Euclidische

Euclidische (German for “Euclidean”) refers to concepts that derive from the work of the ancient Greek mathematician Euclid, especially those presented in his treatise Elements. The most common usage is in Euclidean geometry, the study of plane and solid figures based on a set of axioms and postulates that Euclid formulated around 300 BC. These include the familiar notions of points, lines, angles, and circles, as well as the parallel postulate, which states that through a point not on a given line there is exactly one line parallel to the given line. From these foundations arise classic results such as the Pythagorean theorem, the sum of interior angles in a triangle, and the properties of regular polygons.

In modern mathematics the term “Euclidean” also designates Euclidean spaces, which are finite‑dimensional vector spaces equipped with the standard inner product, yielding the usual notions of distance and angle. The n‑dimensional Euclidean space ℝⁿ provides the setting for analytical geometry, calculus, and many areas of physics. Its metric is derived from the Euclidean norm, and the geometry of ℝⁿ conforms to the same axioms that Euclid outlined for the plane.

Another related notion is the Euclidean algorithm, an efficient method for computing the greatest common divisor of two integers. Although not a geometric concept, it is named after Euclid because the algorithm appears as Proposition 2 of Book VII in the Elements.

Euclidean ideas dominate everyday intuitive geometry but are complemented by non‑Euclidean geometries, which relax or replace Euclid’s parallel postulate. Together, Euclidean and non‑Euclidean systems form a comprehensive picture of the possible structures of space.