Regular polytope numbers

For each regular polytope (in a Euclidean space in $\mathbb {R} ^{d}\,$,) we may associate a sequence of regular polytope numbers.

There are two classes of regular polytope numbers:

• standard regular d-dimensional polytope numbers (with all layers sharing a common vertex and all the (d-1)-dimensional facets sharing that vertex) where:
  • 2-dimensional regular polygonal numbers share 2 sides
  • d-dimensional simplicial numbers share 3 (d-1)-dimensional simplicial facets
  • d-dimensional orthotope numbers share d (d-1)-dimensional orthotopic facets
  • d-dimensional orthoplicial numbers share 2^d-1 (d-1)-dimensional simplicial facets
  • Dodecahedral numbers share 3 pentagonal faces
  • Icosahedral numbers share 5 triangular faces
  • 24-cell numbers share 6 octahedral cells
  • Hyperdodecahedral (120-cell) numbers share 4 dodecahedral cells
  • Hypericosahedral (600-cell) numbers share 20 tetrahedral cells
• centered regular d-dimensional polytope numbers (with all the n layers centered around a central point, which corresponds to n = 0)

A classical theorem from combinatorial geometry classifies all the regular polytopes in Euclidean spaces.

All figurate numbers are accessible via this structured menu: Classifications of figurate numbers

Schläfli's theorem

Theorem (Schläfli). The only possible Schläfli symbols for a regular polytope in the Euclidean space in $\mathbb {R} ^{d}\,$ are given by the following list:

$d=2:\{n\}\,$, where $n\geq 3\,$ is an arbitrary integer;

$d=3:\{3,3\},\{3,4\},\{4,3\},\{3,5\},\{5,3\};\,$

$d=4:\{3,3,3\},\{3,3,4\},\{4,3,3\},\{3,4,3\},\{3,3,5\},\{5,3,3\};\,$

$d\geq 5:\{3^{d-1}\},\{3^{d-2},4\},\{4,3^{d-2}\}.\,$

For each symbol in the list, there exists a regular polytope with that symbol, and two regular polytopes with the same symbols are similar.

Number of regular convex polytopes in d-dimensional space

Consequently, the number of regular convex polytopes in d-dimensional space, d ≥ 0, gives the sequence:

$\{1,1,\infty ,5,6,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,...\}\,$

A060296 Number of regular convex polytopes in n-dimensional space, or -1 if the number is infinite.

$\{1,1,-1,5,6,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,...\}\,$

Classification of all the regular polytopes in Euclidean spaces

For dimension 0, we have:

• the point

For dimension 1, we have:

• the line segment

For each dimension d ≥ 2, we have:

• $\alpha _{d}\,$, or the regular d-simplex, $\{3^{d-1}\}\,$
• $\beta _{d}\,$, or the regular d-orthoplex, $\{3^{d-2},4\}\,$
• $\gamma _{d}\,$, or the regular d-orthotope, $\{4,3^{d-2}\}\,$

For dimension 2, we also have:

• the regular n-gon, n ≥ 3 (in Euclidean space) (the 1-gon, or henagon, and the 2-gon, or digon, are not possible in Euclidean space)

For dimension 3, we also have:

• the icosahedral complex, or 20-face complex, $\{3,5\}\,$
• the dodecahedral complex, or 12-face complex, $\{5,3\}\,$

For dimension 4, we also have:

• the 24-cell complex, $\{3,4,3\}\,$
• the hypericosahedral complex, or 600-cell complex, $\{3,3,5\}\,$
• the hyperdodecahedral complex, or 120-cell complex, $\{5,3,3\}\,$

References

• H. S. M. Coxeter, Regular polytopes, Dover, 1973. (Google Books)

• Hyun Kwang Kim, On Regular Polytope Numbers, Proceedings of the American Mathematical Society, Vol. 131, No. 1 (Jan., 2003), pp. 65-75.

External links

• S. Plouffe, Approximations de Séries Génératrices et Quelques Conjectures, Dissertation, Université du Québec à Montréal, 1992.
• S. Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 1992.
• Herbert S. Wilf, generatingfunctionology, 2^nd ed., 1994.