Naming the convex uniform polychora

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 hecatonkaiicosachoron 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.