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Euclidean space

Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's Elements, it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any positive integer dimension n, which are called Euclidean n-spaces when one wants to specify their dimension.

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Overview

Definition

Prototypical examples

Affine structure

Metric structure

Isometries

Topology

Axiomatic definitions

Usage

Other geometric spaces

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Map overview Semantic statistics

Euclidean space

Nodes229
Edges228
Triples163
Avg. degree1.99
Density0.008734
Components1

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Euclidean space

Top relations

related to References · 37
Euclidean space → Anton, Berger, Berlin, Cayley, Date, Dover, Dover Publications, Elementary Linear Algebra, Emil, EMS Press, Encyclopedia, Euclidean, Geometric Algebra, Geometry, Grassman, History, Howard, ISBN, John Wiley, Lock-gray-alt-2
is a · 11
Euclidean space → abstraction detached from actual physical locations, affine space, affine space equipped with a metric, affine space over the reals such that the associated vector space is a Euclidean vector space, affine space with an associated real vector space equipped with a non-degenerate quadratic form, complete metric space, complete metric space.OrthogonalityTwo nonzero vectors u and v of E, dimension of its associated vector space.The elements of E are called points, fundamental space of geometry, norm of the translation vector that maps one point to the other, standard way for proving consistency of its definition
related to Curved spaces · 9
Euclidean space → Distances, Euclidean, Generally, However, In, Manifolds, Riemannian, So, This
related to history · 9
Euclidean space → Cartesian, Euclid's Elements, Euclidean, Greeks, In, René Descartes, Their, These, This
related to Axiomatic definitions · 8
Euclidean space → Erlangen, Euclid, Euclid's, Euclidean, Felix Klein, In, The, Two
related to Affine space · 7
Euclidean space → Affine, As, Euclidean, For, In, The, Therefore
related to Cartesian coordinates · 7
Euclidean space → Cartesian, Euclidean, Every Euclidean, Given, Gram, More, Schmidt
related to Lines and segments · 7
Euclidean space → Big, Bigl, Bigr, Euclidean, In, PQ, Since
related to Usage · 7
Euclidean space → Euclidean, Greeks, Higher, It, Since, Space, They
related to Motivation of the modern definition · 6
Euclidean space → Euclidean, Even, For, In, One, The

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Important terminology

euclidean space displaystyle spaces vector two one dimension overrightarrow point called geometry affine points product angle isometry definition basis associated

Entity relationships Subject–Predicate–Object triples

SubjectPredicateObjectConfidenceSrc
Euclidean spaceis afundamental space of geometry0.90text
Euclidean spaceis aabstraction detached from actual physical locations0.90text
Euclidean spaceis aaffine space over the reals such that the associated vector space is a Euclidean vector space0.90text
Euclidean spaceis adimension of its associated vector space.The elements of E are called points0.90text
Euclidean spaceis aaffine space0.90text
Euclidean spaceis anorm of the translation vector that maps one point to the other0.90text
Euclidean spaceis acomplete metric space.OrthogonalityTwo nonzero vectors u and v of E0.90text
Euclidean spaceis acomplete metric space0.90text
Euclidean spaceis astandard way for proving consistency of its definition0.90text
Euclidean spaceis aaffine space with an associated real vector space equipped with a non-degenerate quadratic form0.90text
Euclidean spaceis aaffine space equipped with a metric0.90text
Euclidean spacerelated to Affine spaceEuclidean0.60section

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