Cross-polytope
In geometry, a cross-polytope,[1] hyperoctahedron, orthoplex,[2] staurotope,[3] or cocube is a regular, convex polytope that exists in n-dimensional Euclidean space.The vertices of a cross-polytope can be chosen as the unit vectors pointing along each co-ordinate axis – i.e. all the permutations of (±1, 0, 0, ..., 0).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 Rn: In 1 dimension the cross-polytope is simply the line segment [−1, +1], in 2 dimensions it is a square (or diamond) with vertices {(±1, 0), (0, ±1)}.The hypervolume of the n-dimensional cross-polytope is For each pair of non-opposite vertices, there is an edge joining them.More generally, each set of k + 1 orthogonal vertices corresponds to a distinct k-dimensional component which contains them.The number of k-dimensional components (vertices, edges, faces, ..., facets) in an n-dimensional cross-polytope is thus given by (see binomial coefficient): The extended f-vector for an n-orthoplex can be computed by (1,2)n, like the coefficients of polynomial products.An orthogonal projection can be defined that maps all the vertices equally-spaced on a circle, with all pairs of vertices connected, except multiples of n. The regular polygon perimeter in these orthogonal projections is called a petrie polygon.
squareoctahedron16-cell5-orthoplexgeometryregularconvex polytopedimensional Euclidean spacesimplexesvertex figureconvex hullunit ballℓ1-normline segmentpolyhedraPlatonic solidsdual polytopehypercubeskeletonTurán graphconvex regular 4-polytopes4-polytopesLudwig Schläfliregular polytopeCoxetersimplexinfinite tessellations of hypercubessimplicesvertex figuresSchläfli symbolbinomial coefficientf-vectororthographic projectionsPetrie polygonbipyramidSchläfliCoxeter-DynkindiagramsVertices6-orthoplex7-orthoplex8-orthoplex9-orthoplex10-orthoplexManhattan distanceL1 normKusner's conjectureequidistant setcomplexHilbert spacefacetscomplete multipartite graphscomplete bipartite graphorthogonal projectionregular polygon{3,3,4}{3,3,3,4}{3,3,3,3,4}octagrammiccompound of cube and octahedronList of regular polytopesHyperoctahedral groupMcMullen, PeterWeisstein, Eric W.MathWorldCoxeter, H.S.M.Regular PolytopesDimensionVector spaceEuclidean spaceAffine spaceProjective spaceFree moduleManifoldAlgebraic varietySpacetimeLebesgue coveringInductiveHausdorffMinkowskiFractalDegrees of freedomPolytopesshapesHyperplaneHypersurfaceHyperrectangleDemihypercubeHypersphereHyperpyramidHypercomplex numbersCayley–Dickson constructionn-dimensionsHyperspaceCodimensionuniform polytopesTriangleHexagonPentagonUniform polyhedronTetrahedronDemicubeDodecahedronIcosahedronUniform polychoronPentachoronTesseractDemitesseract24-cell120-cell600-cellUniform 5-polytope5-simplex5-cube5-demicubeUniform 6-polytope6-simplex6-cube6-demicubeUniform 7-polytope7-simplex7-cube7-demicubeUniform 8-polytope8-simplex8-cube