• Thumbnail for N-sphere
    ⁠-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
  • Thumbnail for Sphere
    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
  • Thumbnail for Sphere (venue)
    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
  • Thumbnail for 3-sphere
    "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)...
    28 KB (4,027 words) - 13:24, 23 July 2024
  • generally, the unit n {\displaystyle n} -sphere is an n {\displaystyle n} -sphere of unit radius in ( n + 1 ) {\displaystyle (n+1)} -dimensional Euclidean...
    12 KB (1,911 words) - 13:26, 28 May 2024
  • Thumbnail for Homotopy groups of spheres
    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
  • Thumbnail for The Sphere
    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
  • Thumbnail for Sphere packing
    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
  • 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
  • Thumbnail for Volume of an n-ball
    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...
    5 KB (616 words) - 05:39, 12 June 2024
  • Thumbnail for Bloch sphere
    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
  • Thumbnail for Ball (mathematics)
    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
  • Thumbnail for Great circle
    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
  • Thumbnail for Manifold
    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
  • Thumbnail for Spherical lune
    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)...
    5 KB (576 words) - 14:48, 30 June 2024
  • 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
  • Thumbnail for De Rham cohomology
    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
  • Thumbnail for Antipodal point
    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...
    4 KB (509 words) - 18:35, 31 March 2024
  • Thumbnail for Borsuk–Ulam theorem
    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...
    14 KB (2,431 words) - 17:06, 27 June 2024
  • "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
  • 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
  • Thumbnail for Hairy ball theorem
    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
  • Thumbnail for Riemannian manifold
    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
  • Thumbnail for Armillary sphere
    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
  • 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...
    2 KB (185 words) - 04:02, 6 January 2021
  • Thumbnail for Riemannian submanifold
    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