Euclidean space

A Euclidean space is a zero-curve, infinitely large, real metric space that follows Euclidean geometry, and thus, all of Euclid's Postulates.

Euclidean spaces are constructed from repeated Cartesian products of a real line $\mathbb R^{1}$, and can be modelled with a real coordinate space $\R^n$ (often, Euclidean spaces are denoted with $\R^n$, though $\mathbb{E}^n$ can be used). This means that coordinates in a Euclidean space have to be an ordered tuple of real numbers, the number of numbers is the dimensionality of the space.

Euclidean spaces are typically known as "flat" spaces due to the fact that they have zero curvature.

Distances in a Euclidean space can be determined with the Pythagorean theorem.

Euclidean spaces can be generalised with Hilbert spaces, which can allow spaces to pertain an infinite dimensionality.

Spaces that don't follow Euclidean geometry do not follow one of the postulates, and may have curvature, imaginary dimensions, a different geometry.

See Also

Dimensionality	Negative One	Zero	One	Two	Three	Four	Five	Six	Seven	Eight	Nine	Ten	...	Aleph null
Hyperbolic space $\mathbb H^{n}$	—	—	—	Hyperbolic plane $\mathbb H^{2}$	Hyperbolic realm $\mathbb H^{3}$	Hyperbolic flune $\mathbb H^{4}$	Hyperbolic pentrealm $\mathbb H^{5}$	Hyperbolic hexealm $\mathbb H^{6}$	Hyperbolic heptealm $\mathbb H^{7}$	Hyperbolic octealm $\mathbb H^{8}$	Hyperbolic ennealm $\mathbb H^{9}$	Hyperbolic decealm $\mathbb H^{10}$	...	Hyperbolic omegealm $\mathbb H^{\aleph_0}$
Euclidean space $\R^n$	Null polytope $\emptyset$	Point $\mathbb R^{0}$	Euclidean line $\mathbb R^{1}$	Euclidean plane $\mathbb R^{2}$	Euclidean realm $\mathbb R^{3}$	Euclidean flune $\mathbb R^{4}$	Euclidean pentrealm $\mathbb R^{5}$	Euclidean hexealm $\mathbb R^{6}$	Euclidean heptealm $\mathbb R^{7}$	Euclidean octealm $\mathbb R^{8}$	Euclidean ennealm $\mathbb R^{9}$	Euclidean decealm $\mathbb R^{10}$	...	Euclidean omegealm $\mathbb R^{\aleph_0}$
Hypersphere $\mathbb S^{n}$		Point pair $\mathbb S^{0}$	Circle $\mathbb S^{1}$	Sphere $\mathbb S^{2}$	Glome $\mathbb S^{3}$	Tetrasphere $\mathbb S^{4}$	Pentasphere $\mathbb S^{5}$	Hexasphere $\mathbb S^{6}$	Heptasphere $\mathbb S^{7}$	Octasphere $\mathbb S^{8}$	Enneasphere $\mathbb S^{9}$	Dekasphere $\mathbb S^{10}$	...	Omegasphere $\mathbb S^{\aleph_0}$

Real dimensionality	0	1	2	3	...
Real space $\R^n$	Point $\mathbb R^{0}$	Real line $\mathbb R^{1}$	Real plane $\mathbb R^{2}$	Real realm $\mathbb R^{3}$	...
Real projective space $\mathbb {R}\mathbb{P}^n$	Point pair $\mathbb {R}\mathbb{P}^0$	Real projective line $\mathbb {R}\mathbb{P}^1$	Real projective plane $\mathbb {R}\mathbb{P}^2$	Real projective realm $\mathbb {R}\mathbb{P}^3$	...
Complex space $\C^n$	Point $\mathbb {C}^0$	Complex line $\mathbb {C}^1$	Complex plane $\mathbb {C}^2$	Complex realm $\mathbb{C}^3$	...
Complex projective space $\mathbb {C}\mathbb{P}^n$	Point pair $\mathbb {C}\mathbb{P}^0$	Complex projective line $\mathbb {C}\mathbb{P}^1$	Complex projective plane $\mathbb {C}\mathbb{P}^2$	Complex projective realm $\mathbb {C}\mathbb{P}^3$	...
Quaternionic space $\mathbb H^{n}$	Point $\mathbb {H}^0$	Quaternionic line $\mathbb {H}^1$	Quaternionic plane $\mathbb H^{2}$	Quaternionic realm $\mathbb H^{3}$	...
Quaternionic projective space $\mathbb {H}\mathbb{P}^n$	Point pair $\mathbb {H}\mathbb{P}^0$	Quaternionic projective line $\mathbb {H}\mathbb{P}^1$	Quaternionic projective plane $\mathbb {H}\mathbb{P}^2$	Quaternionic projective realm $\mathbb {H}\mathbb{P}^3$	...
Octonionic space $\mathbb {O}^n$	Point $\mathbb {O}^0$	Octonionic line $\mathbb {O}^1$	Octonionic plane $\mathbb {O}^2$	Octonionic realm $\mathbb {O}^3$	...
Octonionic projective space $\mathbb {O}\mathbb{P}^n$	Point pair $\mathbb {O}\mathbb{P}^0$	Octonionic projective line $\mathbb {O}\mathbb{P}^1$	Octonionic projective plane $\mathbb {O}\mathbb{P}^2$	Octonionic projective realm $\mathbb {O}\mathbb{P}^3$	...
...	...	...	...	...	...