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Regular polytope numbers

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For each regular polytope (in a Euclidean space in ,) 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 2d-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.[1]

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

Schläfli's theorem

Theorem (Schläfli). The only possible Schläfli symbols[2] for a regular polytope in the Euclidean space in are given by the following list:

, where is an arbitrary integer;

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:

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

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:

  • , or the regular d-simplex,
  • , or the regular d-orthoplex,
  • , or the regular d-orthotope,

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,[3] are not possible in Euclidean space)

For dimension 3, we also have:

  • the icosahedral complex, or 20-face complex,
  • the dodecahedral complex, or 12-face complex,

For dimension 4, we also have:

  • the 24-cell complex,
  • the hypericosahedral complex, or 600-cell complex,
  • the hyperdodecahedral complex, or 120-cell complex,


  1. H.S.M.Coxeter, Regular Polytopes, Dover Publications Inc., New York (1973.)
  2. Weisstein, Eric W., Schläfli Symbol, From MathWorld--A Wolfram Web Resource.
  3. Weisstein, Eric W. ,Digon, From MathWorld--A Wolfram Web Resource.


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

External links