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In geometry, a cross-polytope,〔 Chapter IV, five dimensional semiregular polytope ()〕 orthoplex,〔Conway calls it an n-orthoplex for ''orthant complex''. 〕 hyperoctahedron, or cocube is a regular, convex polytope that exists in any number of dimensions. The vertices of a cross-polytope are all the permutations of (±1, 0, 0, …, 0). The cross-polytope is the convex hull of its vertices. Its facets are simplexes of the previous dimension, while the cross-polytope's vertex figure is another cross-polytope from the previous dimension. The ''n''-dimensional cross-polytope can also be defined as the closed unit ball (or, according to some authors, its boundary) in the ℓ1-norm on R''n'': : In 1 dimension the cross-polytope is simply the line segment (+1 ), in 2 dimensions it is a square (or diamond) with vertices . In 3 dimensions it is an octahedron—one of the five convex regular polyhedra known as the Platonic solids. Higher-dimensional cross-polytopes are generalizations of these. The cross-polytope is the dual polytope of the hypercube. The 1-skeleton of a ''n''-dimensional cross-polytope is a Turán graph ''T''(2''n'',''n''). == 4 dimensions == The 4-dimensional cross-polytope also goes by the name hexadecachoron or 16-cell. It is one of six convex regular 4-polytopes. These 4-polytopes were first described by the Swiss mathematician Ludwig Schläfli in the mid-19th century. 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「Cross-polytope」の詳細全文を読む スポンサード リンク
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