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 polyhedron, 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.

= Examples =

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 =

= References =