Euclidean space

Definition

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. For n equal to one or two, they are commonly called respectively Euclidean lines and Euclidean planes. The qualifier "Euclidean" is used to distinguish Euclidean spaces from other spaces that were later considered in physics and modern mathematics.

Related concepts

Abstraction Affine algebraic variety Affine basis Affine coordinates Affine frame Affine property Affine space Albert Einstein Alfred Tarski Algebra Algebraic geometry Algebraic group Algebraic number field Algebraically closed field Analytic geometry Analytic manifold Angle Antimeridian Arccosine Architecture Astronomy Axiom Axiomatic theory Barycentric coordinates Base (topology)Basis (vector space)Bijection Bilinear form Birkhoff's axioms Bounded set Cartesian coordinate system Cartesian coordinates Cartesian frame Cauchy–Schwarz inequality Cayley–Dickson construction Circle Codimension Complete metric Complete metric space Complex number Configuration space (physics)Congruence (geometry)Consistency Coordinate system Coordinate vector Coplanar Cross-polytope Cryptography Curvature of Riemannian manifolds Curved space Cylindrical coordinate system David Hilbert Degrees of freedom Demihypercube Dense subset Diffeomorphism Differentiable manifold Dimension Dimension (mathematics)Dimension (vector space)Dimension of an algebraic variety Dimensional analysis Displacement (geometry)Dot product Eight-dimensional space Ellipsoid Elliptic curve Elliptic geometry Elliptic space Embedding Emil Artin Encyclopedia of Mathematics Equivalence relation Erlangen program Euclid Euclid's Elements Euclid's postulates Euclidean distance Euclidean geometry Euclidean group Euclidean line Euclidean plane Euclidean topology Euclidean vector European Mathematical Society Felix Klein Fermat's Last Theorem Fermat curve Field (mathematics)Figure (geometry)Finite fields Five-dimensional space Flat (geometry)Foundational crisis in mathematics Four-dimensional space Fractal dimension Frame of reference Free module Free vector Function composition Functional analysis G. D. Birkhoff General relativity Geodesic Geodesy Geometric Algebra (book)Geometric transformation Geometry Glide reflection Gram–Schmidt process Graph (discrete mathematics)Gravitational field Gravity Great circle Greek mathematics Group action Group action (mathematics)Group homomorphism Harold Scott MacDonald Coxeter Hausdorff dimension Higher dimension Hilbert's axioms Hilbert space History of geometry Homeomorphic Homeomorphism Hyperbolic geometry Hypercomplex number Hypercube Hyperplane Hyperpyramid Hyperrectangle Hyperspace Hypersurface Identity function Indefinite quadratic form Index (group theory)Inductive dimension Industrial design Infinity Inner product Inner product space Interval (mathematics)Invariance of domain Isometry Isomorphism Isotropic John Wiley & Sons Krull dimension Lebesgue covering dimension Lie group Line (geometry)Line segment Linear algebra Linear isomorphism Linear map Linear span Linear subspace Locally compact Ludwig Schläfli Manifold Marcel Berger Mathematical proof Mathematics Mechanics Metric (mathematics)Metric space Minkowski space Minkowski–Bouligand dimension Modular arithmetic Motion (geometry)N-sphere Navigation Neighborhood Non-Euclidean geometries Non-Euclidean geometry Non-degenerate Normal subgroup Number Number theory One-dimensional space Open ball Open set Open subset Orientation (vector space)Oriented angle Orthodrome Orthogonal group Orthogonality Orthonormal basis Parallel (geometry)Parallel postulate Perpendicular Physical space Physical system Physics Platonic solid Playfair's axiom Point set Points at infinity Polar coordinate system Polytope Position space Positive definite Positive definite bilinear form Postulate Principal value Product topology Projective space Proof (mathematics)Pseudo-Euclidean space Pseudo-Riemannian manifold Pythagorean theorem Quadratic form Rational number Ray (geometry)Real n-space Real number Real vector space Reflection (mathematics)Reflex angle Regular Polytopes (book)René Descartes Riemannian manifold Right angle Rigid transformation Rotation Rotation (mathematics)Scalar multiplication Science Screw axis Semidirect product Seven-dimensional space Shape Simplex Six-dimensional space Smooth function Smooth manifold Space-time Space (mathematics)Spacetime Special orthogonal group Special relativity Sphere Spherical coordinate system Stabilizer subgroup Straight line Subset Subspace topology Symmetric bilinear form Symmetries Synthetic geometry Tangent space Tarski's axioms Technical drawing Theorem Three-dimensional space Topography Topological dimension Topological manifold Topological space Topology Translation (geometry)Triangle Triangle inequality Two-dimensional space Unit of length Unit vector Vector line Vector space W. W. Rouse Ball Zero-dimensional space

35 concepts already in your glossary