In mathematics, an n-sphere or hypersphere is an ⁠ n {\displaystyle n} ⁠-dimensional generalization of the ⁠ 1 {\displaystyle 1} ⁠-dimensional circle and...

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...

mathematics, a hypersphere, 3-sphere, or glome is a 4-dimensional analogue of a sphere, and is the 3-dimensional n-sphere. In 4-dimensional Euclidean space...

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....

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...

generally, the unit n {\displaystyle n} -sphere is an n {\displaystyle n} -sphere of unit radius in ( n + 1 ) {\displaystyle (n+1)} -dimensional Euclidean...

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)...

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...

In geometry, a sphere packing is an arrangement of non-overlapping spheres within a containing space. The spheres considered are usually all of identical...

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...

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...

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 (...

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...

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...

sphere-like region or shell. Sphere may also refer to: Armillary sphere, a physical model of the celestial sphere Celestial sphere, the astronomical description...

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...

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...

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)...

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)...

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...

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...

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...

/ˌpwæ̃kɑːˈreɪ/, French: [pwɛ̃kaʁe]) is a theorem about the characterization of the 3-sphere, which is the hypersphere that bounds the unit ball in four-dimensional...

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...

(mathematics). For example, a higher-dimensional knot is an n-dimensional sphere embedded in (n+2)-dimensional Euclidean space. Archaeologists have discovered...

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...

"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...

topology, it is often denoted as S1 because it is a one-dimensional unit n-sphere. If (x, y) is a point on the unit circle's circumference, then |x| and...

= 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...

length, volume, and curvature are defined. Euclidean space, the n {\displaystyle n} -sphere, hyperbolic space, and smooth surfaces in three-dimensional space...