**OMENCLATURE.** It is, perhaps not surprisingly, a
non-trivial task simply to name the various uniform polychora. We
desire the names to be descriptive, mnemonic, euphonious, and not
outrageously difficult to pronounce, though because the figures
are complicated and involve large numbers of cells, few of the
names trip lightly off the tongue (although, having created them,
I’ve gradually become used to even the longest, much as one
becomes used to concatenating short words into long ones in
German—in which the 120-cell is known as a*
Hekatonkaieikosascheme*). And, perhaps most of all, we want
them to resemble—in extension—the traditional names of
the 18 well-known convex non-prismatic uniform polyhedra, their
3-D counterparts. In this light, the polychoric names offered
below should be regarded as suggestions; differing ideas about
polytope nomenclature are always open to debate. Indeed, I am
indebted to an ongoing discussion elsewhere between Norman W.
Johnson and John Horton Conway on Greek numerical prefixes and
other issues in polytope nomenclature for the names constructed
here.

Just as the five Platonic polyhedra are “basal” to
naming the 13 nonprismatic Archimedean polyhedra, so are the six
regular polychora basal to naming the 40 nonprismatic Archimedean
polychora. The six regular polychora I call the *pentachoron
*(with 5 tetrahedral cells), *tesseract* (with 8 cubic
cells), *hexadecachoron *(with 16 tetrahedral cells),
*icositetrachoron *(with 24 octahedral cells),
*hecatonicosachoron *(with 120 dodecahedral cells), and
*hexacosichoron* (with 600 tetrahedral cells). Nomenclature
of all 46 Wythoffian non-prismatic convex uniform polychora
begins with these.

The cells of a non-prismatic uniform polychoron that lie in the
hyperplanes of its basal regular polychoron can be considered its
“principal” cells. Blended names thus become plentiful
and derive mainly from the numerical counts of the principal
cells; instead of cub/octa/hedron, for example, we have
tesseracti/hexadeca/choron (here the slashes separate
corresponding parts of the names), and instead of one
icosi/dodeca/hedron we have two kinds of quasi-regular
hexacosi/hecatonicosa/chora. Among the uniform polychora we still
have “truncated” regular polychora, but instead of the
“rhombi” polyhedra there are more numerous
“prismato” polychora, in which various Archimedean
prisms (including cubes, as square prisms) assume the role of the
squares of the “rhombi” polyhedra. Sometimes prisms of
two different kinds occur in the same polychoron, giving the
“diprismato” polychora.

It sometimes happens that the vertex figure of a nonregular
uniform polychoron P is itself a uniform polyhedron. Then
rectifying and truncating P produce genuine uniform polychora,
suggesting the alternative names “rectified P” and
“truncated P” for such polychora. In three dimensions,
a similar situation occurs when the name “truncated
cuboctahedron” is used in place of “great
rhombicuboctahedron” (for example).

Some polychora derived from the self-dual regular polychora have
equal numbers of principal cells of two distinct kinds, so in
those cases I use the prefix “dis-” to combine two
identical numerical prefixes strung together, as in
*dispentachoron* instead of “pentapentachoron” and
*disicositetrachoron* instead of
“icositetraicositetrachoron.” If both kinds of
principal cells are congruent, however, then I sum the two
prefixes, as in *decachoron* for the uniform polychoron of
10 (= 5+5) truncated tetrahedra. In three dimensions, this
happens when the tetrahedron is rectified into a
“tetratetrahedron,” better known as the octahedron.

In earlier versions of this table, I used the syllable *kai*
to join numerical syllables into complete prefixes, as in
“icosikaitetrachoron” for the 24-cell. The *kai*
is equivalent to “and” in the number
“four-and-twenty.” Norman Johnson has persuaded me that
this is redundant and can safely be omitted to save space, except
in certain cases where using it avoids confusion. (For example,
the hecatonicosachoron has 120 congruent dodecahedral cells, but
a hecaton*kai*icosachoron might have 100 cells of one kind
and 20 cells of another.)

Some polychora may be distinguished by their principal cells,
such as the *truncated-octahedral tesseractihexadecachoron*
from the “ordinary” *tesseractihexadecachoron*,
and the *truncated-dodecahedral
diprismatohexacosihecatonicosachoron* from the
*rhombicosidodecahedral
diprismatohexacosihecatonicosachoron*. I also use
“small” (optionally) and “great,” the latter
denoting the presence of “great” uniform polyhedra
among the principal cells. (Among the Archimedean polyhedra, I
prefer, for example, the name “great
rhombicuboctahedron” to “truncated cuboctahedron,”
because the polyhedron is really a *rectified* truncated or
“rhombitruncated” cuboctahedron, not a *true*
truncated cuboctahedron. But, following the suggestion of Norman
Johnson, I use the “truncated” versions of the names
for the polyhedra in the List, because these versions predominate
in the literature.) As above, optional portions of names are
[bracketed].

The prismatic uniform polychora formed by the Cartesian product
of two polygons, which have no counterpart in three dimensions, I
call *duoprisms*. Elsewhere they are sometimes called
“double prisms.” We may use any two planar polygons in
E(4) to create a duoprism, as long as their planes do not lie in
the same hyperplane. In particular, if the two polygons are
regular, concentric, have the same edge length, and lie in
absolutely orthogonal planes, then the resulting duoprism will be
uniform (Archimedean). If two different regular polygons are
used, both their names are required to characterize the uniform
duoprism completely, as in *triangular-pentagonal duoprism*;
but if the two polygons are the same, only one name is needed, as
in *hexagonal duoprism*. Such single-polygonal duoprisms
have an extra symmetry, namely, interchanging the two “prism
rings,” that the double-polygonal duoprisms lack. The
tesseract is also the square duoprism, but because its basal
cells are congruent to its lateral cells, it has many more
symmetries than a mere square duoprism might be expected to
have.

I would imagine that to a hypothetical four-dimensional viewer
the visually most interesting uniform duoprisms would be those in
which the two polygons are both regular stars. These could become
intricately nonconvex, and as such they are slightly beyond the
scope of the present work. I will deal with them when I commence
working on the nonconvex uniform polychora.

Any polyhedron at all can be the base of a [four-dimensional]
prism, and if the polyhedron is uniform and the lateral faces of
the prism are all squares (in a general prism, they don’t
have to be), the prism itself is uniform, or Archimedean, as
well. The names of the remaining uniform prismatic polychora
derive directly and trivially from the names of their
three-dimensional base polyhedra. For example, the
four-dimensional prism based on a dodecahedron is simply a
*dodecahedral prism*. Calling such a figure a
“hyperprism” seems redundant, because the dimension of
a prism is always 1 greater than the dimension of its base
polytope. If the base is itself a *p*-gonal prism, then the
resulting figure is merely a square-*p*-gonal duoprism, but
if the base is a *p*-gonal antiprism, then the resulting
figure is a p*-gonal antiprismatic prism*, of which there
are an infinite number. The convex uniform four-dimensional
prisms based on non-prismatic uniform polyhedra comprise a finite
set of 17 besides the tesseract (or “cubic prism”).
Together with the anomalous non-Wythoffian antiprism and the 46
Platonic and convex Archimedean polychora, these bring to 64 the
total of convex Archimedean polychora that stand outside the
infinite prismatic families.

**Jonathan Bowers’s Names**

When Jonathan Bowers set out to find as many uniform polychora as
he could (not just the convex ones), he soon discovered that
there were hundreds of them (and now we know of thousands). Many
are extraordinarily complicated and their names, if written out
in Greekish (Hellenic) style, become quite unmanageably long. So
Jonathan has developed a system of acronymic names for them,
based on their much longer Greekish names. He abbreviates a
suitable Greekish name to an acronym, then inserts some vowels
among the acronym’s letters to make a pronounceable nonsense
word. This becomes the name of the polychoron.

For example, the Bowers name for the dispentachoron is
*rap*, short for *rectified pentachoron*, which is
Norman Johnson’s name for polychoron [2] in my table. The
tesseractihexadecachoron, polychoron [11] in my table, is also a
*rectified tesseract*, and the Bowers name for it is
*rit*. The Bowers system has the great advantage that the
names are usually short and sweet; the disadvantage, of course,
is that the name provides little information about the
polychoron’s structure unless you can reconstruct the
original Greekish name from the acronymic name.

In any case, no tabulation of the uniform polychora can be
complete without listing the Bowers names, so I added them to the
tables during the three weeks from December 12, 1999 through
January 2, 2000.

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

Four
Dimensional Figures Page: Return to initial page

List
Key: Explanations of the various List entries

Multidimensional Glossary: Explanations of some geometrical terms and
concepts

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

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

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

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

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

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

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