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In geometry, a ''d''-dimensional simple polytope is a ''d''-dimensional polytope each of whose vertices are adjacent to exactly ''d'' edges (also ''d'' facets). The vertex figure of a simple ''d''-polytope is a (''d'' − 1)-simplex.〔''Lectures on Polytopes'', by Günter M. Ziegler (1995) ISBN 0-387-94365-X 〕 They are topologically dual to simplicial polytopes. The family of polytopes which are both simple and simplicial are simplices or two-dimensional polygons. For example, a ''simple polyhedron'' is a polyhedron whose vertices are adjacent to 3 edges and 3 faces. And the dual to a simple polyhedron is a ''simplicial polyhedron'', containing all triangular faces.〔Polyhedra, Peter R. Cromwell, 1997. (p.341)〕 Micha Perles conjectured that a simple polytope is completely determined by its 1-skeleton; his conjecture was proven in 1987 by Blind and Mani-Levitska.〔.〕 Gil Kalai provided a later simplification of this result based on the theory of unique sink orientations.〔.〕 == Examples == In three dimensions: * Prisms * Platonic solids: * * tetrahedron, cube, dodecahedron * Archimedean solids: * * truncated tetrahedron, truncated cube, truncated octahedron, truncated cuboctahedron, truncated dodecahedron, truncated icosahedron, truncated icosidodecahedron * Goldberg polyhedron and Fullerenes: * * chamfered tetrahedron, chamfered cube, chamfered dodecahedron ... * In general, any polyhedron can be made into a simple one by truncating its vertices of valence 4 or higher. * * truncated trapezohedrons In four dimensions: * Regular: * * 120-cell, Tesseract * Uniform 4-polytope: * * truncated 5-cell, truncated tesseract, truncated 24-cell, truncated 120-cell * * all bitruncated, cantitruncated or omnitruncated 4-polytopes * * duoprisms In higher dimensions: * ''d''-simplex * hypercube * associahedron * permutohedron * all omnitruncated polytopes 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「simple polytope」の詳細全文を読む スポンサード リンク
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