Polytopes & Foldouts

Jeremy Kahn

Bizarre patterns

If we look at the number of possible regular polytopes in n-space where n = {0 to infinity}), we see a rather bizarre series of numbers:

`(1, 1, infinity, 5, 6, 3, 3, 3, 3, 3, 3, 3, ...)`

which is a rather bizarre series of numbers. One would expect something more elegant, like some asymptotic system that rises to infinity at x = 1. But instead we find this monstrosity.

Why have I applied 0 and 1 to the values of n=1 and n=0? What is a regular shape in 1-space? A segment: one of the basic definitions of regularity is that all the vertices look the same with respect to the number and regularity of lines entering it. Since any shape with more than two vertices would leave at least one middle "vertex" different from the end vertices in that it has vertices on both sides of it. A regular shape in 0-space has only one point, which is by definition both the only shape possible in 0-space and also the only regular shape: its single vertex looks the same with respect to all the other vertices, as well.

Simplexes and "Coning"

One of my favorite parts of higher-dimensional math, these shapes are the only poly-somethings identified without three-chauvinism. A tetrahedron is a 3-simplex, a triangle a 2-simplex, a line a 1-simplex (and a 1-cube; it is critical to the "bizarre pattern" above that the first two "1"s listed are the only possible shapes in those dimensional spaces).

"Coning," as we described it in class (in opposition to "extrusion" or something similar) is increasing the dimensionality needed to describe an object A by choosing a point P on a line L perpendicular to the n-space that contains the entire object A ( such that the point P is not in the n-space) and connecting all points in A to the point P. By starting with a point and repeatedly "coning", we can generate examples of every n-simplex. (The regular ones require a little planning to place P at the right point along L). This sequence is an interesting parallel to the n-cube, which starts with a point and repeatedly "extrudes." Since both procedures can be performed regardless of dimension, it becomes a bit clearer why the n-cube and n-simplex must alwa ys exist in any dimension, though the rarity of other shapes is still odd.

Linguistic Digression

I like the name for pentatope because it is penta, five, + topos, shape, object (both Greek, too, so it is a word the old-school linguists would even approve of). Indeed, there are five tetrahedra involved, and one can count them: One points in off of each face, and there are four faces, so four of them go in to the middle. The fifth is of course the outside one itself, just as the eighth cube in a shadow projection of a hypercube is the outside cube that seems to contain the ent ire object.