euclidinen

Euclidinen is an adjective meaning related to Euclid of Alexandria or to Euclidean geometry. In mathematics, it is used to describe concepts, methods, and spaces that fit the framework and axioms associated with Euclid’s Elements, as opposed to non-Euclidean geometries.

Common uses of the term include Euclidean distance, Euclidean space, and Euclidean plane. The Euclidean distance between two points in n-dimensional space is the length of the straight-line segment connecting them, defined by the square root of the sum of the squared coordinate differences. The Euclidean plane and Euclidean n-space refer to spaces equipped with the standard distance function derived from the Euclidean norm. In analysis and linear algebra, the Euclidean norm, denoted ||x||2, induces the Euclidean metric.

Another well-known Euclidean concept is the Euclidean algorithm, a procedure for computing the greatest common divisor of two integers through repeated division. Euclidean transformations, including rotations, translations, and reflections, preserve distances and angles and form the group of isometries of Euclidean space.

The term is often contrasted with non-Euclidean geometries, where Euclid’s parallel postulate is modified or rejected, leading to different notions of distance, parallelism, and curvature. The origin of the term honors Euclid, whose Elements (circa 300 BCE) is regarded as foundational for the axiomatic, deductive approach to geometry.