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Regular polytope numbers
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 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.^{[1]}
All figurate numbers are accessible via this structured menu: Classifications of figurate numbers
Contents
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,
Notes
- ↑ H.S.M.Coxeter, Regular Polytopes, Dover Publications Inc., New York (1973.)
- ↑ Weisstein, Eric W., Schläfli Symbol, From MathWorld--A Wolfram Web Resource.
- ↑ Weisstein, Eric W. ,Digon, From MathWorld--A Wolfram Web Resource.
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.