so the pentachoron army has only one member, the
pentachoron itself, which is also the general of the army and
the colonel of its regiment of one. Small wonder that it is also
called a
*regular four-dimensional simplex*.

- Among the symmetric projections of a regular pentachoron
into three-space are projection into a regular tetrahedron
with the fifth vertex at the center, joined to the other four
vertices by edges; and into an equilateral-triangular pyramid
whose equatorial triangle has the same edge-length as the
distsance between the two apices. In two-space, the regular
pentachoron projects into a regular pentagon with all its
diagonals, that is, a complete pentagon. This provides a nifty
set of coordinates for its five vertices:

(1,0,1,0);

(*cos*[^{2pi}/_{5}],*sin*[^{2pi}/_{5}],*cos*[^{4pi}/_{5}],*sin*[^{4pi}/_{5}]);

(*cos*[^{4pi}/_{5}],*sin*[^{4pi}/_{5}],*cos*[^{8pi}/_{5}],*sin*[^{8pi}/_{5}]);

(*cos*[^{6pi}/_{5}],*sin*[^{6pi}/_{5}],*cos*[^{2pi}/_{5}],*sin*[^{2pi}/_{5}]);

(*cos*[^{8pi}/_{5}],*sin*[^{8pi}/_{5}],*cos*[^{6pi}/_{5}],*sin*[^{6pi}/_{5}]).

- This pentachoron is inscribed in a glome of radius
*sqrt*(2) centered on the origin, and it has an edge of length*sqrt*(5).

- To construct a general pentachoron, begin with a
tetrahedron of arbitrary shape in three-space. Like all
tetrahedra, it has four vertices, six edges, and four triangular
faces. Then choose any point in four-space that is not corealmic
with the tetrahedron. Connect this point to each vertex of the
tetrahedron with an edge, to each edge of the tetrahedron with a
triangle, and to each face of the tetrahedron with another
tetrahedron. This adds four edges, six triangular faces, and four
more tetrahedra to the count of elements, giving five vertices
(4+1), ten edges (6+4), ten triangular faces (4+6), and five
cells (1+4) altogether. This construction displays a general
pentachoron as the simplest kind of pyramid, namely, a
tetrahedral pyramid, in four-space.

- If, in this construction, one begins with a regular
tetrahedron and positions the fifth vertex on a line
perpendicular to the realm of the tetrahedron and passing through
its center, the distances from this vertex to any of the four
vertices of the tetrahedron will all be equal. One may then slide
the fifth vertex along this line until this common distance
equals the length of the edge of the tetrahedron (there are two
such locations, on either side of the realm of the tetrahedron).
The resulting pentachoron will then be regular.

- The preceding construction leads to a different set of
easy Cartesian coordinates for the vertices of a regular
pentachoron. The even corners of a cube centered at the origin
with faces parallel to the coordinate axes, namely, (1,1,1),
(1,–1,–1), (–1,1,–1), and
(–1,–1,1), define a regular tetrahedron with an edge of
length 2
*sqrt*(2), or*sqrt*(8). Assign a fourth coordinate of zero to each corner of this tetrahedron. Then it is practically trivial to find*w*for which the point (0,0,0,*w*) is distant*sqrt*(8) from any of the four corners. It satisfies the following equation:

*w*^{2}+1+1+1 = 8.

That is,*w*=*sqrt*(5) or –*sqrt*(5). This produces what are probably the typographically simplest vertex coordinates for a regular pentachoron:

(1,1,1,0);

(1,–1,–1,0);

(–1,1,–1,0);

(–1,–1,1,0); and

(0,0,0,*sqrt*[5]).

Of course, this pentachoron is not nicely centered at the origin. To center it at the origin, move the whole pentachoron down the*w*-axis by a distance^{1}/_{5}*sqrt*(5), giving the typographically more intricate coordinates

(1,1,1,–^{1}/_{5}*sqrt*[5]);

(1,–1,–1,–^{1}/_{5}*sqrt*[5]);

(–1,1,–1,–^{1}/_{5}*sqrt*[5]);

(–1,–1,1,–^{1}/_{5}*sqrt*[5]); and

(0,0,0,^{4}/_{5}*sqrt*[5]).

*See also*Realm; Simplex.

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- The surtope of a polygon.
*See also*Surtope.

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- The set of points of Euclidean
*n*-space,*n*>1, that do not change under a rotation. For*n*=2, a pivot is a point called the*center*of the rotation; for*n*=3, a pivot is a line (and is called an*axis*); for*n*=4, a pivot is a plane; and so on. (Note that for*n*=1, a pivot would be the empty space; that is, there is no rotation in a one-space.) In general a pivot is an (*n*–2)-flat in*n*-space. Rotations also exist in non-Euclidean spaces, of course, but their pivots are not necessarily lines, planes, and flats.*See also*Rotation.

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- Given a line
and a point*L*not collinear with**p**, a plane is the set of points of*L**n*-space,*n*>1, collinear withand any point of**p**, together with any other points collinear with these. Points and lines that all lie in the same plane are*L**coplanar*. Since a line is determined by any two distinct points, it is clear that a plane is determined by any three noncollinear points. A plane is a 2-flat.*See also*Flat (*k*-flat).

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- The five convex regular polyhedra, described by the
philosopher Plato of Athens (
*c.*428–*c.*348 BCE), well known to the mathematicians and geometers of ancient Greece, and probably known to mathematicians of earlier civilizations. These include the*regular tetrahedron*, which has four equilateral-triangular faces; the*cube*, which has six square faces; the*regular octahedron*, which has eight equilateral-triangular faces; the*regular dodecahedron*, which has twelve regular-pentagonal faces; and the*regular icosahedron*, which has 20 equilateral-triangular faces. For an illustrated account of these figures, visit my Regular Polyhedra website.*See also*Regular polytope.

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- In Euclidean geometry, an undefined object with zero
dimensions; a zero-space; the body of a monad; a vertex of a
polytope. In coordinate geometry of
*n*-space, an ordered*n*-tuple {*x*_{1},*x*_{2},*x*_{3},...,*x*_{n}}. All geometric objects are ultimately collections of points, usually uncountably many.*See also*Body; Monad; Vertex.

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- A four-dimensional polytope, comprising the empty set, a
finite number of five or more points (its vertices or
*corners*), a finite number of ten or more line segments (its*edges*), a finite number of ten or more polygon bodies (its*faces*), a finite number of five or more polyhedron bodies (its*cells*), and an interior (its*body*) that is bounded by the vertices, edges, faces, and cells. Each face is shared by exactly two cells. Customarily, the faces cannot coincide, and the angle between two cells that share a common face is neither 0° nor 180°. I originally called these figures*polychoremata*(singular:*polychorema*), but Norman Johnson came up with the shorter term polychoron.*See also*Cell; Edge; Face; Polygon; Polyhedron; Polytope; Vertex.

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- A two-dimensional polytope, comprising the empty set, a
finite number of three or more points (its vertices or
*corners*), an equal number of line segments (its edges or*sides*), and an interior (its*body*) that is bounded by the vertices and sides. Each vertex is the end of exactly two sides. This compels the line segments form a*cycle*, that is, a path that runs along the edges and returns to its starting point. The cycle is the polygon’s*perimeter*. Customarily, the vertices cannot coincide, and the angle between two sides that share a common vertex is neither 0° nor 180°.*See also*Edge; Polytope; Side; Vertex.

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- A three-dimensional polytope, comprising the empty set, a
finite number of four or more points (its vertices or
*corners*), a finite number of six or more line segments (its edges), a finite number of four or more polygon bodies (its faces), and an interior (its*body*) that is bounded by the vertices, edges, and faces. Each edge is shared by exactly two faces. Customarily, the edges cannot coincide, and the angle between two faces that share a common edge is neither 0° nor 180°.*See also*Edge; Face; Polygon; Polytope; Vertex.

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- The general
*n*-dimensional figure in the sequence*nullitope*(*n*=–1),*monad*(*n*=0),*dyad*(*n*=1),*polygon*(*n*=2),*polyhedron*(*n*=3),*polychoron*(*n*=4),*polytetron*(*n*=5),*polypenton*(*n*=6), etc. Defining a polytope is not quite as easy as it at first might seem. In doing so, we are motivated by the idea that a polytope in (*n*+1)-space is a figure assembled by cutting a finite number of*n*-dimensional polytopes out of*n*-dimensional paper and gluing them together along their (*n*–1)-dimensional facets so as to bound a region of (*n*+1)-space, with no free facets left over. We want exactly two*n*-dimensional polytopes (which are flat in [*n*+1]-space) to contact along a common facet, and we want the contact to be*snug*, that is, there must be no overhanging parts at the join: We call this the*polytope condition*. The polytope condition mandates that the facets along which two*n*-dimensional polytopes will adjoin be congruent. We also want adjoining*n*-dimensional polytopes to lie in different hyperplanes, so that the dihedral angle between them is neither 0° nor 180°. Finally, we want no subset of the*n*-dimensional polytope to form an (*n*+1)-dimensional polytope, for then the figure would be a*compound*polytope.

- Different interpretations of these terms lead to (at
least) two distinct but ultimately equivalent definitions of a
polytope. In the first of these, the
*recursive*definition of a polyope in Euclidean*n*-space, the elements of an*n*-dimensional polytope are (*n*–1)-dimensional polytopes positioned in*n*-space so that their elements, which are (*n*–2)-dimensional polytopes, are each shared by exactly two of them. The*n*-dimensional polytope is then the union of its elements and the finite space that the elements bound. This definition requires no less than six parts:

[1] A polytope in*n*=–1 dimensions is the*nullitope*. It has no elements.

[2] A polytope in*n*=0 dimensions is a*monad*, whose only element is the nullitope. In locally Euclidean spaces, a monad is a point. These two kinds of polytopes, [1] and [2], anchor the recursion.

[3] A polytope in*n*>0 dimensions is a finite collection of (*n*–1)-dimensional polytopes (called*facets*) imbedded in*n*-space and sharing coincident (*n*–2)-dimensional elements (called*ridges*) so that every ridge is an element of exactly two of the facets. (The [*n*–3]-dimensional elements, if they exist, are called*peaks*of the*n*-dimensional polytope.)

[4] In addition, we insist that no two ridges coincide, which would make it ambiguous which pairs of facets incident at such a ridge adjoin, and that

[5] the dihedral angle between two facets at a ridge, if it can be measured, be neither 0° nor 180°. Finally, we require that

[6] no subset of the elements form an*n*-dimensional polytope according to the terms specified in [1] through [5].

- Note that this definition provides for exactly two monads
in
a dyad, because that is how many vertices of a 1-dimensional
polytope are permitted to share a common empty set.

- The
*set-theoretical*definition of a polytope avoids recursion by defining a polytope as a partially ordered set of shared elements called*j*-faces, where*j*, –2<*j*<*n*+1 is the dimension of the*j*-face. The partial ordering is on the*j*prefix, which is an integer that ranges from –1 to*n*inclusive. The*j*-faces share elements with one another in the following very specific way. There is just one (–1)-face, and it is the empty set, and there is just one*n*-face, and it is either a point (if*n*=0) or the interior of the polytope itself. For*n*>0 and –1<*j*<n, every (*j*–1)-face of a*j*-face that belongs to a (*j*+1)-face belongs to exactly one other*j*-face of that (*j*+1)-face.

- For
*n*<5, polytope names are non-numeric, but for*n*>4, we use numeric Greek roots for their names:*polytetron*,*polypenton*,*polyhexon*,*polyhepton*,*polyocton*,*polyenneon*,*polydecon*, and so on. Plurals of these are formed by changing the suffix -on to -a.

- <MORE TO COME>

- It is essential to realize that the vast majority of
polytopes are far too intricate to visualize or to examine, both
because of the sheer number of their elements and because of the
number of their dimensions. Try, for example, to comprehend how
many corners a 112,787-dimensional measure polytope has, or how
many
symmetries! And infinitely many measure polytopes are
*even more*complicated. Fortunately, it also seems likely that such figures seem to be fundamentally uninteresting: merely enormous piles of blended polytopes, polytopes derived from other polytopes by various well-understood transformations, or polytopes belonging to infinitely large transdimensional polytope families. The most interesting polytopes seem to be the simpler symmetric figures, such as the uniform polytopes, regular polytopes, and stellations and facetings of these, in spaces of dimension less than about ten.

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- Polytopes expressible as Cartesian products of
lower-dimensional polytopes (of dimension 1 or greater). When all
the lower-dimensional polytopes are uniform and have the same
edge length, the resulting Cartesian product is a uniform
prismatic polytope. The Cartesian product of any polytope and a
dyad is called a
*prism*; the polytope is the*base*of the prism, and the facets joining the top and bottom bases are the*lateral*facets. The Cartesian product of two regular polygons of the same edge length is a uniform*duoprism*or*double prism*. A*hyperprism*is a prism of more than three dimensions. A very general kind of convex prism may be formed in*n*-space by constructing the convex hull of two (*n*–1)-dimensional polytopes in parallel hyperplanes.

- In three dimensions, it is convenient to give some of the
symmetric prisms special names. A symmetric prism is one that has
a symmetric polygon for both bases. A prism all of whose lateral
faces are rectangles or squares is an
*orthoprism*, and it is usually the kind of prism that is meant by the term “prism” for a polyhedron. A prism all of whose lateral faces are triangles is an*antiprism*. This may be a very general kind of polyhedron, but the term is often restricted to the figure formed by two congruent, antialigned, parallel regular polygons connected by congruent isosceles triangles. If the vertex figures of an antiprism are crossed trapezoids, it becomes a*retroprism*. A prism whose bases are two isogonal even-sided polygons with unequal sides, situated so that the long sides of either are parallel to the short sides of the other, may have congruent trapezoids or neckties for its lateral faces. Such a prism is either a*loxoprism*(if the faces are trapezoids;*loxo*comes from Greek for “slanted”) or a*retroloxoprism*(if the faces are neckties). If a prism and an antiprism are based on identical polygons oriented the same way, and they have the same height, they can be blended into a*toroprism*: a prism that has only the two kinds of lateral faces, and holes (*toro*comes from Greek for “hole”) where the bases were. The holes may sometimes penetrate the prism from base to base. Many kinds of symmetric prisms and antiprisms may be obtained by symmetrically faceting an orthoprism; all 25 different prisms, antiprisms, loxoprisms, and other combinations appear in this diagram.*See also*Antiprism; Blend; Cartesian product; Duoprism; Faceting; Mix-and-match notation.

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- A polytope in
*n*dimensions formed from a polytope in*n*–1 dimensions in the following manner: Let the (*n*–1)-dimensional polytope, to become the*base*of the pyramid, be embedded in a hyperplane of*n*-space. Choose a point, to become the*apex*(plural*apices*) of the pyramid, in*n*-space not in that hyperplane. Join the apex to each vertex of the base by an edge, to each edge of the base by a triangle, to each face of the base by a three-dimensional pyramid having that face as its base and the apex as its apex, and so on. The facets thus created, themselves all (*n*–1)-dimensional pyramids, are called the*lateral*facets, and the resulting*n*-dimensional polytope itself is the pyramid.

- When the base is a simplex, the pyramid is also a
simplex, of
one greater dimension. When the base is a pyramid, the polytope
is a
*double pyramid*with two apices—the new apex and the apex of the base pyramid—and an*apical edge*that connects them. The base of a double pyramid is the (*n*–2)-dimensional polytope that is the base of the original pyramid. All pyramids and double pyramids are triangles in 2-space, and all double pyramids are tetrahedra in 3-space; but double pyramids are non-trivial in 4-space and higher-dimensional spaces. A triangular double pyramid in 4-space is a pentachoron.

- The most general pyramid in
*n*-space has a*j*-dimensional apical polytope and a*k*-dimensional base polytope, where*j*+*k*=*n*–1 and*j*is less than or equal to*k*. The*j*-flat of the apical polytope and the*k*-flat of the base polytope should be positioned so that they are not parallel and do not intersect, that is, so that they are*skew*. The elements of the general pyramid are constructed by joining each vertex of the apical polytope to each element of the base polytope, and*vice versa*. The first kind of pyramid described above is the general case with*j*=0 and*k*=*n*–1; the double pyramid is the only possible general case with*j*=1 and*k*=*n*–2. Jonathan Bowers’s name for a general pyramid is a*duopyramid*, this term being modified by the names of the apical and base polytopes, the apical polytope being a polygon or a polytope of higher dimension. For example, a duopyramid in 5-space whose apex is a triangle and whose base is a square is a*triangular-square duopyramid*.

- As with pyramids in 3-space, such as right pyramids based
on
regular polygons, the most interesting duopyramids are those that
possess some symmetry. For example, in 5-space the two base
polygons can both be regular and situated perpendicular to and
centered on the ends of the shortest line segment that connects
their respective skew planes. The resulting
*right doublepolygonal duopyramid*is isogonal, no matter what regular polygon serves as apex and base. If the apex and base are both triangles, the resulting doubletriangular duopyramid is a 5-dimensional simplex, or*hexatetron*. More generally, if the apex and base are*j*-dimensional and*k*-dimensional simplexes, respectively, then the resulting duopyramid will be a (*j*+*k*+1)-dimensional simplex. In particular, if the apex and base are both identical equilateral triangles, the distance between them can be adjusted so that all 15 edges have the same length, and the resulting hexatetron is regular. In this manner we can always construct a regular (*j*+*k*+1)-dimensional simplex as a duopyramid with a regular*j*-dimensional simplex as apex and a regular*k*-dimensional simplex as base.*See also*Bipyramid; Duopyramid.

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*See under*Truncation.

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- Given a plane
and a point*P*not coplanar with**p**, a realm is the set of points of*P**n*-dimensional space,*n*>3, collinear withand any point of**p**, together with any other points collinear with these. Points, lines, planes, and other figures that all lie in the same realm are*P**corealmic*. Since a plane is determined by any three noncollinear points, it is clear that a realm is determined by any four noncoplanar points. A realm is a 3-flat.*See also*Flat (*k*-flat).

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- The operation of creating a new polytope
r
from another polytope*P*by using the midpoints of the edges of*P*as the vertices of r*P*. The facets of r*P*then become (1) all the facets of*P*rectified, and (2) all the vertex figures of*P*. Rectification, which is a special kind of truncation, works when the dimension of*P*is two or more. When it is two,*P*is a polygon of*P**n*sides, and ris, trivially, a polygon of*P**n*vertices. (If polygonhas edges that intersect each other at their midpoints, r*P*will have its corresponding vertices coincident, which cannot be permitted under some definitions of a polygon.) Rectification becomes nontrivial for polytopes of dimension three or more.*P*

- Practically any polytope can be rectified to yield a new
polytope, but the most interesting polytopes to rectify are the
regular polytopes and certain kinds of uniform polytopes.
Rectification of any regular polytope
always yields a uniform polytope r*R*(when the dimension of*R*is two, r*R*is regular, and is merely a smaller version of*R*: uniform polygons must be regular by definition). The rectification r*R*of a uniform polytope*U*will be uniform if the facets of*U*are regular and its vertex figure is uniform.*U* - If
is a regular polyhedron, its rectification r*R*is either a regular polyhedron (in the case of the regular tetrahedron, which rectifies into an octahedron) or a quasi-regular polyhedron (in all other cases). Specifically,*R*

r[tetrahedron] = octahedron

r[cube] = r[octahedron] = cuboctahedron

r[dodecahedron] = r[icosahedron] = icosidodecahedron

r[small stellated dodecahedron] = r[great dodecahedron] = dodecadodecahedron

r[great stellated dodecahedron] = r[great icosahedron] = great icosidodecahedron.

In four-space, the 16 regular polychora yield 16 different uniform rectified polychora, some of which (those whose vertex figures are Archimedean prisms) may be rectified a second time to yield further uniform polychora.

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- An opposite isometry of
*n*-space that leaves the points of a hyperplane invariant. The hyperplane is called the*mirror*of the reflection. A reflection is the simplest nontrivial isometry of*n*-space; two successive reflections in the same mirror leave the space unchanged, so they are equal to the*identity*isometry. A reflection and the identity form the simplest possible symmetry group, the*bilateral symmetry group*of order 2.

- All other isometries may be expressed as combinations of
reflections in various mirrors. For example, a
*rotation*is the combination of successive reflections in two intersecting mirrors; a*translation*is the combination of reflections in two parallel mirrors; a*screw displacement*is the combination of reflections in four mirrors, two intersecting and two parallel, both of the latter perpendicular to the former; and so on. A reflection reverses sense in orientable spaces (it changes an object into its “mirror image”), which is why it is an*opposite*isometry.*See also*Isometry; Rotation; Translation.

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- The set of all
*n*-dimensional uniform polytopes (and, if need be, uniform compound polytopes) that share the same set of edges; a 1-regiment. When prefixed with a dimension number*k*, –1<*k*<*n*, a*k*-regiment is the set of all uniform polytopes (and, if need be, uniform compound polytopes) that share the same set of elements of dimension(and, of course, fewer). Special names for particular values of *k*are*army*for a 0-regiment and a*company*for a 2-regiment. Even in four-space, companies usually have only two or three members. Enumerating and characterizing the regiments of four-space is perhaps the most interesting part of the search for uniform polychora.

- The smallest polytope containing all the members of a
regiment is called its
*colonel*.*See also*Mix-and-match notation.

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- In most concise terms, any polytope
whose symmetry group is transitive on its flags. That is, given any two flags of*R*, there is a symmetry of*R*that carries one flag onto the other. A*R**flag*is a collection of elements ofthat comprises one vertex, one edge incident with that vertex, one face incident with that edge, one cell incident with that face, and so on, stopping at one facet incident at he one ridge in the flag. This marvelous definition encompasses not only finite Euclidean polytopes but also infinite polytopes, tilings and honeycombs, and skew polytopes, in Euclidean, elliptic, spherical, projective, and hyperbolic spaces.*R*

- In zero-space, a regular polytope is the monad. If
*n*>0, a regular polytope in*n*-space is a polytope whose symmetry group is transitive on all of its*k*-dimensional elements, –1<*k*<*n*,. So all dyads are regular, because every dyad is symmetric by reflection in its midpoint, and this operation carries either end point into the other end point (these are the only nontrivial elements of a line segment). But in spaces of dimension greater than one, regularity becomes a surprisingly restrictive property. Of all the kinds of polygons in two-space, for example, only those that are equilateral and equiangular are regular. We do not exclude star-polygons from being regular, but we usually exclude regular*Riemannian polygons*, which are polygons with coincident vertices and sides that surround their centers more than once.

- The above definition is equivalent to the property that
the facets of any regular polytope are themselves regular
polytopes, arranged the same way around every vertex.

- Of the infinite number of kinds of polyhedra in
three-space, only nine are regular (five convex and four
nonconvex; see my Web page Regular
Polyhedra). The five convex regular polyhedra are also called
the
*Platonic*solids, since they were described by the Greek philosopher Plato; the four nonconvex regular polyhedra are also called the*Kepler-Poinsot*polyhedra, since they were discovered by Johannes Kepler and Louis Poinsot. There are 16 regular polychora (six convex and ten star-polychora). These were discovered by Ludwig Schläfli and Edmund Hess, and following tradition they can be called the. In*n*-space for all*n*>4 there are only three regular polytopes (all convex); these were discovered by Ludwig Schläfli in the mid-1850s, and following the same tradition they can be called the*Schläflian polytopes*.*See also*Hecatonicosachoron, regular; Hexacosichoron, regular; Hexadecachoron, regular; Icositetrachoron, regular; Isogonal polytope; Isotope; Line segment; Monad; Pentachoron, regular; Platonic solids; Polygon; Polyhedron; Polytope; Riemannian polygon; Skew polytope; Tesseract; Uniform polytope.

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- A contraction of the term
*retrograde antiprism:*A Colohimian antiprism whose vertex figure is a crossed trapezoid, or*necktie*. Any Colohimian antiprism has a regular star-polygon {*p*/*d*} for its base and congruent equilateral triangles as its lateral faces. If*p*/*d*< 3, then two kinds of Colohimian antiprisms are possible, the usual one, whose vertex figure is a trapezoid, and the retrograde one, whose vertex figure is a necktie. The latter figure is the Colohimian (*p*/*d*)-grammatic retroprism.*See also*Antiprism.

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- Uniform polytopes whose vertex figures are topologically
similar, so they can be faceted in pretty much the same way.
Typically, such vertex figures are various kinds of prisms and
antiprisms, or figures with tetrahedral symmetry.
*See also*Conjugate polytopes.

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- An (
*n*–2)-dimensional element of an*n*-dimensional polytope. Two facets of the polytope adjoin along every ridge.

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- An exotic polygon that consists of multiple coincident copies
of a single polygon so arranged that after completion of a cycle
of sides of one copy, the path around the polygon jumps to the
next copy and continues in this manner through each copy in turn,
concluding by jumping to the initial vertex of the first copy.
For example, label the vertices of a regular pentagon with the
ten numbers {1, 2, ..., 10} in order around the pentagon, so that
each vertex gets the numbers
*j*and*j*+5. Consider the ten-sided polygon whose vertices are {1, 2, ..., 10} in order, cycling twice around the pentagon. This is a*regular Riemannian decagon of order 2*. It is equiangular and equilateral, and thus it is formally a regular polygon, although most definitions of a regular polygon exclude such figures because they are not “true” polygons. Polytopes of dimension greater than two whose faces or vertex figures include Riemannian polygons are*Riemannian polytopes*. The surtope of a Riemannian polytope is its underlying polytope. A regular Riemannian decagon is the exactly truncation of an ordinary pentagram, in which the five stump sides alternate with the remnants of the pentagon sides. Many other regular Riemannian polygons are also truncated regular star-polygons.*See also*Exotic polytopes; Regular polytope.

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- A direct isometry of
*n*-space,*n*>1, that leaves the points of an (*n*–2)-flat invariant. The (*n*–2)-flat is called the*pivot*of the rotation. In one-space there is no rotation because the pivot is the empty space; in two-space rotation takes place around a point called the*center*of the rotation; in three-space rotation takes place around a line called the*axis*of the rotation; in four-space rotation takes place around a plane called the*axis plane*of the rotation; and so on. Every rotation is the composition of successive reflections in two intersecting mirrors, the*angle*of the rotation being twice the dihedral angle between the mirrors and the pivot of the rotation being the (*n*–2)-flat that is the intersection of the mirrors. Note that the order in which the reflections are executed matters; if the order is reversed, the resulting rotation reverses its orientation (from clockwise to counterclockwise or*vice versa*).*See also*Dihedral angle; Mirror; Reflection.

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- A facet of a polygon.
*See also*Facet.

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- The simplest possible polytope in
*n*-space, the analogue of a triangle in the plane and a tetrahedron in three-space. A simplex in*n*-space is regular if and only if all its edges are the same length. All the lower-dimensional elements of a simplex are themselves simplexes. If*Choose*(*k*,*n*), with integers*k*and*n*such that –1<*k*<*n*+1, is the choice function, that is, the number of different ways to choose*k*distinct items from a set of*n*distinct items, then an*n*-dimensional simplex has*Choose*(*k*+1,*n*+1)*k*-dimensional elements. (Note that by convention there is just one way to choose zero items from any set of items.)

- In other words, the number of
*k*-dimensional elements of an*n*-dimensional simplex is given by the coefficient of*a*^{k+1}in the binomial expansion of (*a*+1)^{n+1}. For example, the binomial expansion of (*a*+1)^{8}is*a*^{8}+ 8*a*^{7}+ 28*a*^{6}+ 56*a*^{5}+ 70*a*^{4}+ 56*a*^{3}+ 28*a*^{2}+ 8*a*+ 1.

From this, a seven-dimensional simplex has

one hepton (itself, an octahexon),

eight hexons (heptapenta: hence the name*octahexon*),

28 pentons (hexatetra),

56 tetrons (pentachora),

70 cells (tetrahedra),

56 faces (triangles),

28 edges,

eight vertices, and

one nullitope (element of dimension –1).

- When the number of dimensions exceeds six, even a simplex
gets complicated: The symmetry group of a regular
*n*-dimensional simplex is the permutation group of*n*+1 items, with order (*n*+1)! So, for example, a regular octahexon has 40,320 symmetries.*See also*Hyperspherical simplex; Polytope (particularly for how to name polytopes, including simplexes).

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- A finite collection of facets of dimension
*k*with all the properties of a (*k*+1)-dimensional polytope except that they are embedded in a space of more than*k*+1 dimensions. For example, a skew polygon in three-space is a cycle of line segments that lie in three-space joined end to end. Since*k*-dimensional facets are thin in spaces of (*k*+2) or more dimensionals, it is clear that a skew polytope cannot enclose a portion of the space it is embedded in. Lacking an interior, a skew polytope cannot be convex. If some facets of a skew polytope intersect, it is a*skew star-polytope*.

- Two numbers play an important role in the properties of a
skew polytope: The dimension
*k*of the facets and the dimension*n*>*k*+1 of the space in which the facets are embedded. The facets may be ordinary polytopes of dimension*k*or skew polytopes embedded in spaces of*k*dimensions. Skew polytopes with rigid facets are usually nonrigid, being members of a continuum of topologically equivalent skew polytopes related by a continuous deformation in which the dihedral angles between adjoining facets change continuously. Triangular hotels are examples of rigid skew polytopes.

- A skew polytope is
*regular*if its facets fulfill the criteria of regularity, that is, if the symmetry group of the skew polytope is transitive on its flags. There are uncountably many regular skew polygons in three-space.

- <MORE TO COME>

*See also*Dihedral angle; Hotel; Infinite polytope; Polytope.

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- A collection of points together with a
*topology*, that is, a rule specifying what the basic open sets in the space are.

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- The pieces into which Euclidean
*n*-dimensional space is divided by the hyperplanes of the facets of a convex polytope. Some stellachunks have finite extent in*n*-space, whereas others extend to infinity. Aggregates and stellations are constructed by joining finite stellachunks, usually symmetric sets of them, together.

- For example, the five lines that contain the edges of a
regular pentagon divide the plane into 16 stellachunks. One of
these, the core polygon, is the pentagon itself. The others are
the five triangles that make up the points of a regular
pentagram, and two kinds of infinite stellachunks, five of each.
*See also*Aggregate and Aggregation; Stellalayer; Stellation.

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- A set of stellachunks that just covers (that is, hides from
the exterior) a core polytope (which does not count as a
stellalayer) or another stellalayer. The stellachunks in a
stellalayer are so chosen that removing any one of them exposes
part of the core or stellalayer underneath to the exterior. The
outermost stellalayer is the one farthest from the core that
contains no infinite stellachunks.

- In the example of the pentagon under Stellachunks, the
five isosceles triangles comprise the pentagon’s one finite
stellalayer.
*See also*Aggregate and Aggregation; Stellachunks; Stellation.

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- The operation of producing a star-polytope, called a
*stellation*, from a convex*core*polytope by extending the facets of the core polytope in their realms until they meet other extended facets of the core polytope. Usually, identical facets of the core are extended identically, so that some or all of the symmetry of the core polytope is retained by the stellation. A stellation that has fewer symmetries than the core polytope is a*subsymmetric*stellation; otherwise it is a*fully symmetric*stellation.

- Stellation also has a more restrictive definition,
namely, the transformation of a convex polygon in the plane into
a star-polygon by extending its edges until they meet other
extended edges. For example, a regular pentagram
{
^{5}/_{2}} is a fully symmetric stellation of a regular pentagon (5). When a regular polyhedron or regular polychoron has pentagonal faces, the star-polyhedron or star-polychoron formed by stellating the pentagons into pentagrams is called a*small stellated*or simply a*stellated*polyhedron or polychoron. Thus, if we stellate the pentagons of a regular dodecahedron into pentagrams, the resulting star-polyhedron is a small stellated dodecahedron. If we stellate the pentagons of a great dodecahedron into pentagrams, the resulting star-polyhedron becomes a stellated great dodecahedron, more usually called a great stellated dodecahedron. To avoid confusion between the general concept of stellation and this restricted definition of stellation, the latter is often called*edge-stellation*, because the edges of the core polytope are extended until they meet other edges.

- In the plane, the most general kind of stellation is
edge-stellation. In three-space, the corresponding operation is
face-stellation, or greatening, and in four-space, it is
cell-stellation, or aggrandizement. In spaces of dimension
greater
than four, we employ the terms
*tetron-stellation*,*penton-stellation*,*hexon-stellation*, and so forth, when necessary.

- Stellation was throughly described for the regular
icosahedron by H. S. M. Coxeter, P. Du Val, H. T. Flather & J. F.
Petrie in the 1938 publication
*The Fifty-Nine Icosahedra*(reprinted in 1999 by Tarquin Publications). They enumerated and illustrated the 59 externally different irreducible symmetric icosahedral aggregates, which are a particular kind of stellation whose faces have no regions lying in the interior of the figure.*See also*Aggrandizement; Aggregate and Aggregation; Greatening; Hecatonicosachoron, regular; Hexacosichoron, regular; Regular polytope.

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- The three-dimensional exterior of a polychoron or other
four-dimensional figure bounded by a closed three-dimensional
manifold. Since a
*surface*is the two-dimensional exterior of a three-dimensional figure, such as a polyhedron, the term*surcell*derives from boosting the dimensionality of a “face” to that of a “cell.” A glome is the surcell of a gongyl.

- For polytopes in spaces of dimension greater than four,
we continue the face-cell analogy: The
*surtetron*is the four-dimensional exterior of a polytetron or a closed four-manifold in five-space, the*surpenton*is the five-dimensional exterior of a polypenton or a closed five-manifold in six-space, and so on.*See also*Glome; Gongyl; Polychoron; Polyhedron.

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- The simple polytope made up of the exterior facetlets of an
arbitrary polytope.

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- An isometry that permutes the features of a geometric figure,
that is, carries any feature into another equivalent feature. For
example, any rotation of a sphere about a line passing through
its center permutes the sphere’s great circles, so the
sphere is said to be
*symmetric under*such rotations.

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- The complete set of symmetries of a geometric figure. It
is clear that if
*A*and*B*are symmetries of a figure, then so is the isometry*AB*, that is,*A*followed by*B*. It is also clear that the identity*I*, the*trivial isometry*(which leaves every point of the entire space unchanged), is a symmetry of any figure, and that for every symmetry*A*there exists a symmetry*A*^{–1}that undoes, or reverses,*A*. Finally, it is also clear that performing three symmetries consecutively on a figure yields the same result regardless of whether it is denoted (*AB*)*C*or*A*(*BC*) or*ABC*. Any set of operations that has these properties (closure, identity, inverse, associativity) under the composition operation forms a group by definition.

- If for any two
*k*-dimensional elements of a polytope there is a symmetry of the polytope that will carry the first element into the second element, the polytope’s symmetry group is said to be*transitive*on the*k*-dimensional elements.

- The
*order*of a symmetry group is the number of distinct operations in it. The*trivial symmetry group*consists of only the identity operation, so it has order 1. A polytope whose symmetry group is trivial is said to be*asymmetric*; other polytopes are*symmetric*. The*bilateral symmetry group*has order 2, including the identity element and a reflection in a mirror. Applying a reflection a second time reverses the reflection and gives back the identity. Although all order-2 groups are*isomorphic*(that is, have the same structure), they do not necessarily represent the same symmetries. For example, a half-turn about an axis is a different isometry from a reflection in a mirror, yet either operation reverses itself when applied a second time and thus gives rise to the same order-2 symmetry group.

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- Filling
*n*-space with*n*-dimensional polytopes so that their facets meet exactly and no part of*n*-space remains unfilled. When*n*=2, a tessellation is called a*tiling*; when*n*>2, a tessellation is called a*honeycomb*. In many references, however, the three terms are used interchangeably. The individual polytopes of a tessellation are called*cellets*. The cellets of a tessellation may overlap, but no facet may belong to more than two cellets, and no two cellets may overlap exactly. It is permitted for an (*n*–1)-dimensional tessellation to serve as a cellet of an*n*-dimensional tessellation. In this case, the interior of the cellet is either of the half-*n*-spaces into which its hyperplane divides*n*-space.

- As with polytopes, if every vertex of a tessellation is
surrounded alike, the tessellation is
*isogonal*; if in addition the cellets are all uniform polytopes, the tessellation itself is*uniform*. If the cellets are all congruent, the tessellation is*isotopic*, and if in addition the cellets are all regular polytopes, the tessellation itself is*regular*(being isotopic and having regular cellets forces the vertex figures of the tessellation to be congruent). The vertex figure of a tessellation of*n*-space is a tessellation of a small (*n*–1)-hypersphere centered at the vertex; if the tessellation is regular, so is the vertex figure. If the tessellation is uniform, the vertex figure may also be the*n*-dimensional polytope whose vertices are those of the hyperspherical vertex figure and with corresponding elements.

- There is only one regular tessellation of the line, and
that is the
*apeirogon*. There are three regular tessellations of the Euclidean plane: the*square tiling*, or*checkerboard tiling*; the*triangular tiling*; and the*hexagonal tiling*. Euclidean three-dimensional space has just one regular tessellation, the*cubic honeycomb*, but four-dimensional space has three: the*tesseractic honeycomb*, the*hexadecachoric honeycomb*, and the*icositetrachoric honeycomb*. In Euclidean five-space and spaces of higher dimension, there is only one regular tessellation, the*measure polytope honeycomb*. A circle has infinitely many regular tessellations, each corresponding to a regular polygon {^{p}/_{q}}. The hyperbolic plane has infinitely many regular tilings (for example, by convex*p*-gons,*q*at a vertex, where^{1}/_{p}+^{1}/_{q}<^{1}/_{2}), whereas the sphere has nine regular tilings, spherical versions of the nine regular polyhedra. Tessellating Euclidean spaces and hyperbolic spaces requires countably infinitely many cellets, but tessellating elliptic and spherical spaces requires only a finite number of cellets.*See also*Apeirogon; Hexadecachoron; Honeycomb; Icositetrachoron, regular; Measure polytope; Polytope; Regular polytope; Tesseract; Tesseract army; Tiling; Uniform polytope; Vertex figure.

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- The four-dimensional measure polytope. Its elements include
one nullitope, 16 vertices, 32 edges all the same length, 24
congruent square faces, 8 congruent cubic cells, and itself as
its body. It was discovered by Ludwig Schläfli in the
mid-1850s; its Schläfli symbol is {4,3,3}. Euclidean
four-space
may be filled uniformly and completely by tesseracts without
overlapping, joined cell to cell, just as three-space can be
filled with cubes and the plane can be filled with squares. This
tessellation is called the
*tesseractic honeycomb*; its Schläfli symbol is {4,3,3,4}. Bulk, or four-dimensional volume, may be defined as the number of unit tesseracts and fractions thereof required to fill a given region of four-space. The unit tesseract is the vertex figure of the regular tessellation of four-space by icositetrachora. The vertex figure of a unit tesseract is a regular tetrahedron of edge*sqrt*(2).*See also*Bulk; Honeycomb; Measure polytope; Polytope; Regular polytope; Tessellation; Tesseract army.

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- The set of uniform polychora and uniform compound polychora
that have the 16 vertices of a tesseract. This small army
(mix-and-match designation 16/1), of which the tesseract is the
general, includes two regiments, the
*tesseract regiment*(designated 16/1/1), of which the tesseract is the colonel and only member, and the*doublecross regiment*(16/1/2), of which the regular compound of two hexadecachora (four-dimensional cross polytopes: hence the name “doublecross”) is the colonel. The latter regiment includes two distinct uniform compounds of the octahemioctachoron besides the doublecross compound.

- We may easily enumerate the members of the tesseract army
by examining their vertex figures. First, however, we note that
the 16 vertices of the army can be connected by a total of
16*15/2 = 120 edges. These fall into four different sets by
length, so there can be at most four regiments in the tesseract
army:

A: 32 edges of the tesseract (length 1)

B: 2*24=48 diagonals of the faces (length*sqrt*[2])

C: 4*8=32 diagonals of the cells (length*sqrt*[3])

D: eight main diagonals of the tesseract (length 2)

- The eight main diagonals obviously do not form a
polychoron, so there is no regiment using edge set D. A
completist might consider these diagonals to be a symmetric
compound of eight digons (or eight of infinitely many kinds of
hosohedra or hosochora), but these kinds of figures are not
usually considered valid uniform polychora or compounds.

- The 32 cell diagonals also do not form uniform polychora
or compounds, although they may be used in combination with edges
of other lengths form compounds of nonuniform polychora. The
angle between any two of the four cell diagonals that meet at a
vertex is 2*
*Arcsin*(*sqrt*[6]/6) = 48.1896851...°. This is not the dihedral angle of any regular polygon or star-polygon, so the edges cannot make a regular face as required for a uniform polychoron.

- Digression: Among those compounds of nonuniform polychora
is one of 16 tetrahedral spikes. Each spike is an irregular
pentachoron with a regular tetrahedron, formed from six B edges,
as its base, and four isosceles pyramids, each formed from three
B edges and three C edges, as its lateral cells. The apex of the
spike is formed by the four C edges that come together at each
corner of the tesseract: so, 16 vertices means 16 spikes.
Interestingly, the isosceles triangles formed by two C edges and
a B edge, which are the lateral faces of the spikes, are the
vertex figures of truncated octahedra (the C edges representing
the two hexagons and the B edge representing the square at each
vertex of a truncated octahedron). Looking further, we find that
the lateral cells of the spikes are the vertex figures of
truncated icositetrachora, and the bases are the vertex figures
of tesseracts. These polychora are the cellets of a
*truncated icositetrachoric honeycomb*, the uniform tessellation of four-space by truncated icositetrachora and tesseracts. It derives from the regular icositetrachoric honeycomb by simply replacing its icositetrachora with edge-inscribed truncated icositetrachora and taking up the free cubes with tesseracts. The compound of 16 spikes is the vertex figure of a corresponding compound of some number (I have yet to determine just how many there are, but it must be*at least*16) of identical, distinct truncated icositetrachoric honeycombs vertex-inscribed in an icositetrachoric honeycomb. The original tesseract is the vertex figure of the “enclosing” icositetrachoric honeycomb. Note that the 16 truncated icositetrachoric honeycombs have the*same*set of vertices as the icositetrachoric honeycomb; they are*not*edge-inscribed in it. The faces of the truncated icositetrachoric honeycombs are (1) the diametral squares of the octahedral cells of the icositetrachoric honeycomb (each square belongs to two different honeycombs of the compound), and (2) the diametral hexagons of the icositetrachoric cellets. End of digression.

- The 32 edges of the tesseract of course make up the
tesseract itself, but because the vertex figure of a tesseract is
a regular tetrahedron, which has no facetings, there are no
polychora that are the tesseract’s edge-facetings. So the
tesseract stands alone as the colonel and only member of its
regiment (as well as being the general of its army).

- This leaves edge set B as the only hope for a nontrivial
regiment in the tesseract army, and indeed, these are the edges
of the doublecross regiment. The B edges form no less than eight
mutually intersecting octahedra, each octahedron being
perpendicular to one of the eight main diagonals of the tesseract
and passing through its center. Each edge belongs to just two
octahedra, so unfortunately these eight octahedra cannot form a
polychoron by themselves; in any polychoron, each edge belongs to
three or more cells. But they can be thought of collectively as a
uniform compound of eight octahedral dichora (with each
octahedron being thought of as two coincident octahedra: a
*dichoron*). Because a dichoron is not considered a true polychoron, such a compound would not be considered a valid uniform compound. The eight octahedra fall into two sets of four mutually perpendicular octahedra, each set being the four central octahedra of a hexadecachoron. These define the two hexadecachora of the doublecross compound.

- An alternative derivation of the doublecross compound is
to alternate the vertices of the tesseract, that is, divide the
16 vertices into two sets of eight in the following manner: Color
one vertex blue. This vertex has four neighbors that are one
edge-length removed from it. Color them red. Color blue any
uncolored neighbors of the four red vertices that are one
edge-length away from them. This colors six more vertices blue.
Color red the four uncolored vertices one edge-length away from
any blue vertices. Finally, color the remaining uncolored vertex
blue. The eight red vertices and the eight blue vertices each
define a hexadecachoron, and the compound of the two
hexadecachora is the doublecross compound.

- Between them, the two hexadecachora have 32 tetrahedral
cells. The eight vertices of either hexadecachoron lie directly
above eight tetrahedra of the other, so we may color eight
tetrahedra of one blue and eight tetrahedra of the other red to
correspond with the colors of the vertices above them. (The blue
tetrahedra will all have red vertices, and
*vice versa*.) This leaves eight tetrahedra of either hexadecachoron uncolored. These 16 uncolored hexadecachora lie in intersecting pairs in the eight cubes of the case tesseract. Each pair of tetrahedra is a well-known compound polyhedron called a*stella octangula*. Altogether, the 32 tetrahedra occupy 24 realms, the realms of the cells of a regular icositetrachoron. This is the core polychoron of the compound.

- The vertex figure of a unit hexadecachoron is a unit
octahedron, which is convex, so the doublecross compound is the
colonel of its regiment, the doublecross regiment 16/1/2. An
octahedron has one subsymmetric faceting whose faces are vertex
figures of uniform polyhedra, namely, the uniform polyhedron
known as a tetrahemihexahedron, or three-dimensional demicross
polytope. Its faces are four of the octahedron’s eight
triangles and the octahedron’s three diametral squares. It
is the vertex figure of the octahemioctachoron, or
four-dimensional demicross polytope. The octahemioctachoron is
the uniform star-polychoron with the least number of cells,
namely, an alternating set of eight of the 16 tetrahedra of the
hexadecachoron and one set of four mutually perpendicular
octahedra of the kind described previously. If we replace both
hexadecachora by two octahemioctachora that have the same
vertices and edges, the resulting compound is potentially
uniform. For a compound to be uniform, it is necessary that its
symmetry group be transitive on
*all*of its vertices, irrespective of which component any vertex might be an element of.

- Because there are two ways to replace a hexadecachoron by
an octahemioctahedron, we must consider the following three
cases:

[1] The eight tetrahedra of each fall beneath the eight vertices of the other;

[2] The eight tetrahedra of each pair up in stella octangulae with the eight tetrahedra of the other;

[3] The eight tetrahedra of one fall beneath the eight vertices of the other, but not*vice versa*.

- It should be clear that the first two compounds are
uniform, while the third compound is not uniform. In the first
two cases, any symmetry that maps a vertex of one component onto
a vertex of the other will also map the tetrahedra of the first
component onto the tetrahedra of the other. But in the third
case, such a mapping maps the tetrahedra of one component onto
the tetrahedral “holes” of the other, and it is thus
not a symmetry of the compound. The symmetry group of the
compound is therefore not transitive on the vertices, and so it
is not a uniform compound. We can describe these as [1] the
*ortho*compound of two octahemioctachora, [2] the*para*compound, and [3] the*meta*compound. In the para compound, the vertices of each component are situated above the tetrahedral “holes” of the other. The ortho and para compounds of two octahemioctachora are the other two members of the doublecross regiment. So we can call them the*orthodoubledemicross compound*and the*paradoubledemicross compound*. Likewise, the third, nonuniform, member of this set of compounds becomes the*metadoubledemicross compound*.

- The existence of these compounds implies the existence of
interesting uniform compound tessellations among the vertices and
edges of the icositetrachoric honeycomb. The hexadecachoron of
edge
*sqrt*(2) is the vertex figure of the tesseractic honeycomb, and sure enough, there is a way to emplace*three*(not just two) tesseractic honeycombs uniformly within the vertices and edges of an icositetrachoric honeycomb. Each vertex of the icositetrachoric honeycomb, however, belongs to only two of the three tesseractic honeycombs, which is why the vertex figure is a compound of two hexadecachora. There is a regular compound of three tesseracts in an icositetrachoron, and those three tesseracts can propagate by translation to infinity to become the three tesseractic honeycombs in the icositetrachoric honeycomb. The faces of the three tesseractic honeycombs are all the diametral squares of all the octahedra (each of the three squares in each octahedron belongs to a different tesseractic honeycomb) of the icositetrachoric honeycomb.

- Just as the hexadecachoron is the vertex figure of the
tesseractic honeycomb, so is its faceting, the
octahemioctachoron, the vertex figure of a honeycomb that has the
same vertices, edges, and faces. But because the
octahemioctachoron has four octahedral cells that pass through
its center, the corresponding honeycomb has cellets that are
themselves honeycombs of three-space, specifically, cubic
honeycombs. The tessellation of which the octahemioctahedron is
the vertex figure is the tesseractic honeycomb with exactly half
its tesseracts removed (color the tesseracts alternately black
and white, then remove, say, the white ones), with their free
cubic cells taken up by four infinite tiers of cubic honeycombs
parallel to the four cell-realms of any tesseract remaining in
the honeycomb. Three such
*demitesseractic honeycombs*can overlie one another uniformly in the ortho, para, and meta configurations, to produce three distinct uniform compound tessellations.*See also*Army; Bulk; Compound polytope; Cross polytope; Demicross polytope; Faceting; Hexadecachoron, regular; Honeycomb; Icositetrachoron, regular; Mix-and-match notation; Realm; Regiment; Regular polytope; Uniform polytope; Tessellation; Tesseract; Transitivity.

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- An adjective describing a figure of
*n*dimensions in*n*-dimensional space.*See also*Flat (*k*-flat); Thin.

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- An adjective describing a figure of fewer than
*n*–1 dimensions in*n*-dimensional space. A line segment is thin in three-space (and any higher-dimensional space), flat in two-space, and thick in one-space. A polygon is thin in four-space (and any higher-dimensional space), flat in three-space, and thick in two-space. No countable collection of thin figures can bound a region of*n*-space.*See also*Flat (*k*-flat); Thick.

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- An infinite collection of polygons adjoining along their
facets, that fills the plane completely. If the polygons are
regular and the symmetry group of the tiling is transitive on its
vertices (that is, the corners are “surrounded alike”),
the tiling is
*uniform*. If in addition the polygons are congruent, the tiling is*regular*. If the polygons overlap, so that the tiling fills the plane more than once, the tiling is a*star-tiling*. Examples of tilings include the familiar regular tilings of equilateral triangles, squares, and regular hexagons.*See also*Apeirogon; Honeycomb; Polygon; Tessellation.

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- Sets of polytopes that have members in
*n*-spaces for different values of*n*. For example, simplexes, measure polytopes, and cross polytopes are transdimensional polytope families with members in*n*-space for all integral*n*, –2<*n*. Pentagonal polytopes and ditrigonary polytopes have members in two-, three-, and four-space. Gosset polytopes have members in spaces of dimension two through eight.

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- A formal way of saying that the elements of a figure are
“all alike”: If the symmetry group of a figure includes
enough symmetries to map any element into any other element of
the same kind, the symmetry group is said to be
*transitive*on that set of elements. For example, by definition the symmetry group of an isogonal polytope is transitive on its set of vertices; and likewise by definition the symmetry group of an isotope is transitive on its set of facets.

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- A direct isometry of
*n*-space,*n*>0, in which every point is moved the same distance in a particular direction. A translation may be expressed as the product of reflections in two parallel mirrors, in which case the distance of the translation is twice the distance between the mirrors and the direction of the translation is perpendicular to the mirrors. (The distance and direction determine the*translation vector*of a translation.) The order in which the reflections are performed determines whether the translation is backward or forward along its direction. The product of two translations in different directions for different distances is another translation, in the direction and distance determined by the vector sum of the original two translation vectors. Translations cannot be among the symmetries of a finite polytope, but are often among the symmetries of an apeirogon, tiling, honeycomb, or infinite polytope.*See also*Apeirogon; Honeycomb; Infinite polytope; Isometry; Mirror; Reflection; Tiling.

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- Loosely speaking, the operation that creates a new polytope
t
from a polytope*P*by cutting off its corners and replacing them with the corresponding vertex figures as new facets (called*P**stump*facets). The facets of a truncated polytope tare thus (1) the stump facets, and (2) the facets of*P*truncated. If not all the corners are removed, the truncation is*P**partial*. There are various kinds of truncations, depending on the locations of the*truncating hyperplanes*, that is, the hyperplanes of the stump facets, and just how the stump facets interact with one another.

- In
*bitruncation*, the truncating hyperplanes are located so deep inside the polytope that the stump facets themselves truncate one another. On the other hand, in*quasitruncation*, the stump facets extend through one another instead of truncating one another. Ifis a convex polytope, bitruncation of*P*produces another convex polytope, but quasitruncation of*P*produces a star-polytope.*P*

- For example, a suitable choice of truncating lines will
truncate a square {4} into a regular octagon {8}, and another
choice of truncating lines will quasitruncate a square into a
regular octagram
{
^{8}/_{3}}. A bitruncated square, however, is merely a smaller square. In three-space, a suitable choice of truncating planes will make the uniform polyhedron known as a*truncated cube*from a cube, and another choice of truncating planes will make a*quasitruncated cube*from a cube. In the former, the faces are regular octagons and triangles, and in the latter, the faces are regular octagrams and triangles. Another choice of truncating planes will bitruncate a cube into a*truncated octahedron*. Its faces are squares (the bitruncated squares of the cube) and regular hexagons (triangles truncating one another).

- In spaces of dimension higher than three, bitruncation
and quasitruncation yield a variety of nontrivial uniform
polytopes when applied to regular and to other uniform polytopes.
The dual operation to truncation (which is essentially the
replacement of vertices by facets) is apiculation (which is
essentially the replacement of facets by vertices).
*See also*Alteration; Alternation; Apiculation.

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- For
*n*=2, a regular polygon. For*n*>2, a polytope whose facets are uniform polytopes and whose symmetry group is transitive on its vertices. Convex nonregular uniform polytopes are also called*Archimedean*polytopes, because it was Archimedes who first posed and solved the problem of finding all the convex polyhedra whose faces are regular polygons and whose vertices are “surrounded alike.” All regular polytopes are uniform, but because we have changed the restriction that the facets be regular to merely being uniform, most uniform polytopes are not regular.

- In three-space, all the uniform polyhedra are known;
besides the infinite families of prisms and antiprisms, there are
75 uniform polyhedra. Of these, nine are regular, 13 are
Archimedean, and 53 are nonconvex and nonregular. The latter were
systematically described by H. S. M. Coxeter, M. S.
Longuet-Higgins, and J. C. P. Miller in 1954, and thus we call
them the
*Colohimian polyhedra*, Colohimi being an acronym formed from the surnames of the describers. Their enumeration was proved complete by S. P. Sopov in 1970 and independently by John Skilling in 1975.*See also*Colohimian polyhedra.

- The uniform polytopes in spaces of dimension greater than
three are not yet known completely. The Uniform Polychora Project
is an informal enterprise organized by Jonathan Bowers, Norman
Johnson, and me to collect information about uniform polychora
(and uniform polytopes in spaces of dimension greater than four)
with the aim of enumerating them and eventually demonstrating
that the enumeration is complete. So far we have found 8190
uniform polychora outside the infinite families of prismatic
polychora (the vast majority of which were discovered by Bowers),
of which only 64 are convex and the other 8126 are nonconvex. The
higher the dimension, the less we know about the uniform
polytopes in those spaces. The convex uniform polychora are
tabulated at my Uniform
Polychora website; that there are 64 convex nonprismatic
uniform polychora was established by John Horton Conway and
Michael Guy in the mid-1960s. The convex uniform polytopes with
regular facets were completely enumerated by Thorold Gosset. In
four-space, these are the convex uniform polychora whose cells
are combinations of tetrahedra, octahedra, and icosahedra. In
spaces of dimension greater than four, these are the convex
polytopes whose facets are regular simplexes and cross polytopes.
*See also*Isogonal polytope; Regular polytope; Vertex figure.

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- A 0-dimensional element of a polytope, also called a
*corner*; plural*vertices*.*See also*Facet.

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- Loosely speaking, the (
*n*–1)-dimensional polytope formed at the stump when a corner is truncated from an*n*-dimensional polytope or compound polytope; the cross-section of an*n*-dimensional polytope or compound polytope very close to a vertex. Also called a*verf*. The vertex figure shows exactly how a polytope’s facets fit together to surround a vertex.

- This definition, unfortunately, is far
too loose for most polytopes, for which there is no particularly
symmetric position, center, or distance from the vertex at which
to place the sectioning hyperplane. In uniform polytopes and
compounds, however, the edges are all of the same length, so the
other ends of all the edges incident at a vertex lie on a
hypersphere, centered on the vertex, whose radius equals the edge
length. This hypersphere intersects the circumhypersphere of the
uniform polytope, which is centered at the center of symmetry, so
the vertices of the vertex figure will all lie on the hypercircle
that is the intersection of the two hyperspheres. The hyperplane
of this hypercircle becomes the hyperplane of the vertex figure.
Since the vertex figures at all the vertices are congruent, the
vertex figure of a uniform polytope or compound has a natural
definition: The (
*n*–1)-dimensional polytope or compound polytope whose vertices are the other ends of the edges incident at a vertex, and whose higher-dimensional elements are themselves the vertex figures of the corresponding elements incident at that vertex. Additionally, for any Euclidean uniform polytope, the circumradius of the vertex figure must be less than the polytope’s edge length (otherwise the edges could not “reach” the vertex).

- Vertex figures are very important in discovering
uniform polytopes. For example, the vertex figures of all the
uniform polytopes and uniform compound polytopes in a regiment
must be facetings of the vertex figure of the colonel of the
regiment; so by constructing all possible facetings of the
colonel, we capture all the uniform polytopes and compounds in
the regiment (including, perhaps, the vertex figures of exotic
figures outside some definitions of a uniform polytope or
compound). Some definitions of a polytope exclude figures that
have compound vertex figures. We may also test for the existence
of a uniform polytope by attempting to construct its vertex
figure. If the vertex figure does not close, the polytope cannot
close, either, and is thus nonexistent. In this sense, the
properties of a uniform polytope are reflected in the simpler
properties of its vertex figure; a vertex figure is an
(
*n*–1)-dimensional “image” of its parent uniform polytope.

- For example, the vertex figure of a regular polygon
{
*p*/*q*} of unit edge length is the diagonal that joins the vertex behind a particular vertex to the vertex ahead of that vertex. This diagonal, conveniently enough, has length 2*cos*(*q**pi/*p*). The vertex figure of a uniform polyhedron, in turn, is a polygon, sometimes regular but usually irregular, whose vertices all lie on a circle and whose sides are the vertex figures of the faces (which are all regular polygons) of the polyhedron that meet at that vertex. The vertex figures of uniform polyhedra have either three, four, five, six, or (in one instance) eight sides.

- Continuing in this vein, the vertex figures of uniform
polychora are polyhedra all of whose vertices lie on a sphere,
whose faces are the vertex figures of the cells that meet at each
vertex, and whose circumradius is less than one. These include
tetrahedra, wedges, all the regular polyhedra, various kinds of
prisms and antiprisms, and other polyhedra less easily described;
and a huge number of facetings of these. In enumerating the
convex uniform polychora in the 1960s, John Horton Conway and
Michael Guy computer-checked all the
*convex*polyhedra under these constraints to find those that actually give rise to convex uniform polychora. At the conclusion of their work, they showed that there were two infinite families of convex uniform polychora (the uniform duoprisms and the uniform antiprismatic prisms) and 64 others (the Platonic and Archimedean polychora). These polychora, with all vertex figures illustrated, are tabulated at the Uniform Polychora website. The nonconvex uniform polychora have not yet been completely enumerated, but the count of those outside the infinite families has reached 8126.

- In spaces of more than four dimensions, the uniform
polytopes have not yet been completely enumerated, even in the
convex case. The number of possible vertex figures that need to
be checked increases with the dimension of the space, making even
computer searches quite time-consuming.

- In more general, less symmetric polytopes, the vertex
figures at the vertices of a polytope are not congruent, and
since the edges incident at a vertex may approach from any
direction, the definition that works for uniform polytopes can
break down. A general vertex figure does not necessarily lie in a
hyperplane, although for a convex polytope it can be made to.
Thus, for arbitrary polytopes in locally Euclidean spaces, the
vertex figure is defined as a hyperspherical polytope or compound
polytope whose vertices are the sections of the edges incident at
the vertex by a small hypersphere (with radius less than the
length of the shortest edge incident at the vertex) centered on
the vertex. The higher-dimensional elements (edges, faces, etc.)
of the vertex figure are the vertex figures of the corresponding
higher-dimensional elements (faces, cells, etc.) of the polytope
incident at the vertex. In this definition, the hyperspherical
dihedral angles of the vertex figures equal the dihedral angles
of the polytope.

- Honeycombs and tilings have vertex figures, too, of
course. For these kinds of figures in
*n*-space, the vertex figures are the*n*-dimensional polytopes whose vertices are located at the sections of the edges incident at a vertex by a small hypersphere centered on the vertex.*See also*Antiprismatic prism; Compound polytope; Convex polytope; Dihedral angle; Duoprism; Exotic polytopes; Honeycomb; Hypercircle; Hypersphere; Tiling; Truncation; Uniform polytope.

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- The three-dimensional content of a solid figure, such as a
polyhedron. More generally, the content of a three-dimensional
manifold embedded in a metric space.
*See also*Content of a figure.

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- A method of locating the vertices of uniform polytopes,
developed by W. A. Wythoff and used by H. S. M. Coxeter, M. S.
Longuet-Higgins & J. C. P. Miller to discover all the uniform
star-polyhedra. Wythoff’s construction is intimately related
to the symmetry groups of polytopes, because the symmetry group
of a uniform polytope is transitive on its vertices. Every
symmetry of a polytope is an isometry that leaves the center of
symmetry of the polytope (and perhaps other points of
*n*-space) invariant. Being an isometry, a symmetry can be expressed as the composition of zero (if and only if it is the identity operation) or more reflections in mirrors that pass through the center of symmetry. So all the symmetries of a polytope can be expressed as combinations of reflections in a set of mirrors that intersect in the center of symmetry called the*complete kaleidoscope*of the symmetry group.*See also*Isometry; Kaleidoscope; Reflection; Symmetry.

- Now construct a hypersphere of arbitrary positive radius
centered on the center of symmetry. Each mirror intersects the
hypersphere in a great hypercircle. These great hypercircles
divide the hypersphere into a number of congruent hyperspherical
regions, the shapes of which, for all but the simplest symmetry
groups, are
*hyperspherical simplexes*. These are the*fundamental regions*of the symmetry group; each fundamental region is bounded by the*n*hyperplanes of a fundamental kaleidoscope of the polytope’s symmetry group.*See also*Hypercircle; Hypersphere; Hyperspherical simplex.

- The dihedral angles of the fundamental regions are the
same
as the angles between the hyperplanes that form their boundaries.
These must be fractional multiples of 180°; otherwise, the
symmetry group would have uncountable order. (To see why, simply
observe the fate of a point of the hypersphere under repeated
alternating reflections in two mirrors that intersect in a
dihedral angle that is an irrational multiple of 180°. It
never returns to where it started from, and after an infinite
number of such reflections its images form a fractal circle
around the axis that is the intersection of the two mirrors.

- A symmetry of the symmetry group maps a fundamental
region
exactly into another fundamental region (or, if it is the
identity, it maps every fundamental region into itself). Clearly,
because a symmetry preserves distance, two symmetries that map
the a particular region into two different regions must be
different symmetries; and equally clearly, two symmetries that
map a fundamental region into the same fundamental region
must be the same symmetry. Therefore,
*the number of fundamental regions in the hypersphere exactly equals the order of the symmetry group*, that is, the number of different symmetries.

- The vertices of a uniform polytope behave much like the
fundamental regions of its symmetry group. In particular, there
can be at most one vertex of a uniform polytope in a fundamental
region; if a fundamental region had two such vertices, no
symmetry of the polytope could carry either vertex into the other
(recall that the fundamental regions are bounded by great
hypercircles that are sections of the symmetry group’s
mirror hyperplanes), which contradicts the definition of a
uniform polytope. Wythoff’s construction is to find all the
different locations for a vertex within the fundamental regions
of all the different symmetry groups that yield uniform
polytopes.

- <MORE TO COME>

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- A regular finite skew 2
*p*-gon in three-space whose vertices lie in two congruent regular*p*-gons centered above each other in parallel planes. The*p*-gons may be*aligned*, in which case their vertices lie exactly opposite each other, or*antialigned*, in which case the vertices of either lie evenly between the vertices of the other. The edges of the zigzag polygon join a vertex in one*p*-gon to a vertex in the other*p*-gon cyclically, so that the path formed by the edges zigzags evenly between the two rings of vertices. The symmetry that carries a vertex of the zigzag polygon into an adjacent vertex is a rotatory-reflection through an angle*q*pi/*p*, where*q*is an integer between –*p*/2 and +*p*/2, around the axis that passes through the centers of the two rings of vertices. The symmetry that turns an edge around onto itself is a half-turn around the axis determined by the centroid of the vertices and the midpoint of the edge. If the zigzag is antialigned and*q*=1, the edges will not intersect; otherwise they will. Regular zigzag polygons can be aligned only when*p*is odd. The vertices of a regular aligned zigzag belong to a right prism based on a regular*p*-gon; the vertices of a regular antialigned zigzag belong to a right antiprism based on a regular*p*-gon.*See also*Skew polytope.

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Click on the underlined text to access various portions of the
Convex Uniform Polychora List:

Four
Dimensional Figures Page: Return to initial page

Nomenclature: How the convex
uniform polychora are named

List
Key: Explanations of the various List entries

Section
1: Convex uniform polychora based on the pentachoron
(5-cell): polychora #1–9

Section
2: Convex uniform polychora based on the tesseract
(hypercube) and hexadecachoron (16-cell): polychora
#10–21

Section
3: Convex uniform polychora based on the icositetrachoron
(24-cell): polychora #22–31

Section
4: Convex uniform polychora based on the hecatonicosachoron
(120-cell) and hexacosichoron (600-cell): polychora
#32–46

Section
5: The anomalous non-Wythoffian convex uniform polychoron:
polychoron #47

Section
6: Convex uniform prismatic polychora: polychora #48–64
and infinite sets

Section
7: Uniform polychora derived from glomeric tetrahedron
B_{4}: all duplicates of prior
polychora