-dimensional sphere to any non-negative integer n {\displaystyle n} . The n {\displaystyle n} -sphere is the setting for n {\displaystyle n} -dimensional...
37 KB (7,130 words) - 10:34, 22 August 2024
A sphere (from Greek σφαῖρα, sphaîra) is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. Formally, a sphere is the...
41 KB (5,328 words) - 16:32, 24 August 2024
homotopy sphere is an n-manifold that is homotopy equivalent to the n-sphere. It thus has the same homotopy groups and the same homology groups as the n-sphere...
1 KB (172 words) - 18:09, 27 May 2024
Sphere (also called Sphere at the Venetian Resort) is a music and entertainment arena in Paradise, Nevada, United States, east of the Las Vegas Strip....
77 KB (6,176 words) - 21:52, 21 August 2024
"surface"). Topologically, a 3-sphere is an example of a 3-manifold, and it is also an n-sphere. In coordinates, a 3-sphere with center (C0, C1, C2, C3)...
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generally, the unit n {\displaystyle n} -sphere is an n {\displaystyle n} -sphere of unit radius in ( n + 1 ) {\displaystyle (n+1)} -dimensional Euclidean...
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The n-dimensional unit sphere — called the n-sphere for brevity, and denoted as Sn — generalizes the familiar circle (S1) and the ordinary sphere (S2)...
82 KB (7,972 words) - 00:30, 24 August 2024
In geometry, a sphere packing is an arrangement of non-overlapping spheres within a containing space. The spheres considered are usually all of identical...
28 KB (3,414 words) - 06:13, 21 May 2024
The Sphere (officially Große Kugelkaryatide N.Y., also known as Sphere at Plaza Fountain, WTC Sphere or Koenig Sphere) is a monumental cast bronze sculpture...
24 KB (2,230 words) - 21:37, 29 June 2024
homology sphere is an n-manifold X having the homology groups of an n-sphere, for some integer n ≥ 1 {\displaystyle n\geq 1} . That is, H 0 ( X , Z ) = H n (...
11 KB (1,529 words) - 01:47, 27 May 2024
the region enclosed by a sphere or hypersphere. An n-ball is a ball in an n-dimensional Euclidean space. The volume of a n-ball is the Lebesgue measure...
28 KB (5,444 words) - 18:27, 31 March 2024
exotic sphere is a differentiable manifold M that is homeomorphic but not diffeomorphic to the standard Euclidean n-sphere. That is, M is a sphere from...
29 KB (3,844 words) - 19:54, 8 August 2024
sphere-like region or shell. Sphere may also refer to: Armillary sphere, a physical model of the celestial sphere Celestial sphere, the astronomical description...
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Ball (mathematics) (redirect from Solid sphere)
spaces in general. A ball in n dimensions is called a hyperball or n-ball and is bounded by a hypersphere or (n−1)-sphere. Thus, for example, a ball in...
12 KB (1,654 words) - 14:45, 10 August 2024
intersection of a sphere and a plane passing through the sphere's center point. Any arc of a great circle is a geodesic of the sphere, so that great circles...
7 KB (972 words) - 05:55, 19 March 2024
In quantum mechanics and computing, the Bloch sphere is a geometrical representation of the pure state space of a two-level quantum mechanical system (qubit)...
23 KB (3,793 words) - 22:40, 16 July 2024
Spherical lune (section n-sphere lunes)
tessellation of the sphere by lunes. A n-gonal regular hosohedron, {2,n} has n equal lunes of π/n radians. An n-hosohedron has dihedral symmetry Dnh, [n,2], (*22n)...
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Then the statement is Every homotopy sphere (a closed n-manifold which is homotopy equivalent to the n-sphere) in the chosen category (i.e. topological...
10 KB (1,285 words) - 04:55, 26 May 2024
De Rham cohomology (section The n-sphere)
objects: For the n-sphere, S n {\displaystyle S^{n}} , and also when taken together with a product of open intervals, we have the following. Let n > 0, m ≥ 0...
19 KB (2,923 words) - 15:42, 22 August 2024
In mathematics, two points of a sphere (or n-sphere, including a circle) are called antipodal or diametrically opposite if they are the endpoints of a...
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Manifold (section n-Sphere as a patchwork)
normal bundle. The n-sphere Sn is a generalisation of the idea of a circle (1-sphere) and sphere (2-sphere) to higher dimensions. An n-sphere Sn can be constructed...
68 KB (9,498 words) - 05:38, 16 July 2024
function from an n-sphere into Euclidean n-space maps some pair of antipodal points to the same point. Here, two points on a sphere are called antipodal...
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Spherical measure (redirect from Sphere measure)
"natural" Borel measure on the n-sphere Sn. Spherical measure is often normalized so that it is a probability measure on the sphere, i.e. so that σn(Sn) = 1...
5 KB (698 words) - 05:38, 23 December 2021
Inversive geometry (redirect from Sphere inversion)
orthogonal to the unit sphere. Hence we are led to consider the (n − 1)-spheres with equation x 1 2 + ⋯ + x n 2 + 2 a 1 x 1 + ⋯ + 2 a n x n + 1 = 0 , {\displaystyle...
30 KB (4,378 words) - 04:07, 25 July 2024
Hairy ball theorem (category Spheres)
even-dimensional n-spheres. For the ordinary sphere, or 2‑sphere, if f is a continuous function that assigns a vector in R3 to every point p on a sphere such that...
14 KB (1,803 words) - 09:01, 23 August 2024
length, volume, and curvature are defined. Euclidean space, the n {\displaystyle n} -sphere, hyperbolic space, and smooth surfaces in three-dimensional space...
59 KB (8,676 words) - 01:17, 20 August 2024
proof of the sphere theorem did not conclude that M was necessarily diffeomorphic to the n-sphere. This complication is because spheres in higher dimensions...
5 KB (594 words) - 16:47, 31 July 2024
did this through looking at real vector bundles V → S n {\displaystyle V\to S^{n}} over a sphere and studied the properties of the associated disk bundle...
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armillary sphere (variations are known as spherical astrolabe, armilla, or armil) is a model of objects in the sky (on the celestial sphere), consisting...
32 KB (4,235 words) - 16:43, 15 August 2024
submanifolds. For example, the n-sphere S n = { x ∈ R n + 1 : ‖ x ‖ = 1 } {\displaystyle S^{n}=\{x\in \mathbb {R} ^{n+1}:\lVert x\rVert =1\}} is an embedded...
2 KB (292 words) - 19:38, 24 June 2024