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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:
R n: Euclidean space with addition N 0 0 abelian R n: n: R ×: nonzero real numbers with multiplication N Z 2 – abelian R: 1 R + positive real numbers with multiplication N 0 0 abelian R: 1 S 1 = U(1) the circle group: complex numbers of absolute value 1 with multiplication; Y 0 Z: R: abelian, isomorphic to SO(2), Spin(2), and R/Z: R: 1 Aff(1)
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 ]
hence has Betti number 1 in dimensions 0 and n, and all other Betti numbers are 0. Its Euler characteristic is then χ = 1 + (−1) n ; that is, either 0 if n is odd, or 2 if n is even. The n dimensional real projective space is the quotient of the n sphere by the antipodal map. It follows that its Euler characteristic is exactly half that of ...
The set of n-tuples of real numbers equipped with the dot product is a Euclidean space of dimension n. Conversely, the choice of a point called the origin and an orthonormal basis of the space of translations is equivalent with defining an isomorphism between a Euclidean space of dimension n and R n {\displaystyle \mathbb {R} ^{n}} viewed as a ...
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.
Alexander horned sphere − A particular embedding of a sphere into 3-dimensional Euclidean space. Antoine's necklace − A topological embedding of the Cantor set in 3-dimensional Euclidean space, whose complement is not simply connected. Irrational winding of a torus/Irrational cable on a torus; Knot (mathematics) Linear flow on the torus