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 (Â±1, 0, 0, â¦, 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 &#x2113;1-norm on rn: :\{x\in\mathbb r^n : \|x\|_1 \le 1\}. in 1 dimension the cross-polytope is simply the line segment [&minus;1, +1], in 2 dimensions it is a square (or diamond) with vertices {(Â±1, 0), (0, Â±1)}. in 3 dimensions it is an octahedron&mdash;one of the five convex regular polyhedra known as the platonic solids. higher-dimensional cross-polytopes are generalizations of these. {| style="margin-left: auto; margin-right: auto" |style="text-align: center;"|120px|a 2-dimensional cross-polytope |width="50px"| |style="text-align: center;"|120px|a 3-dimensional cross-polytope |width="50px"| |style="text-align: center;"|120px|a 4-dimensional cross-polytope |- |style="text-align: center;"|2 dimensionssquare | |style="text-align: center;"|3 dimensionsoctahedron | |style="text-align: center;"|4 dimensions16-cell |} the cross-polytope is the dual polytope of the hypercube. the 1-skeleton of a n-dimensional cross-polytope is a turÃ¡n graph t(2n,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 polychora were first described by the swiss mathematician ludwig schlÃ¤fli in the mid-19th century. == higher dimensions == the cross polytope family is one of three regular polytope families, labeled by coxeter as Î²n, the other two being the hypercube family, labeled as Î³n, and the simplices, labeled as Î±n. a fourth family, the infinite tessellations of hypercubes, he labeled as Î´n'
Four-dimensional_geometry Multi-dimensional_geometry Polytopes