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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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A

Accretion

See Apiculation.
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Aggrandizement

Stellation of a core polychoron P 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 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 R, the grand R has the same kinds of cells as R, only larger and more interpenetrating. See also Hecatonicosachoron, regular; Hexacosichoron, regular; Regular polytope; Stellation; Symmetry.
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Aggregate and Aggregation

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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Alteration

A kind of faceting that creates a new polytope or compound polytope aP 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 aP. 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 aP may become a compound of two polytopes. In this case, either polytope will be an alternated P, that is, hP. Among symmetric polytopes, alteration frequently produces a new uniform polytope or a compound of two new uniform polytopes from another uniform or isogonal polytope.

For example, an altered triangle aT is simply the same triangle T with its vertices in reverse order; an altered regular pentagon a{5} is a regular pentagram {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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Alternation

A kind of partial truncation in which exactly half of the vertices of a symmetric polytope P 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 hP 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 2p sides, h{2p}, 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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Antiprism

A polyhedron whose faces consist of two congruent, parallel p-gons and a belt of 2p 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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Antiprismatic prism

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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Apeirogon

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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Apiculation

The operation of constructing a new polytope vP 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 vP 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 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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Archimedean polytopes

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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Area

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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Army

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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Axis

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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B

Ball

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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Bipyramid

A polytope bB in n-space derived from a base polytope B in (n–1)-space in the following way: The hyperplane containing B divides n-space into two regions. Choose two points, a1 and a2, called apices (singular: apex), one in each region, and construct bB by (1) discarding the body of B, and (2) joining a1 and a2 to every remaining element of B to make the elements of bB. So each vertex of B makes an edge of bB with a1 and with a2; each edge of B makes a triangle with a1 and a2; each face of B makes a three-dimensional pyramid with 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 polytope P is this kind of symmetric bipyramid, based on the dual of P.

If the base polytope B is isotopic, any symmetric bipyramid sbB 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, sbB will also be convex (but a less constrained bipyramid need not be convex). The 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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Bitruncation

See under Truncation.
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Blend

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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Body

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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Bulk

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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C

Captain

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

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

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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Case polytope

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

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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Cell

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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Cell-stellation

See Aggrandizement .
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Center of symmetry

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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Chiral figures

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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Collinearity

See Line .
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Colohimian polyhedra

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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Colonel

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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Company

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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Compound polytope

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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Conjugate polytopes

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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Content of a figure

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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Convex polytope

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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Coplanarity

See Plane.
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Copycat polytopes

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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Corealmism

See Realm.
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Corner

A vertex of a polytope. See also Vertex.
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Cross polytope

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 2ⁿ (n–1)-dimensional simplexes, and an n-dimensional cross polytope has 2n 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 (2a+1)ⁿ. For example, the binomial expansion of (2a+1)⁷ is

128a⁷ + 448a⁶ + 672a⁵ + 560a⁴ + 280a³ + 84a² + 14a + 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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D

Demicross polytope

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 2n demicross polytopes, n rectified cross polytopes, and 2^n–1 rectified simplexes, of n–1 dimensions. The facets of a truncated demicross polytope are 2n demicross polytopes, n truncated cross polytopes, and 2^n–1 truncated simplexes, of n–1 dimensions. See also Ico regiment.
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Demihypercube

See Half measure polytope.
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Density

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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Dihedral angle

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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Dimension of a space

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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Dimpling

Inward apiculation. See also Apiculation.
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Ditrigonary star-polychora

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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Duality

The dual dP 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 dP; to every ridge of P there corresponds an edge of dP; to every peak of P there corresponds a face of dP; ...; to every face of P there corresponds a peak of dP; to every edge of P there corresponds a ridge of dP; and to every vertex of P there corresponds a facet of dP.

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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Duoprism

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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Duopyramid

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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Dyad

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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E

Edge

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

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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Element of a polytope

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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Empty set, the

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

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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Enantiomorphic figures

See Chiral figures.
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End [point]

A facet or vertex of a dyad. See also Dyad; Facet; Point.
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Euclidean n-space

A space of n dimensions with the Euclidean metric: the distance between two points

a = {a₁, a₂, a₃,..., a_n} and
b = {b₁, b₂, b₃,..., b_n}

is given by the general Pythagorean rule

D[a,b] = sqrt[(a₁–b₁)²+(a₂–b₂)²+(a₃–b₃)²+...+(a_n–b_n)²].

In Euclidean n-space, given a hyperplane H and a point p not in that hyperplane, there is exactly one hyperplane that passes through p parallel to H.
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Exotic polytopes

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:

Small complex hexacosihecatonicosachoron

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

Grand 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

Great complex dishecatonicosachoron

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

Stellated 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

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 &#147all 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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F

Face

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

See Greatening.
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Facet

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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Faceting

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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Flag

See under Regular polytope.
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Flat (k-flat)

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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Free facet

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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G

General

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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Glide reflection

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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Glome

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 2pi²r³. 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 pi²r³.

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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Gongyl

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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Grand polychora

See Aggrandizement.
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Greatening

Stellation of a polyhedron P 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 face-stellation.

A more restrictive definition of greatening requires that the expanded faces be similar to the original faces. Then, given a regular polyhedron R, the great R 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. See also Regular polytope; Stellation.
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H

Half measure polytope

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: 2n (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 H in 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 H, and (2) n rectified (n–2)-dimensional simplexes, which are the vertex figures of the (n–1)-dimensional demihypercube facets of H. 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.

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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Hecatonicosachoron, regular

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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Hexacosichoron, regular

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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Hexadecachoron, regular

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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Homogon

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

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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Homotope

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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Honeycomb

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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Hotel

Any of a family of regular skew polytopes whose facets are k-dimensional measure polytopes embedded in 2k-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 2k-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 p² 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 (2k–1)-space, whose facets are among the (k–1)-simplexes of an irregular (2k–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 2sqrt(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 2sqrt(pi*q/p) and whose (other four) zigzag edges are all of length sqrt(2).
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Hyperball

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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Hypercircle

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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Hypercube

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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Hyperplane

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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Hypersphere

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 pi³, and the pentabulk (5-dimensional content) of a unit tetraglome (n=5) is numerically the largest for any n-dimensional hypersphere, at ⁸/₁₅ pi². See also Content of a figure; Glome; Gongyl; Hyperball; Hypercircle.
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Hyperspherical simplex

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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I

Ico regiment

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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Icositetrachoron, regular

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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Infinite polytope

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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Isogonal polytope

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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Isometry

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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Isomorphic polytopes

See Conjugate polytopes.
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Isotope

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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J

Johnson antiprisms

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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Join

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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K

Kaleidoscope

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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L

Length

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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Line, straight

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 point x has coordinates {x₁, x₂, x₃,..., x_n}, a line is the set of points whose coordinates are expressed as

{t(b₁–a₁)+a₁, t(b₂–a₂)+a₂, t(b₃–a₃)+a₃,..., t(b_n–a_n)+a_n}

where t ranges from –infinity to +infinity and

a = {a₁, a₂, a₃,..., a_n} and
b = {b₁, b₂, b₃,..., 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 through a and b. In the above parametrization, t=0 corresponds to the point a and 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 and a and b are vectors extending from the origin to the points a and b. For any value of t the expression yields a vector that extends from the origin to a point somewhere on the line through the points a and b.
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Line segment

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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M

Measure polytope

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 2n (n–1)-dimensional measure polytopes, and an n-dimensional measure polytope has 2ⁿ 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 2ⁿ 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

a⁷ + 14a⁶ + 84a⁵ + 280a⁴ + 560a³ + 672a² + 448a + 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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Metric space

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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Mirror

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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Mix-and-match notation

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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Monad

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

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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N

n-space

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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Nullitope, the

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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O

Orthogonality

Perpendicularity.
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Orthoplex

See Cross polytope.
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Orthotope

See Measure polytope.
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P

Peak

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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Pentachoron, regular

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 2sqrt(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:

w²+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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Periphery

The surtope of a polygon. See also Surtope.
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Pivot

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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Plane

Given a line L and a point p not collinear with L, a plane is the set of points of n-space, n>1, collinear with p and any point of L, together with any other points collinear with these. Points and lines that all lie in the same plane are 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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Platonic solids

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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Point

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 {x₁, x₂, x₃,..., x_n}. All geometric objects are ultimately collections of points, usually uncountably many. See also Body; Monad; Vertex.
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Polychoron

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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Polygon

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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Polyhedron

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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Polytope

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.

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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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Prismatic polytopes

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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Pyramid

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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Q

Quasitruncation

See under Truncation.
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R

Realm

Given a plane P and a point p not coplanar with P, a realm is the set of points of n-dimensional space, n>3, collinear with p and 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 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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Rectification

The operation of creating a new polytope rP from another polytope P by using the midpoints of the edges of P as the vertices of rP. The facets of rP 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 n sides, and rP is, trivially, a polygon of n vertices. (If polygon P has edges that intersect each other at their midpoints, rP 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.

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 R always yields a uniform polytope rR (when the dimension of R is two, rR is regular, and is merely a smaller version of R: uniform polygons must be regular by definition). The rectification rU of a uniform polytope U will be uniform if the facets of U are regular and its vertex figure is uniform.

If R is a regular polyhedron, its rectification rR 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[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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Reflection

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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Regiment

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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Regular polytope

In most concise terms, any polytope R 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 flag is a collection of elements of R that 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.

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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Retroprism

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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Revert polytopes

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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Ridge

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

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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Rotation

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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S

Side

A facet of a polygon. See also Facet.
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Simplex

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

a⁸ + 8a⁷ + 28a⁶ + 56a⁵ + 70a⁴ + 56a³ + 28a² + 8a + 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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Skew polytope

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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Space

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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Stellachunks

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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Stellalayer

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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Stellation

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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Surcell

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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Surtope

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

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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Symmetry group

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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T

Tessellation

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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Tesseract

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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Tesseract army

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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Thick

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

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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Tiling

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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Transdimensional polytope families

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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Transitivity

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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Translation

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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Truncation

Loosely speaking, the operation that creates a new polytope tP from a polytope P by cutting off its corners and replacing them with the corresponding vertex figures as new facets (called stump facets). The facets of a truncated polytope tP are thus (1) the stump facets, and (2) the facets of P truncated. If not all the corners are removed, the truncation is 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. If P is a convex polytope, bitruncation of P produces another convex polytope, but quasitruncation of P produces a star-polytope.

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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U

Uniform polytope

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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V

Vertex

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

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 2cos(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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Volume

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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W

Wythoff’s construction

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.

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Zigzag polygon

A regular finite skew 2p-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 qpi/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 B₄: all duplicates of prior polychora