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A space endowed with a non-Euclidean elliptic geometry.

Let K be a field of arbitrary characteristic. Let v:K->R union {infty} be defined by the following properties: 1. v(x)=infty<=>x=0, 2. v(xy)=v(x)+v(y) forall x,y in K, and 3. ...

A more common way to describe a Euclidean ring.

The two-dimensional Euclidean space denoted R^2.

A non-Abelian group, also sometimes known as a noncommutative group, is a group some of whose elements do not commute. The simplest non-Abelian group is the dihedral group ...

A metric topology induced by the Euclidean metric. In the Euclidean topology of the n-dimensional space R^n, the open sets are the unions of n-balls. On the real line this ...

A ring without zero divisors in which an integer norm and an associated division algorithm (i.e., a Euclidean algorithm) can be defined. For signed integers, the usual norm ...

A Euclidean motion of R^n is an affine transformation whose linear part is an orthogonal transformation.

The Euclidean metric is the function d:R^n×R^n->R that assigns to any two vectors in Euclidean n-space x=(x_1,...,x_n) and y=(y_1,...,y_n) the number ...

Four-dimensional geometry is Euclidean geometry extended into one additional dimension. The prefix "hyper-" is usually used to refer to the four- (and higher-) dimensional ...