cp's OEIS Frontend

It appears that the stereographic projection of the Platonic solids requires respectively 4, 6, 8, 6, 10, different colors to represent them. - Eric Desbiaux, Feb 15 2009

Sorted by number of 3-dimensional cells.

Also, number of vertices of the regular 4-dimensional polyhedra. - Douglas Boffey, Aug 12 2012

Sorted by number of 2-dimensional faces.

Also the number of edges in the regular 4-dimensional polytopes [Douglas Boffey, Aug 12 2012]

Sorted by number of 3-dimensional faces.

Andrew Weimholt suggests a related sequence, namely "total number of vertices in all finite n-dimensional regular polytopes, or 0 if the number is infinite, includes both convex and non-convex, beginning: 1, 2, 0, 106, 2453, 48, 83, 150, 281, 540, ... and writes that the sequence of just the non-convex cases (0, 0, -1, 56, 1680, 0, 0, 0, ..., where "-1" indicates infinity as zero is otherwise employed) is not as interesting, since it's all zeros from a(5) on.

Only included for completeness (see other sequences in crossrefs). This is one of those entries that I think should be included since it is such an elementary sequence in geometry, but where the terms do otherwise not really make a very spectacular sequence.

The ordering of the polytopes in the enumeration of the element counts is somewhat arbitrary, though based on invariants inherently associated with these polytopes. For example, ordering first by vertex-count, then by edge-count, then by face-count and then by cell-count might have been as good a choice as the one used for the sequence.

For n = 1: there is only a 0-dimensional simplex.

For n = 2: the two points may coincide or may form a 1-dimensional simplex.

For n = 3: the three points may coincide or may form a 2-dimensional simplex.

For n = 2^(k+1), where k is a positive integer: a(n) = k + (k+2) + (k-1) + (k-1) = 4*k: k polygons (one for each factor > 2), k+2 simplexes (one for each factor), k-1 cubes (one for each even factor > 4, the cubes for 2 and 4 are a simplex and polygon, respectively), and k-1 orthoplexes (one for each even factor > 4, orthoplexes with 1, 2, and 4 vertices are already counted).

For prime numbers greater than 3 (n = p > 3, where p is prime): a(n) is always 3:

(1) the 0-dimensional polytope (all points coinciding), (2) a 2-dimensional p-gon, where p is a prime n, and (3) a (p-1)-dimensional simplex.

For even numbers which are not powers of 2: a(n) = 2*(number of factors) + (number of even factors) - 3 + adjustments. The adjustments are as follows: -1 if n is a multiple of 3; -1 if n is a multiple of 4; +1 for each positive integer k such that 2^(k+2) is a factor of n; +1 for each factor of n which is in the set (12,20,24,120,600). With the exception of factors 1 and 2, every factor contributes a simplex and a polygon. Even factors add a third polytope which is an orthoplex. Factors 1 and 2 only add a zero-dimensional and one-dimensional simplex respectively and so a total of three is subtracted (-1 for each of factors 1 + 2 and -1 for the even factor 2). The polygon and the simplex to which the factor of 3 maps are identical leading to an adjustment of -1. The polygon and the 2-dimensional "cube" that a factor of 4 maps to are identical also leading to a -1 adjustment. Factors which are powers of 2 greater than 4 and factors which correspond to a polytope peculiar to 3 or 4 dimensions each add one more possible polytope.

For nonprime odd numbers which are multiples of 3: a(n) = 2*(the number of factors) - 2. Each factor maps to a polygon and a simplex, but for the factor 3 the polygon is the simplex, and the factor 1 maps to a single coincident point.

For nonprime odd numbers which are not multiples of 3: a(n) = 2*(the number of factors) - 1. Each factor > 1 maps to a polygon and a simplex and the factor 1 maps to a single coincident point.