Over the years, a kind of unofficial terminology has evolved among people who play with polytopes. For example, in increasing order of dimensionality, there are four venerable names for the different kinds of polytope elements: vertex (plural: vertices), edge, face, and cell. This series has recently been indefinitely extended with teron, peton, exon, etc. There are now also names for polytope elements going the
other way, in decreasing order of dimensionality: facet, ridge, and peak. A
facet of an
n-dimensional polytope is any of its (
n-1)-dimensional elements, its “flat sides.” In four-space (and only in four-space), facets are three-dimensional polytopes (polyhedra) more usually called “cells.” A
ridge is an (
n-2)-dimensional element (if there is one) of an
n-dimensional polytope, and a
peak is an (
n-3)-dimensional element (if there is one). In four-space, ridges are the faces (not to be confused with “facets”!), while 4D peaks are the edges. In any polytope of dimension greater than 2 (in which facets, ridges, and peaks all exist), exactly two facets adjoin at each ridge, whereas three or more facets must always be incident around a peak.
Star-polytopes in general, and star-polychora in particular, need some more terms. I’ve already used the term “surcell” in previous posts. The surcell was named as a pun on the word “surface”: the “-face” part is two-dimensional, so to obtain the analogous three-dimensional term, change “-face” to “-cell.” In
n dimensions, the corresponding term I use is “surtope.” This then begets the infinitude of names of the other kinds of surtopes. In the present 4D context, "surcell" denotes the simple polychoron (no intersecting cells, no self-intersecting faces, etc.; but coincident elements may be OK in certain exceptional circumstances) formed by just the
external cellets of a star-polychoron. In the plane, on the other hand, we already have a term that means the same as a “suredge” (of a polygom), namely,
periphery. Thus we have, in order of increasing dimension starting with 2, the following sequence of surtopes: periphery, surhedron (or surface), surchoron (or surcell), surteron, surpeton, surexon, etc. Inasmuch as the term “surface” may apply to any kind of two-dimensional manifold, flat or curved, such as the surface of a sphere, I created the term “surhedron” especially for the external surface of a polyhedron. Likewise, a “surcell” can be any three-dimensional manifold, such as the surcell of a glome (sphere in four dimensions), so I created the term “surchoron” specifically for the external surcell of a polychoron. I hope this doesn’t come across as too nitpicky or confusing. I should have been using "surchoron" instead of "surcell" all along.
The surtope of a simple polytope is the polytope itself, of course. But the surtope of a star-polytope is not the same figure as the star-polytope itself. It’s just the simple “shell” of exterior topelets that exactly encloses the star-polytope. A
cellet, by the way, is a piece of a cell bounded by other cells of the polytope that intersect or adjoin the cell. We add the suffix -let to the name of a polytope element to get the name of a piece of the element, as in “edgelet, “facelet,” “facetlet,” “ridgelet,” “peaklet,” “teronlet,” “petonlet,” and so forth. “Topelet” would be the general term.
The surhedron of a star-polyhedron, the surchoron of a star-polychoron, and in general the surtope of a star-polytope are usually polytopes with a lumpy, bumpy topography loaded with several kinds of peaks and valleys. The term “peak” itself is already in use (see above), so I use the term
point, as in “a five-pointed star has five points” (not to be confused with the usual concept of a mathematical/geometrical point, or with a vertex) for a configuration of facetlets about a vertex.. This configuration must be, loosely speaking, “lumplike” (locally convex) so that the point “sticks out” from within its neighborhood. In four-space, points are polyhedral, defined by the surhedron of the vertex figure at that vertex. So, in particular, there must be at least four cells/cellets at a point, and we speak of tetrahedral points, dodecahedral points, and so forth. Depending on the vertex-figure polyhedron, points may have lateral grooves and even holes in them.
The opposite of a point is a
dimple. Whereas a point “sticks out,” a dimple “sticks in.” Otherwise, a dimple is just like a point. In four-space, dimples, like points, are polyhedral. A ridge that “sticks out” I call an
arete, which is a kind of sharply defined ridge in mountaineering. And a ridge that “sticks in” I call, simply, a
valley. In 4D and higher spaces, a peak that “sticks out” I call a
crest, and a peak that “sticks in” I call, again simply, a
depression. Here’s a little chart summarizing these terms for 3D and 4D:
Polyhedra:
Points (convex vertex) and dimples (concave vertex)
Aretes (convex edge) and valleys (concave edge)
Two facelets (
i.e., faces of a surhedron) meet at an arete or a valley
Three or more facelets meet about a point or a dimple
Polychora:
Points (convex vertex) and dimples (concave vertex)
Crests (convex edge) and depressions (concave edge)
Aretes (convex face) and valleys (concave face)
Two cellets (
i.e., cells of a surchoron) meet at an arete or valley
Three or more cellets meet in a crest or depression
Four or more cellets meet about a point or dimple
To get specific, the stellated hecatonicosachoron’s surchoron has 120 dodecahedral points separated by 720 pentagonal valleys. The great hecatonicosachoron’s surchoron is bult by sticking 720 stellachunks into the valleys, which raises the 720 valleys into crests that surround 1200 equit depressions (three congruent cellets around an edge). And to build the icosahedral hecatonicosachoron’s surchoron we stick 1200 stellachunks (of a shape to be described below) into the equit depressions. This models the D aggrandizement of the hecatonicosachoron, which is a regular polychoron whose cells are 120 icosahedra. Its Schlaefli symbol is {5,3,5/2}.
Restating the situation, the B aggrandizement of the hecatonicosachoron was created by adding 120 dodecahedral pyramids (the points) onto the underlying A aggrandizement (the hecatonicosachoron itself). This covered the hecatonicosachoron completely and left a surcell composed of 1440 pentagonal pyramids (two per valley), which models the exterior of the stellated hecatonicosachoron. The C aggrandizement was modeled by adding 720 pentagonal double pyramids to this surchoron, into the valleys between the dodecahedral points. This covered the B aggrandizement completely and left a surchoron composed of 3600 tetrahedral wedges (3D "triangular double pyramids"), which models the exterior of the great hecatonicosachoron.
The D aggrandizement is created by aggrandizing the 120 great dodecahedral cells of the great hecatonicosachoron into the regular icosahedra that have the same vertices and edges. We model this by adding 1200 flattened pentachoric stellachunks into the triangular depressions of the C aggrandizement surchoron. Here is a picture of the net of one such stellachunk.
The three tetrahedral-wedge-shaped cells are colored teal; these are the same shape (and color) as the cellets of the underlying great hecatonicosachoron in my earlier post. They blend out when the D stellalayer is added, leaving a surchoron composed of 2400 (two per stellachunk) light yellow flattened tetrahedra. (In the picture of the net, one of them is completely covered by the other cells.) These have the shape of the flattened tetrahedra that may be placed into the dimples of a great dodecahedron to change it into a regular icosahedron. The D stellalayer models the
icosahedral hecatonicosachoron, so called because it has 120 interpenetrating icosahedra as its cells (of which just those flattened tetrahedra would be visible on the exterior to a 4D model maker).
All this is fairly well known to us polytopers. But now, here is “humanity’s first look” at the compound of five icosahedral hecatonicosachora:
The picture shows the usual level 0.555 3D cross section of the compound, taken by a sectioning realm perpendicular to an axis of icosahedral symmetry. The figure is colored in the usual five colors, one for each component. The closeup
zooms in on a center of fivefold symmetry. Note that a rotation of 2pi/5 of the cross section about this or any fivefold axis will send each component into a component of a different color: All five components are congruent. I was surprised at how complicated this section turned out to be as a physical polyhedron model: Stella4D calculated that it requires a total of 3860 nets to build, although many of them are one- and two-piece “sniv nets.” I cannot be certain from the account in
Regular Polytopes, but I’m pretty sure it was my old mentor, H. S. M. Coxeter himself, who discovered the vertex-regular and cell-regular compounds of five and ten regular star-polychora. Sadly, he died some four years ago,
just before desktop computer technology had advanced to the point where sectioning movies, in full color, could easily be made. He would certainly have been thrilled to see his figures “in the flesh.