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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 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.

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, $\{5,3,3\}\,$ 