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In mathematics, an 'abstract polytope' is a combinatorial structure with properties similar to those shared by a more classical polytope. Abstract polytopes include the polygons, the platonic solids and other polyhedra, tessellations of the plane and higher-dimensional spaces, and of other manifolds such as the torus or projective plane, and many other objects (such as the 11-cell and the 57-cell) that don't fit well into any "normal" space.
More precisely, an abstract polytope is an incidence geometry defined on different types of objects,
satisfying certain axioms, supposed to represent the vertices, edges and so on — the faces — of the polytope. A linear "order" is imposed on the set of types.
More precisely, an abstract polytope P is a poset with a rank function (having range {-1, 0, ..., n} that satisfies the following four properties:
1. It has a unique minimal face (F_-1) and a unique maximal face (F_n)
2. Every flag (i.e maximal chain) has exactly n+2 elements
Given faces F,G of P with F < G, the section G/F = {H | F 3. It is strongly connected (every section is connected)
4. All sections of rank one have a diamond shape

Contents

Examples

See also

References

Examples

★ The hemicube has vertices:
::''V'' = {1,2,3,4}
:edges:
::''E'' = {a = 12,b = 23,c = 13,d = 14,e = 24,f = 34}
:and faces:
::''F'' = {A = 1234 = abfd,B = 1243 = aefc, C = 1324 = cbed}
:with the following incidences:
::1a,1c,1d,1A,1B,1C,2a,2b,2e,2A,2B,2C,3b,3c,3f,3A,3B,3C,
::4d,4e,4f,4A,4B,4C,aA,aB,bA,bC,cB,cC,dA,dC,eB,eC,fA,fB.
:Its skeleton is the complete graph ''K''₄.

★ Any ordinary polytope (cube, simplex) is an abstract polytope, of course.

See also

★ 11-cell and 57-cell - two four-dimensional abstract regular polytopes

★ Regular polytope

★ Graded poset

References

★ Peter McMullen, Egon Schulte, Abstract Regular Polytopes, Cambridge University Press, 2002. ISBN 0-521-81496-0