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In geometry, an orthant〔(Advanced linear algebra By Steven Roman, Chapter 15 )〕 or hyperoctant is the analogue in ''n''-dimensional Euclidean space of a quadrant in the plane or an octant in three dimensions. In general an orthant in ''n''-dimensions can be considered the intersection of ''n'' mutually orthogonal half-spaces. By permutations of half-space signs, there are 2''n'' orthants in ''n''-dimensional space. More specifically, a closed orthant in R''n'' is a subset defined by constraining each Cartesian coordinate to be nonnegative or nonpositive. Such a subset is defined by a system of inequalities: :ε1''x''1 ≥ 0 ε2''x''2 ≥ 0 · · · ε''n''''x''''n'' ≥ 0, where each ε''i'' is +1 or −1. Similarly, an open orthant in R''n'' is a subset defined by a system of strict inequalities :ε1''x''1 > 0 ε2''x''2 > 0 · · · ε''n''''x''''n'' > 0, where each ε''i'' is +1 or −1. By dimension: #In one dimension, an orthant is a ray. #In two dimensions, an orthant is a quadrant. #In three dimensions, an orthant is an octant. John Conway defined the term ''n''-orthoplex from orthant complex as a regular polytope in n-dimensions with 2''n'' simplex facets, one per orthant.〔J. H. Conway, N. J. A. Sloane, ''The Cell Structures of Certain Lattices'' (1991) ()〕 ==See also== * Cross polytope (or orthoplex) - a family of regular polytopes in n-dimensions which can be constructed with one simplex facets in each orthant space. * Measure polytope (or hypercube) - a family of regular polytopes in n-dimensions which can be constructed with one vertex in each orthant space. * Orthotope - Generalization of a rectangle in n-dimensions, with one vertex in each orthant. 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「orthant」の詳細全文を読む スポンサード リンク
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