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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 ...
convex, Hanner polytope. 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 ...
The Euler characteristic χ was classically defined for the surfaces of polyhedra, according to the formula. where V, E, and F are respectively the numbers of v ertices (corners), e dges and f aces in the given polyhedron. Any convex polyhedron 's surface has Euler characteristic. This equation, stated by Euler in 1758, [2] is known as Euler's ...
Polyhedral combinatorics is a branch of mathematics, within combinatorics and discrete geometry, that studies the problems of counting and describing the faces of convex polyhedra and higher-dimensional convex polytopes. Research in polyhedral combinatorics falls into two distinct areas. Mathematicians in this area study the combinatorics of ...
In mathematics, and more specifically in polyhedral combinatorics, a Goldberg polyhedron is a convex polyhedron made from hexagons and pentagons. They were first described in 1937 by Michael Goldberg (1902–1990). They are defined by three properties: each face is either a pentagon or hexagon, exactly three faces meet at each vertex, and they ...
In five-dimensional geometry, a cantellated 5-orthoplex is a convex uniform 5-polytope, being a cantellation of the regular 5-orthoplex. There are 6 cantellation for the 5-orthoplex, including truncations.
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.
Equivalently, a convex set or a convex region is a subset that intersects every line into a single line segment (possibly empty). [1][2] 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.
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