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In mathematics, an n-sphere or hypersphere is an -dimensional generalization of the -dimensional circle and -dimensional sphere to any non-negative integer . The circle is considered 1-dimensional, and the sphere 2-dimensional, because the surfaces themselves are 1- and 2-dimensional respectively, not because they ...
There are 4 symmetry classes of reflection on the sphere, and three in the Euclidean plane. A few of the infinitely many such patterns in the hyperbolic plane are also listed. (Increasing any of the numbers defining a hyperbolic or Euclidean tiling makes another hyperbolic tiling.) Point groups:
It is true, however, that every locally Euclidean space is T 1. An example of a non-Hausdorff locally Euclidean space is the line with two origins. This space is created by replacing the origin of the real line with two points, an open neighborhood of either of which includes all nonzero numbers in some open interval centered at zero. This ...
Surface Class Surface Charts n-spheres: n-sphere S n: Hopf chart. Hyperspherical coordinates. Sphere S 2: Spherical coordinates. Stereographic chart Central projection chart Axial projection chart Mercator chart. 3-sphere S 3: Polar chart. Stereographic chart Mercator chart. Euclidean spaces: n-dimensional Euclidean space E n: Cartesian chart ...
The regular finite polygons in 3 dimensions are exactly the blends of the planar polygons (dimension 2) with the digon (dimension 1). They have vertices corresponding to a prism ({n/m}#{} where n is odd) or an antiprism ({n/m}#{} where n is even). All polygons in 3 space have an even number of vertices and edges.
Euclidean space, R n; n-sphere, S n; n-torus, T n; Real projective space, RP n; Complex projective space, CP n; Quaternionic projective space, HP n; Flag manifold; Grassmann manifold; Stiefel manifold; Lie groups provide several interesting families. See Table of Lie groups for examples. See also: List of simple Lie groups and List of Lie group ...
The n-dimensional model is the celestial sphere of the (n + 2)-dimensional Lorentzian space R n+1,1. Here the model is a Klein geometry : a homogeneous space G / H where G = SO( n + 1, 1) acting on the ( n + 2) -dimensional Lorentzian space R n +1,1 and H is the isotropy group of a fixed null ray in the light cone .
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 ...