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The universal cover of a connected topological space X is a simply connected space Y with a map f:Y->X that is a covering map. If X is simply connected, i.e., has a trivial ...

The Alexander invariant H_*(X^~) of a knot K is the homology of the infinite cyclic cover of the complement of K, considered as a module over Lambda, the ring of integral ...

The n×n square matrix F_n with entries given by F_(jk)=e^(2piijk/n)=omega^(jk) (1) for j,k=0, 1, 2, ..., n-1, where i is the imaginary number i=sqrt(-1), and normalized by ...

The imaginary part I[z] of a complex number z=x+iy is the real number multiplying i, so I[x+iy]=y. In terms of z itself, I[z]=(z-z^_)/(2i), where z^_ is the complex conjugate ...

The problem of determining how many nonattacking kings can be placed on an n×n chessboard. For n=8, the solution is 16, as illustrated above (Madachy 1979). In general, the ...

A lattice graph, also known as a mesh graph or grid graph, is a graph possessing an embedding in a Euclidean space R^n that forms a regular tiling. Examples include grid ...

The 12-faced convex deltahedra also known as the Siamese dodecahedron, which is also Johnson solid J_(84). It is implemented in the Wolfram Language as ...

A square matrix U is a special unitary matrix if UU^*=I, (1) where I is the identity matrix and U^* is the conjugate transpose matrix, and the determinant is detU=1. (2) The ...

A stereohedron is a convex polyhedron that is isohedrally space-filling, meaning the symmetries of a tiling of copies of a stereohedron take any copy to any other copy. The ...

Whirls are figures constructed by nesting a sequence of polygons (each having the same number of sides), each slightly smaller and rotated relative to the previous one. The ...