(vertices, edges, faces, ..., facets) in an n-dimensional cross-polytope is thus given by (see binomial coefficient): There are many possible orthographic projections that can show the cross-polytopes as 2-dimensional graphs. Petrie polygon projections map the points into a regular 2n-gon or lower order regular polygons. A second projection takes the 2(n−1)-gon petrie polygon of the lower dimension, seen as a bipyramid, projected down the axis, with 2 vertices mapped into the center. The vertices of an axis-aligned cross polytope are all at equal distance from each other in the Manhattan distance (L1 norm). Kusner's conjecture states that this set of 2d points is the largest possible equidistant set for this distance. Generalized orthoplex Regular complex polytopes can be defined in complex Hilbert space called generalized orthoplexes (or cross polytopes), β = 2{3}2{3}...2{4}p, or ... Real solutions exist with p = 2, i.e. β = βn = 2{3}2{3}...2{4}2 = {3,3,..,4}. For p > 2, they exist in . A p-generalized n-orthoplex has pn vertices. Generalized orthoplexes have regular simplexes (real) as facets. Generalized orthoplexes make complete multipartite graphs, β make Kp,p for complete bipartite graph, β make Kp,p,p for complete tripartite graphs. β creates Kpn. 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. Related polytope families Cross-polytopes can be combined with their dual cubes to form compound polytopes: In two dimensions, we obtain the octagrammic star figure , In three dimensions we obtain the compound of cube and octahedron, In four dimensions we obtain the compound of tesseract and 16-cell. See also List of regular polytopes Hyperoctahedral group, the symmetry group of the cross-polytope Citations References pp. 121-122, §7.21. see illustration Fig 7.2B p. 296, Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n≥5) External links Polytopes Multi-dimensional geometry