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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.
Additionally, a Hamiltonian path exists between two vertices u and v if and only if they have different colors in a 2-coloring of the graph. Both facts are easy to prove using the principle of induction on the dimension of the hypercube, and the construction of the hypercube graph by joining two smaller hypercubes with a matching.
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
A cube can be considered a multi-dimensional generalization of a two- or three-dimensional spreadsheet. For example, a company might wish to summarize financial data by product, by time-period, and by city to compare actual and budget expenses. Product, time, city and scenario (actual and budget) are the data's dimensions. [3]
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.
The same principle can be applied to the All-Reduce operations, but instead of concatenating the messages, it performs a reduction operation on the two messages. So it is a Reduce operation, where all processing units know the result. Compared to a normal reduce operation followed by a broadcast, All-Reduce in hypercubes reduces the number of ...
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.
The cross-polytope family is one of three regular polytope families, labeled by Coxeter as β n, the other two being the hypercube family, labeled as γ n, and the simplex family, labeled as α n. A fourth family, the infinite tessellations of hypercubes, he labeled as δ n. [5]