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9-cube. In geometry, a 9-cube is a nine- dimensional hypercube with 512 vertices, 2304 edges, 4608 square faces, 5376 cubic cells, 4032 tesseract 4-faces, 2016 5-cube 5-faces, 672 6-cube 6-faces, 144 7-cube 7-faces, and 18 8-cube 8-faces . It can be named by its Schläfli symbol {4,3 7 }, being composed of three 8-cubes around each 7-face.
Rectified 9-cubes. In nine-dimensional geometry, a rectified 9-cube is a convex uniform 9-polytope, being a rectification of the regular 9-cube . There are 9 rectifications of the 9-cube. The zeroth is the 9-cube itself, and the 8th is the dual 9-orthoplex. Vertices of the rectified 9-cube are located at the edge-centers of the 9-orthoplex.
convex. In geometry, a demienneract or 9-demicube is a uniform 9-polytope, constructed from the 9-cube, with alternated vertices removed. It is part of a dimensionally infinite family of uniform polytopes called demihypercubes . E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as HM 9 for a 9-dimensional half measure ...
Equivalently, a convex set or a convex region is a subset that intersects every line into a single line segment (possibly empty). For example, a solid cube is a convex set, but anything that is hollow or has an indent, for example, a crescent shape, is not convex. The boundary of a convex set in the plane is always a convex curve.
In geometry, a hypercube is an n -dimensional analogue of a square ( n = 2) and a cube ( n = 3 ). It is a closed, compact, convex figure whose 1- skeleton consists of groups of opposite parallel line segments aligned in each of the space's dimensions, perpendicular to each other and of the same length. A unit hypercube's longest diagonal in n ...
Regular polyhedron. A regular polyhedron is a polyhedron whose symmetry group acts transitively on its flags. A regular polyhedron is highly symmetrical, being all of edge-transitive, vertex-transitive and face-transitive. In classical contexts, many different equivalent definitions are used; a common one is that the faces are congruent regular ...
The truncated cuboctahedron is the convex hull of a rhombicuboctahedron with cubes above its 12 squares on 2-fold symmetry axes. The rest of its space can be dissected into 6 square cupolas below the octagons, and 8 triangular cupolas below the hexagons. A dissected truncated cuboctahedron can create a genus 5, 7, or 11 Stewart toroid by ...
Catalan solid. A toroidal polyhedron. In geometry, a polyhedron ( pl.: polyhedra or polyhedrons; from Greek πολύ (poly-) 'many', and ἕδρον (-hedron) 'base, seat') is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices . A convex polyhedron is a polyhedron that bounds a convex set.
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