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The tetrahedral group T_d is the point group of symmetries of the tetrahedron including the inversion operation. It is one of the 12 non-Abelian groups of order 24. The ...

The first example discovered of a map from a higher-dimensional sphere to a lower-dimensional sphere which is not null-homotopic. Its discovery was a shock to the ...

A collection of faces of an n-dimensional polytope or simplicial complex, one of each dimension 0, 1, ..., n-1, which all have a common nonempty intersection. In normal three ...

Given a differential p-form q in the exterior algebra ^ ^pV^*, its envelope is the smallest subspace W such that q is in the subspace ^ ^pW^* subset ^ ^pV^*. Alternatively, W ...

A pair of quantities (a, b) where ordering is significant, so (a, b) is considered distinct from (b, a) for a!=b.

In an exterior algebra ^ V, a top-dimensional form has degree n where n=dimV. Any form of higher degree must be zero. For example, if V=R^4 then alpha=e_1 ^ e_2 ^ e_3 ^ e_4 ...

Given an m×n matrix A and a p×q matrix B, their Kronecker product C=A tensor B, also called their matrix direct product, is an (mp)×(nq) matrix with elements defined by ...

A projection of a figure by parallel rays. In such a projection, tangencies are preserved. Parallel lines project to parallel lines. The ratio of lengths of parallel segments ...

A real normed algebra, also called a composition algebra, is a multiplication * on R^n that respects the length of vectors, i.e., |x*y|=|x|*|y| for x,y in R^n. The only real ...

A simple directed graph is a directed graph having no multiple edges or graph loops (corresponding to a binary adjacency matrix with 0s on the diagonal). The number of simple ...