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The above -sphere exists in (+) -dimensional Euclidean space and is an example of an -manifold. The volume form ω {\displaystyle \omega } of an n {\displaystyle n} -sphere of radius r {\displaystyle r} is given by
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 can be realized non-degenerately in some non-Euclidean spaces, such as on the surface of a sphere or torus. For example, digon can be realised non-degenerately as a spherical lune . A monogon {1} could also be realised on the sphere as a single point with a great circle through it. [ 7 ]
The symmetry group of the sphere (n=3) or hypersphere. so(n) n(n−1)/2 SO(n) special orthogonal group: real orthogonal matrices with determinant 1 Y 0 Z n=2 Z 2 n>2 Spin(n) n>2 SO(1) is a single point and SO(2) is isomorphic to the circle group, SO(3) is the rotation group of the sphere. so(n) n(n−1)/2 SE(n)
An example of uniform tiling in the Archeological Museum of Seville, Sevilla, Spain: rhombitrihexagonal tiling Regular tilings and their duals drawn by Max Brückner in Vielecke und Vielflache (1900) This table shows the 11 convex uniform tilings (regular and semiregular) of the Euclidean plane, and their dual tilings.
The sum of the angles of a spherical triangle is not equal to 180°. A sphere is a curved surface, but locally the laws of the flat (planar) Euclidean geometry are good approximations. In a small triangle on the face of the earth, the sum of the angles is only slightly more than 180 degrees. A sphere with a spherical triangle on it.
Direct projection of 3-sphere into 3D space and covered with surface grid, showing structure as stack of 3D spheres (2-spheres) In mathematics, a hypersphere or 3-sphere is a 4-dimensional analogue of a sphere, and is the 3-dimensional n-sphere. In 4-dimensional Euclidean space, it is the
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 ...