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The hypercubes are one of the few families of regular polytopes that are represented in any number of dimensions. [8] The hypercube (offset) family is one of three regular polytope families, labeled by Coxeter as γ n. The other two are the hypercube dual family, the cross-polytopes, labeled as β n, and the simplices, labeled as α n.
In this tiling of the plane by congruent squares, the green and violet squares meet edge-to-edge as do the blue and orange squares. In geometry, Keller's conjecture is the conjecture that in any tiling of n-dimensional Euclidean space by identical hypercubes, there are two hypercubes that share an entire (n − 1)-dimensional face with each other.
In graph theory, the hypercube graph Q n is the graph formed from the vertices and edges of an n-dimensional hypercube. For instance, the cube graph Q 3 is the graph formed by the 8 vertices and 12 edges of a three-dimensional cube. Q n has 2 n vertices, 2 n – 1 n edges, and is a regular graph with n edges touching each vertex.
Alternation of the n-cube yields one of two n-demicubes, as in this 3-dimensional illustration of the two tetrahedra that arise as the 3-demicubes of the 3-cube.. In geometry, demihypercubes (also called n-demicubes, n-hemicubes, and half measure polytopes) are a class of n-polytopes constructed from alternation of an n-hypercube, labeled as hγ n for being half of the hypercube family, γ n.
A hypercube is basically a multidimensional mesh network with two nodes in each dimension. Due to similarity, such topologies are usually grouped into a k-ary d-dimensional mesh topology family, where d represents the number of dimensions and k represents the number of nodes in each dimension. [1] Different hypercubes for varying number of nodes
The two general forms of the hypercube honeycombs are the regular form with identical hypercubic facets and one semiregular, with alternating hypercube facets, like a checkerboard. A third form is generated by an expansion operation applied to the regular form, creating facets in place of all lower-dimensional elements.
In geometry, a tesseract or 4-cube is a four-dimensional hypercube, analogous to a two-dimensional square and a three-dimensional cube. [1] Just as the perimeter of the square consists of four edges and the surface of the cube consists of six square faces , the hypersurface of the tesseract consists of eight cubical cells , meeting at right ...
The term unit cube or unit hypercube is also used for hypercubes, or "cubes" in n-dimensional spaces, for values of n other than 3 and edge length 1. [1] [2]Sometimes the term "unit cube" refers in specific to the set [0, 1] n of all n-tuples of numbers in the interval [0, 1].