Abstract polytope
From Wikipedia, the free encyclopedia
The hemicube is constructed from the cube by treating opposite edges (likewise faces and corners) as really the same edge. It has 3 faces, 6 edges, and 4 corners.
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.
[edit] Examples
- The hemicube has vertices:
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- V = {1,2,3,4}
- edges:
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- E = {a = 12,b = 23,c = 13,d = 14,e = 24,f = 34}
- and faces:
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- F = {A = 1234 = abfd,B = 1243 = aefc, C = 1324 = cbed}
- with the following incidences:
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- 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 K4.
- Any ordinary polytope (cube, simplex) is an abstract polytope, of course.
[edit] See also
- 11-cell and 57-cell - two four-dimensional abstract regular polytopes
- Regular polytope
- Graded poset
[edit] References
- Peter McMullen, Egon Schulte, Abstract Regular Polytopes, Cambridge University Press, 2002. ISBN 0-521-81496-0
