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The elongated dodecahedron, also known as the extended rhombic dodecahedron and rhombo-hexagonal dodecahedron, is a space-filling polyhedron and primary parallelohedron ...

A surface S is n-embeddable if it can be placed in R^n-space without self-intersections, but cannot be similarly placed in any R^k for k<n. A surface so embedded is said to ...

An equalizer of a pair of maps f,g:X->Y in a category is a map e:E->X such that 1. f degreese=g degreese, where degrees denotes composition. 2. For any other map e^':E^'->X ...

A functor between categories of groups or modules is called exact if it preserves the exactness of sequences, or equivalently, if it transforms short exact sequences into ...

Every polynomial equation having complex coefficients and degree >=1 has at least one complex root. This theorem was first proven by Gauss. It is equivalent to the statement ...

The surface generated by a twisted curve C when rotated about a fixed axis A and, at the same time, displaced parallel to A so that the velocity of displacement is always ...

A group automorphism is an isomorphism from a group to itself. If G is a finite multiplicative group, an automorphism of G can be described as a way of rewriting its ...

Given two groups G and H, there are several ways to form a new group. The simplest is the direct product, denoted G×H. As a set, the group direct product is the Cartesian ...

A hole in a mathematical object is a topological structure which prevents the object from being continuously shrunk to a point. When dealing with topological spaces, a ...

The tangent space at a point p in an abstract manifold M can be described without the use of embeddings or coordinate charts. The elements of the tangent space are called ...