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Regular polytope
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===Abstract polytopes=== {{main|Abstract polytope}} [[Image:Hemicube.svg|right|frame|The [[Hemi-cube (geometry)|Hemicube]] is derived from a cube by equating opposite vertices, edges, and faces. It has 4 vertices, 6 edges, and 3 faces.]] Grünbaum also discovered the [[11-cell]], a four-dimensional [[Dual polyhedron|self-dual]] object whose facets are not icosahedra, but are "hemi-icosahedra" — that is, they are the shape one gets if one considers opposite faces of the icosahedra to be actually the ''same'' face {{harv|Grünbaum|1976}}. The hemi-icosahedron has only 10 triangular faces, and 6 vertices, unlike the icosahedron, which has 20 and 12. This concept may be easier for the reader to grasp if one considers the relationship of the cube and the hemicube. An ordinary cube has 8 corners, they could be labeled A to H, with A opposite H, B opposite G, and so on. In a hemicube, A and H would be treated as the same corner. So would B and G, and so on. The edge AB would become the same edge as GH, and the face ABEF would become the same face as CDGH. The new shape has only three faces, 6 edges and 4 corners. The 11-cell cannot be formed with regular geometry in flat (Euclidean) hyperspace, but only in positively curved (elliptic) hyperspace. A few years after Grünbaum's discovery of the [[11-cell]], [[H. S. M. Coxeter]] independently discovered the same shape. He had earlier discovered a similar polytope, the [[57-cell]] (Coxeter 1982, 1984). By 1994 Grünbaum was considering polytopes abstractly as combinatorial sets of points or vertices, and was unconcerned whether faces were planar. As he and others refined these ideas, such sets came to be called '''[[abstract polytope]]s'''. An abstract polytope is defined as a [[partially ordered set]] (poset), whose elements are the polytope's faces (vertices, edges, faces etc.) ordered by ''containment''. Certain restrictions are imposed on the set that are similar to properties satisfied by the classical regular polytopes (including the Platonic solids). The restrictions, however, are loose enough that regular tessellations, hemicubes, and even objects as strange as the 11-cell or stranger, are all examples of regular polytopes. A geometric polytope is understood to be a ''realization'' of the abstract polytope, such that there is a one-to-one mapping from the abstract elements to the corresponding faces of the geometric realisation. Thus, any geometric polytope may be described by the appropriate abstract poset, though not all abstract polytopes have proper geometric realizations. The theory has since been further developed, largely by {{harvtxt|McMullen|Schulte|2002}}, but other researchers have also made contributions. ====Regularity of abstract polytopes==== Regularity has a related, though different meaning for [[abstract polytope]]s, since angles and lengths of edges have no meaning. The definition of regularity in terms of the transitivity of flags as given in the introduction applies to abstract polytopes. Any classical regular polytope has an abstract equivalent which is regular, obtained by taking the set of faces. But non-regular classical polytopes can have regular abstract equivalents, since abstract polytopes do not retain information about angles and edge lengths, for example. And a regular abstract polytope may not be realisable as a classical polytope. ''All polygons'' are regular in the abstract world, for example, whereas only those having equal angles and edges of equal length are regular in the classical world. ====Vertex figure of abstract polytopes==== The concept of ''vertex figure'' is also defined differently for an [[abstract polytope]]. The vertex figure of a given abstract ''n''-polytope at a given vertex ''V'' is the set of all abstract faces which contain ''V'', including ''V'' itself. More formally, it is the abstract section : ''F''<sub>''n''</sub> / ''V'' = {''F'' | ''V'' ≤ ''F'' ≤ ''F''<sub>''n''</sub>} where ''F''<sub>''n''</sub> is the maximal face, i.e. the notional ''n''-face which contains all other faces. Note that each ''i''-face, ''i'' ≥ 0 of the original polytope becomes an (''i'' − 1)-face of the vertex figure. Unlike the case for Euclidean polytopes, an abstract polytope with regular facets and vertex figures ''may or may not'' be regular itself – for example, the square pyramid, all of whose facets and vertex figures are regular abstract polygons. The classical vertex figure will, however, be a realisation of the abstract one.
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