Euklidiske

Euklidiske is an adjective used to describe things related to the ancient Greek mathematician Euclid of Alexandria or to the body of geometry and mathematics that bears his name. In Norwegian and other Scandinavian languages, the form euklidisk/euklidiske is used to modify nouns such as geometry, space, or algorithms.

Euklidisk geometry refers to the geometry established by Euclid in his Elements. It is built on a small set of axioms and postulates, from which the properties of flat (Euclidean) space follow. Central ideas include points, lines, planes, triangles, similarity and congruence, and the Pythagorean theorem.

Euclidean space, or Euclidean n-space, is the standard mathematical model of ordinary space. It consists of n-dimensional real coordinate space R^n equipped with the Euclidean metric d(x,y) = sqrt(sum (x_i - y_i)^2). This metric induces lengths, angles, and volumes and is derived from the inner product.

Euklidiske algorithms: The Euclidean algorithm computes the greatest common divisor of integers by repeated division with remainder. It is essential in number theory and underpins many algorithms used in computer algebra, cryptography, and algorithmic number theory.

Relation to non-Euclidean geometries: The term euklidisk is often contrasted with non-Euclidean geometries, such as hyperbolic and spherical geometry, which arise when the parallel postulate is modified or rejected. These geometries show that Euclidean assumptions are not the only possibility for consistent geometric systems.