Explanations of List Entries

HE LIST. The first line for each convex uniform polychoron is an ASCII/HTML version of the Coxeter-Dynkin notation that describes the position of a vertex of the polychoron relative to the glomeric tetrahedron (a glome is my term for a four-dimensional hypersphere: the four-dimensional element of the series that begins, “point, dyad, circle, sphere,…,” from the Latin glomus, a “ball,” or more specifically, a “ball of string,” referring to the Hopf fibration of this 3-mainfold) defined by the reflection realms of its symmetry group. There are altogether 15 different places for the vertex: at one of the 4 vertices, on one of the 6 edges, on one of the 4 faces, or entirely within the tetrahedron. For the two self-dual basal polychora, the additional symmetry of the glomeric tetrahedron reduces this to 9. In three dimensions, the snub polyhedra are found by placing vertices into alternating spherical (Möbius and Schwarz) triangles, but this procedure does not extend readily to n dimensions for n>3. There are almost always too many constraints on a vertex to ensure that all the resulting “snub facets” will be uniform polyhedra. (This is why, for example, the snub cuboctahedron cannot tile space in the company of regular tetrahedra and octahedra.) Archimedean snub polytopes and antiprisms are thus rare and anomalous in spaces above E(3). (Though if the condition of uniformity of just the snub facets be dropped, symmetric “pseudosnub” polytopes, with irregular “snub facets,” are plentiful.) In E(4), two convex snub polychora, neither of which is a “perfect” (i.e., chiral: occurring in left- and right-handed, or laevo and dextro forms) snub or antiprism analogue, exist. Also counting as snubs are the prisms trivially generated by the snub cuboctahedron and snub icosidodecahedron, which aren’t chiral, either. Indeed, it is interesting that, whereas two convex Archimedean polyhedra, the snub cuboctahedron and snub icosidodecahedron, are chiral, none of the convex uniform polychora is. (This contrasts with the situation for nonconvex uniform polychora, among which chiral snubs form a decided majority! But they are outside the scope of this tabulation.)

All 15 positions for a vertex yield different uniform polychora in the hexacosihecatonicosachoric assemblage, but in the tesseractihexadecachoric assemblage three of the 15 positions duplicate polychora derived from the icositetrachoron, because of the close relationship between the tesseract, hexadecachoron, and icositetrachoron. This is similar to the situation in three dimensions, where several of the tetrahedral uniform polyhedra are also found among the cuboctahedral polyhedra. All 9 positions for the (self-dual) pentachoric and icositetrachoric assemblages yield distinct uniform polychora, and there is even a “snub analogue” in the icositetrachoric assemblage, whose existence ultimately stems from the observation that the regular icosahedron also happens to be the snub tetrahedron.

HTML forces compromises in representing the Coxeter-Dynkin graphic notation. Here (o) denotes a circled node, o denotes an uncircled node, and ( ) denotes an empty node (for snub and snublike polytopes). A full explanation of the notation would take too long for inclusion here, but I may supply one if there’s a demand for it.

The second line, indented, gives my suggested “Greekish” (Hellenized) name for the polychoron and, in brackets, its number in the List unless the polychoron is listed in a more appropriate section elsewhere or is a member of an infinite set.

Following the “Greekish” canonical name are alternative names for the polychoron that have previously been published or have arisen in recent discussions about names for these figures. Norman W. Johnson has named the Wythoffian transformations represented by the nine symmetrically distinct Coxeter-Dynkin diagrams and applied them to the basal regular polychora in creating his own names for these figures. They appear here with his permission. Norman’s naming philosophy has the great advantage of keeping the polychoric names concise, so I expect his names, in some form, will eventually prevail over the formal “Greekish” names I’ve invented. All this and much more will appear in his forthcoming book on uniform polytopes from Cambridge University Press. Also included among the alternative names for the regular polychora are ones devised by John Horton Conway, which are used with his permission. I have taken the liberty of naturally combining Johnson’s adjectives with Conway’s nouns to arrive at further alternative names for the Archimedean polychora. As further alternative names, I have added Jonathan Bowers’s acronymic names for the polychora and their derivations (see the Nomenclature section for an explanation of how these are constructed).

Following the alternative names is the symmetry group of the polychoron, but only if it differs from the usual symmetry group of the other polychora in its assemblage. The names of the groups are straightforward extensions of the names of the groups in three dimensions. For example, the symmetry group of the dodecahedron, icosahedron, icosidodecahedron, and so forth, is denoted [3,5] or [5,3] and is called the diploid icosahedral group. Its four-dimensional analogue is [3,3,5] or [5,3,3], the diploid hexacosichoric group.

Next comes the Schläfli symbol (and alternatives) for the polychoron, and following the Schläfli symbol is a short table of the elements of dimension 3 through 0 (cells, faces, edges, and vertices) that comprise the polychoron, together with self-explanatory comments on the roles of the faces in the polychoron. All elements are either regular or uniform polytopes, so adjectives such as “equilateral,” “regular,” “uniform,” and “Archimedean” are understood here.

Finally, following the elements is a short description of the polychoron’s vertex figure, which is usually an irregular polyhedron whose faces indicate the disposition of the cells around each of the polychoron’s vertices. (In any uniform polytope, the vertex figures are all congruent, so we speak of the vertex figure of a uniform polytope. In particular, the vertex figure of a uniform polychoron is the polyhedron formed by all the vertices connected to a given vertex by edges. Exercise: These vertices all lie in the same realm. Why? Hint: They not only lie in the same realm, they all lie on the surface of a sphere. A realm is a three-dimensional hyperplane in any space of four or more dimensions, as in the series point, line, plane, realm, etc.) The faces of the vertex figure are themselves the vertex figures of the cells: In the case of convex uniform polychora, these are most often triangles and quadrilaterals, occasionally pentagons. The tabulated lengths of the vertex figure’s edges, which for p>3 are the (shortest) diagonals of the p-gonal polychoric faces, are based on a unit edge for the polychoron. As might be expected, “sqrt(p)” denotes the square root of p; other notations should be self-explanatory.

For completeness, I list the prismatic Archimedean polychora. Indeed, these (Section 6) are a bit overrepresented here, but I thought it useful to cover the Coxeter-Dynkin notation for all the different cases. Prismatic uniform polytopes (in any dimension) may be characterized as those whose Coxeter-Dynkin graphs are disconnected.