 TOPICS  # Euclidean

The term Euclidean refers to everything that can historically or logically be referred to Euclid's monumental treatise The Thirteen Books of the Elements, written around the year 300 B.C.

The Euclidean geometry of the plane (Books I-IV) and of the three-dimensional space (Books XI-XIII) is based on five postulates, the first four of which are about the basic objects of plane geometry (point, straight line, circle, and right angle), which can be drawn by straightedge and compass (the so-called Euclidean tools of geometric construction). Euclid's fifth postulate, also known as the parallel postulate, is modified in so-called non-Euclidean geometry.

The ratios of segment lengths represent numbers, and this makes sense since the geometric shapes remain unchanged when placed elsewhere in the plane by rotation, translation, or, more generally, by a rigid motion (a so-called Euclidean motion). The geometric congruence of figures is in fact verified by superposition. This is the starting point of Descartes' algebraic approach to geometry in the so-called Euclidean plane, and also the far origin of the modern notions of Euclidean metric and Euclidean topology. All these concepts can be extended to three or more dimensions, in the abstract context known as Euclidean space.

The Euclidean algorithm is the constructive procedure described by Euclid for proving the existence of the greatest common divisor of two positive integers, stated in Proposition 2 of Book VII, which is the first of four books on numbers and arithmetic. The definition of Euclidean ring arises in modern commutative algebra as the generalization of this procedure from the ring of integers to other abstract rings.

Portions of group theory are also rooted in Euclid's mathematics, through the classification of geometric transformations developed by Felix Klein (namely, the transformation group), and especially the algebraic characterization of constructibility realized in Galois theory. The latter is based on the notion of constructible number or Euclidean number, which is defined as the length of a segment which can be constructed from a segment of unit length by straightedge and compass alone.

Compass, Constructible Number, Elements, Euclidean Algorithm, Euclidean Geometry, Euclidean Metric, Euclidean Motion, Euclidean Number, Euclidean Plane, Euclidean Ring, Euclidean Space, Euclidean Topology, Geometric Congruence, Geometric Construction, Non-Euclidean Geometry, Parallel Postulate, Straightedge

This entry contributed by Margherita Barile

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Barile, Margherita. "Euclidean." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/Euclidean.html