1,176
pages

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 , and can be modelled with a real coordinate space (often, Euclidean spaces are denoted with , though 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.

Dimensionality Negative One Zero One Two Three Four Five Six Seven Eight Nine Ten ... Aleph null
Hyperbolic space

Hyperbolic plane

Hyperbolic realm

Hyperbolic flune

Hyperbolic pentrealm

Hyperbolic hexealm

Hyperbolic heptealm

Hyperbolic octealm

Hyperbolic ennealm

Hyperbolic decealm

... Hyperbolic omegealm

Euclidean space

Null polytope

Point

Euclidean line

Euclidean plane

Euclidean realm

Euclidean flune

Euclidean pentrealm

Euclidean hexealm

Euclidean heptealm

Euclidean octealm

Euclidean ennealm

Euclidean decealm

... Euclidean omegealm

Hypersphere

Point pair

Circle

Sphere

Glome

Tetrasphere

Pentasphere

Hexasphere

Heptasphere

Octasphere

Enneasphere

Dekasphere

... Omegasphere

Real dimensionality 0 1 2 3 ...
Real space

Point

Real line

Real plane

Real realm

...
Real projective space

Point pair

Real projective line

Real projective plane

Real projective realm

...
Complex space

Point

Complex line

Complex plane

Complex realm

...
Complex projective space

Point pair

Complex projective line

Complex projective plane

Complex projective realm

...
Quaternionic space

Point

Quaternionic line

Quaternionic plane

Quaternionic realm

...
Quaternionic projective space

Point pair

Quaternionic projective line

Quaternionic projective plane

Quaternionic projective realm

...
Octonionic space

Point

Octonionic line

Octonionic plane

Octonionic realm

...
Octonionic projective space

Point pair

Octonionic projective line

Octonionic projective plane

Octonionic projective realm

...
... ... ... ... ... ...