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Generalized sphere of dimension n (mathematics)
In mathematics, an n-sphere or hypersphere is an n {\displaystyle n} -dimensional generalization of the 1 {\displaystyle 1} -dimensional circle and
N-sphere
Set of points equidistant from a center
A sphere (from Ancient Greek σφαῖρα (sphaîra) 'ball') is a surface analogous to the circle, a curve. In solid geometry, a sphere is the set of points that
Sphere
Entertainment venue in the Las Vegas Valley, United States
Sphere (also known as Sphere at the Venetian Resort or Las Vegas Sphere) is a music and entertainment arena in Paradise, Nevada, United States, east of
Sphere_(venue)
Topological manifold whose homology coincides with that of a sphere
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 (
Homology_sphere
Mathematical object
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
3-sphere
Smooth manifold that is homeomorphic but not diffeomorphic to a sphere
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
Exotic_sphere
How spheres of various dimensions can wrap around each other
structure of spheres viewed as topological spaces, forgetting about their precise geometry. The n-dimensional unit sphere — called the n-sphere for brevity
Homotopy_groups_of_spheres
Sphere with radius one, usually centered on the origin of the space
generally, the unit n {\displaystyle n} -sphere is an n {\displaystyle n} -sphere of unit radius in ( n + 1 ) {\displaystyle (n+1)} -dimensional
Unit_sphere
Geometrical structure
In geometry, a sphere packing is an arrangement of non-overlapping spheres within a containing space. The spheres considered are usually all of identical
Sphere_packing
Representation of a quantum mechanical system
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)
Bloch_sphere
Area on a sphere bounded by two semicircles joined at antipodal points
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)
Spherical_lune
Size of a mathematical 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
Volume_of_an_n-ball
Concept in algebraic topology
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
Homotopy_sphere
Volume space bounded by a sphere
n-ball (an (n-1)-sphere) is: A n ( r ) = d V n d r = 2 π n 2 Γ ( n 2 ) r n − 1 , {\displaystyle A_{n}(r)={\frac {dV_{n}}{dr}}={\frac {2\pi ^{\frac {n}{2}}}{\Gamma
Ball_(mathematics)
Topological space that locally resembles Euclidean space
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
Manifold
Sculpture in New York City
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
The_Sphere
Study of angle-preserving transformations
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
Inversive_geometry
Cohomology with real coefficients computed using differential forms
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 {\displaystyle
De_Rham_cohomology
How many linearly independent smooth nowhere-zero vector fields can be on an n-sphere
independent smooth nowhere-zero vector fields can be constructed on a sphere in n {\displaystyle n} -dimensional Euclidean space. A definitive answer was provided
Vector_fields_on_spheres
Whether a manifold which is a homotopy sphere is a sphere
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
Generalized Poincaré conjecture
Generalized_Poincaré_conjecture
Smooth manifold with an inner product on each tangent space
length, volume, and curvature are defined. Euclidean space, the n {\displaystyle n} -sphere, hyperbolic space, and smooth surfaces in three-dimensional space
Riemannian_manifold
Theorem in geometric topology
/ˌpwæ̃kɑːˈreɪ/, French: [pwɛ̃kaʁe]) is a theorem about the characterization of the 3-sphere (the hypersphere that bounds the 4-ball in four-dimensional space). Originally
Poincaré_conjecture
Manifold with the same rational homotopy groups as a sphere
n {\displaystyle n} -sphere is an n {\displaystyle n} -dimensional manifold with the same rational homotopy groups as the n {\displaystyle n} -sphere
Rational_homotopy_sphere
Pair of diametrically opposite points on a circle, sphere, or hypersphere
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
Antipodal_point
Topics referred to by the same term
sphere-like region or shell. Sphere may also refer to: Armillary sphere, a physical model of the celestial sphere Celestial sphere, the astronomical description
Sphere_(disambiguation)
Manifold with the same rational homology groups as a sphere
n {\displaystyle n} -sphere is an n {\displaystyle n} -dimensional manifold with the same rational homology groups as the n {\displaystyle n} -sphere
Rational_homology_sphere
Spherical geometry analog of a straight line
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
Great_circle
Theorem in differential topology
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
Hairy_ball_theorem
Theorem in Riemannian geometry
{\displaystyle (1,4]} then M {\displaystyle M} is homeomorphic to the n-sphere. (To be precise, we mean the sectional curvature of every tangent 2-plane
Sphere_theorem
Geometric concept
mathematics What is the maximum possible kissing number for n-dimensional spheres in (n + 1)-dimensional Euclidean space? More unsolved problems in mathematics
Kissing_number
Theory in algebraic topology
n + 1 , X n ) → H n ( X n , X n − 1 ) → H n − 1 ( X n − 1 , X n − 2 ) → ⋯ , {\displaystyle \cdots \to {H_{n+1}}(X_{n+1},X_{n})\to {H_{n}}(X_{n},X_{n-1})\to
Cellular_homology
Counts 0s of a vector field on a differentiable manifold using its Euler characteristic
even-dimensional n-sphere having no sources or sinks. Let M {\displaystyle M} be a differentiable manifold, of dimension n {\displaystyle n} , and v {\displaystyle
Poincaré–Hopf_theorem
Euclidean n-sphere become denser than non-poles). No efficient algorithm for the ideal (i.e., uniform) vector quantization of the Euclidean n-sphere is known
Pyramid_vector_quantization
Three-dimensional packing problem
packing of m>1 equal spheres in a sphere setting a new density record The best known packings of equal spheres in a sphere (complete up to N = 900), Packomania
Sphere_packing_in_a_sphere
Type of topological space
P n {\displaystyle \mathbb {RP} ^{n}} has the topology that is obtained by identifying antipodal points of the unit n {\displaystyle n} -sphere,
Real_projective_space
Fiber bundle of the 3-sphere over the 2-sphere, with 1-spheres as fibers
any natural number n, an n-dimensional sphere, or n-sphere, can be defined as the set of points in an ( n + 1 ) {\displaystyle (n+1)} -dimensional space
Hopf_fibration
Circle with radius of one
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
Unit_circle
Perimeter of a circle or ellipse
the locus corresponding to the edge of a disk. The circumference of a sphere is the circumference, or length, of any one of its great circles. The circumference
Circumference
Continuous surjection satisfying a local triviality condition
degree n + 1 {\displaystyle n+1} cohomology class in the total space of the bundle. In the case n = 1 {\displaystyle n=1} the sphere bundle is called a circle
Fiber_bundle
Topological invariant in mathematics
the n-sphere by the antipodal map. It follows that its Euler characteristic is exactly half that of the corresponding sphere – either 0 or 1. The n-dimensional
Euler_characteristic
Pathological embedding of the sphere in 3D space
The Alexander horned sphere is a pathological embedding of the 2-sphere into 3-dimensional Euclidean space. The topological object was discovered by J
Alexander_horned_sphere
Thing in mathematics and theoretical physics
then L ∩ (P ∪ N) = ∅, puncturing the union of quasi-sphere and counter-sphere. One example is the unit hyperbola that forms a quasi-sphere of the hyperbolic
Quasi-sphere
Theorem in topology
S n {\displaystyle S^{n}} is the n-sphere and B n {\displaystyle B^{n}} is the n-ball: If g : S n → R n {\displaystyle g:S^{n}\to \mathbb {R} ^{n}} is
Borsuk–Ulam_theorem
{\hat {n}}} in S n − 1 {\displaystyle S^{n-1}} , where S n {\displaystyle S^{n}} is the n-sphere (the surface of a ( n + 1 ) {\displaystyle (n+1)} -dimensional
Mean_width
Model of objects in the sky consisting of a framework of rings
armillary sphere (variations are known as spherical astrolabe, armilla, or armil) is a model of objects in the sky (on the celestial sphere), consisting
Armillary_sphere
Branch of differential geometry and differential topology
nontrivial); this implies, for example, that the only n-sphere that admits a symplectic form is the 2-sphere. A parallel that one can draw between the two subjects
Symplectic_geometry
Finding the smallest circle that contains all given points
The corresponding problem in n-dimensional space, the smallest bounding sphere problem, is to compute the smallest n-sphere that contains all of a given
Smallest-circle_problem
n, in homotopy". All definitions below consider a topological space X. A hole in X is, informally, a thing that prevents some suitably placed sphere from
Homotopical_connectivity
Sphere that contains a set of objects
of points, a bounding sphere, enclosing sphere or enclosing ball for that set is a d {\displaystyle d} -dimensional solid sphere containing all of these
Bounding_sphere
Manifold or algebraic variety of dimension n in a space of dimension n+1
Euclidean space of dimension n. This hypersurface is also a smooth manifold, and is called a hypersphere or an (n – 1)-sphere. A hypersurface that is a smooth
Hypersurface
"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
Spherical_measure
Type of n-manifold in topology
an n-manifold that cannot be expressed as a non-trivial connected sum of two n-manifolds. Non-trivial means that neither of the two is an n-sphere. A
Prime_manifold
Antipodally symmetric probability distribution on the n-sphere
Christopher Bingham, is an antipodally symmetric probability distribution on the n-sphere. It is a generalization of the Watson distribution and a special case of
Bingham_distribution
field of topology, a sphere bundle is a fiber bundle in which the fibers are spheres S n {\displaystyle S^{n}} of some dimension n. Similarly, in a disk
Sphere_bundle
Topological operation of turning a sphere inside-out without creasing
In differential topology, sphere eversion is a theoretical process of turning a sphere inside out in a three-dimensional space (the word eversion means
Sphere_eversion
Branch of mathematics
non-orientable when it is 0. The n-sphere admits a nowhere-vanishing continuous unit vector field if and only if n is odd. (For n = 2, this is sometimes called
Algebraic_topology
Maximally symmetric Lorentzian manifold with a positive cosmological constant
manifold with constant positive scalar curvature. It is analogue of an n-sphere, with a Lorentzian metric in place of the Riemannian metric of the latter
De_Sitter_space
Differential operator in mathematics
over an n-sphere of radius R {\displaystyle R} , and A n − 1 {\displaystyle A_{n-1}} is the hypervolume of the boundary of a unit n-sphere. There is
Laplace_operator
Special mathematical functions defined on the surface of a sphere
spherical harmonics are special functions defined on the surface of a sphere. They are often employed in solving partial differential equations in many
Spherical_harmonics
Method of proof involving paradoxical properties of infinite sums
Oriented n {\displaystyle n} -manifolds have an addition operation given by connected sum, with identity the n {\displaystyle n} -sphere. If A + B {\displaystyle
Eilenberg–Mazur_swindle
Study of angle-preserving transformations of a geometric space
flat, although often in the literature no distinction is maintained. The n-sphere is a locally conformally flat manifold that is not globally conformally
Conformal_geometry
SI derived unit of solid angle
to describe light and particle beams. Other multiples are rarely used. n-sphere Spat (angular unit) IAU designated constellations by area Stutzman, Warren
Steradian
Study of mathematical knots
(mathematics). For example, a higher-dimensional knot is an n-dimensional sphere embedded in (n+2)-dimensional Euclidean space. Knot theory can also be extended
Knot_theory
Homotopy invariant of maps between n-spheres
topology, the Hopf invariant is a homotopy invariant of certain maps between n-spheres. In 1931 Heinz Hopf used Clifford parallels to construct the Hopf map
Hopf_invariant
Algebraic construct classifying topological spaces
define the nth homotopy group, the base-point-preserving maps from an n-dimensional sphere (with base point) into a given space (with base point) are collected
Homotopy_group
Geometry of the surface of a sphere
geometry of the two-dimensional surface of a sphere or the n-dimensional surface of higher dimensional spheres. Long studied for its practical applications
Spherical_geometry
Mathematical concept
In R n {\displaystyle \mathbb {R} ^{n}} , the hyper surface ∏ i = 1 n x i = 1 {\displaystyle \prod _{i=1}^{n}x_{i}=1} is a hyperbolic affine sphere centered
Affine_sphere
Straight line segment that passes through the centre of a circle
for circles and spheres. However, they are special cases of a more general definition that is valid for any kind of n {\displaystyle n} -dimensional object
Diameter
Nonempty compact connected metric space
n-dimensional continuum. An n-sphere is a space homeomorphic to the standard n-sphere in the (n + 1)-dimensional Euclidean space. It is an n-dimensional homogeneous
Continuum_(topology)
Section of a sphere
In geometry, a spherical cap or spherical dome is a portion of a sphere or of a ball cut off by a plane. It is also a spherical segment of one base, i
Spherical_cap
Mathematical theory
set of two points. Explicitly, the nth space in the sphere spectrum is the n-dimensional sphere Sn, and the structure maps from the suspension of Sn
Sphere_spectrum
Algebraic structure associated with a topological space
n : A n → B n {\displaystyle f_{n}:A_{n}\to B_{n}} such that f n − 1 ∘ d n = e n ∘ f n {\displaystyle f_{n-1}\circ d_{n}=e_{n}\circ f_{n}} for all n.
Homology_(mathematics)
Rational function of the form (az + b)/(cz + d)
defined in spaces of dimension n > 2 as the bijective conformal orientation-preserving maps from the n-sphere to the n-sphere. Such a transformation is the
Möbius_transformation
Geometry founded on spheres
Lie sphere geometry is a geometrical theory of planar or spatial geometry in which the fundamental concept is the circle or sphere. It was introduced by
Lie_sphere_geometry
Optimization performance test
Sphere function of two variables In mathematical optimization, the sphere function is a convex function used as a performance test problem for optimization
Sphere_function
Concept in topology
continuous map from the n {\displaystyle n} -sphere S n {\displaystyle S^{n}} to itself (in the case n = 1 {\displaystyle n=1} , this is called the winding
Degree of a continuous mapping
Degree_of_a_continuous_mapping
listings see Category:Manifolds and its subcategories. Euclidean space, Rn n-sphere, Sn n-torus, Tn Real projective space, RPn Complex projective space, CPn Quaternionic
List_of_manifolds
Concept in theoretical astrophysics
{\displaystyle n_{e}=n_{p}} ): N n = n e n p β n ( T e ) = n e 2 β n ( T e ) , {\displaystyle N_{n}=n_{e}n_{p}\beta _{n}(T_{e})=n_{e}^{2}\beta _{n}(T_{e}),}
Strömgren_sphere
Model of n-dimensional hyperbolic geometry
spherical distance is inherited from Euclidean distance when the n-sphere is embedded in (n+1)-dimensional Euclidean space. Other models of hyperbolic space
Hyperboloid_model
Particular mapping that projects a sphere onto a plane
stereographic projection is a perspective projection of the sphere, through a specific point on the sphere (the pole or center of projection), onto a plane (the
Stereographic_projection
Mathematical theories
Bn → E on the n-skeleton of B. For every (n + 1)-simplex Δ in B, σn can be restricted to the boundary ∂Δ (which is a topological n-sphere). Because p sends
Obstruction_theory
Thought experiment in statistical physics
single nA-sphere and a single n B {\displaystyle n_{B}} -sphere, but instead ( N n A ) = N ! n A ! n B ! {\displaystyle {\binom {N}{n_{A}}}={\frac {N!}{n_{A}
Gibbs_paradox
Topics referred to by the same term
(NEVPT) n-entity n-flake n-gram n-group n-monoid n-player game n-skeleton n-slit interferometer n-slit interferometric equation n-sphere n-vector n-vector
N-
small sphere around the origin with the singular, complex hypersurface x 1 k 1 + ⋯ + x n k n = 0 {\displaystyle x_{1}^{k_{1}}+\cdots +x_{n}^{k_{n}}=0}
Brieskorn_manifold
Dense arrangement of congruent spheres in an infinite, regular arrangement
In geometry, close-packing of equal spheres is a dense arrangement of congruent spheres in an infinite, regular arrangement (or lattice). Carl Friedrich
Close-packing of equal spheres
Close-packing_of_equal_spheres
Topics referred to by the same term
attribute of the Lightweight Directory Access Protocol Symmetric group or Sn n-sphere or Sn sn (elliptic function), one of Jacobi's elliptic functions Sigma
SN
Mathematical space with two coordinates
each-other. Two-dimensional spaces can also be curved, for example the sphere and hyperbolic plane, sufficiently small portions of which appear like the
Two-dimensional_space
Property of a mathematical space
surface of a sphere. A two-dimensional Euclidean space is a two-dimensional space on the plane. The inside of a cube, a cylinder or a sphere is three-dimensional
Dimension
Topics referred to by the same term
Euclidean space R4 Exotic sphere, a differentiable n-manifold, homeomorphic but not diffeomorphic to the ordinary n-sphere Exotic atom, an atom with one
Exotic
Mathematics glossary
Convention: Throughout the article, I denotes the unit interval, Sn the n-sphere and Dn the n-disk. Also, throughout the article, spaces are assumed to be reasonable;
Glossary of algebraic topology
Glossary_of_algebraic_topology
Geometric space with four dimensions
first as a point, then as a growing sphere (until it reaches the "hyperdiameter" of the hypersphere), with the sphere then shrinking to a single point and
Four-dimensional_space
Probability distribution
with n-variate normal distribution over the unit (n-1)-sphere. Given a random variable X ∈ R n {\displaystyle {\boldsymbol {X}}\in \mathbb {R} ^{n}} that
Projected_normal_distribution
Operation combining two oriented knots
2-sphere in the 3-sphere which is not tame. In the smooth category, the n-sphere is known not to knot in the n + 1-sphere provided n ≠ 3. The case n =
Knot_(mathematics)
Theorem limiting types of conformal mappings in Euclidean space of dimension > 2
dimensions holds exactly when the conformal manifold is isometric with the n-sphere or projective space. Local versions of the result also hold: The Lie algebra
Liouville's theorem (conformal mappings)
Liouville's_theorem_(conformal_mappings)
Model particles in statistical mechanics
statistical mechanics, hard spheres are widely used as model particles in fluids and solids. They are defined simply as impenetrable spheres that cannot overlap
Hard_spheres
Online database of integer sequences
sequences corresponding to unsolved problems, such as "How many n-spheres can touch another n-sphere of the same size?" A001116 lists the first ten known solutions
On-Line Encyclopedia of Integer Sequences
On-Line_Encyclopedia_of_Integer_Sequences
Sphere touching all of a polyhedron's vertices
In geometry, a circumscribed sphere of a polyhedron is a sphere that contains the polyhedron and touches each of the polyhedron's vertices. The word circumsphere
Circumscribed_sphere
Construction for n-dimensional noise functions
computed over 2, 3, 4, or possibly 5 dimensions. For higher dimensions, n-spheres around n-simplex corners are not densely enough packed, reducing the support
Simplex_noise
Index of articles associated with the same name
state. In the theory of manifolds, an n-manifold is irreducible if any embedded (n − 1)-sphere bounds an embedded n-ball. Implicit in this definition is
Irreducibility_(mathematics)
Model of the extended complex plane plus a point at infinity
In mathematics, the Riemann sphere, named after Bernhard Riemann, is a model of the extended complex plane (also called the closed complex plane): the
Riemann_sphere
N SPHERE
N SPHERE
Male
Spanish
Spanish form of Latin Romanus, ROMÃN means "Roman."
Female
Irish
Irish Gaelic name CAILÃN means "girl."
Male
Irish
Old Irish Gaelic name BRADÃN means "salmon."
Male
Irish
Variant spelling of Irish Gaelic Lomán, LOMMÃN means "little bare one."Â
Male
Spanish
Spanish form of Hebrew Shimown, SIMÓN means "hearkening."
Male
Hebrew
Tiberian form of Hebrew Qeynan, QÊNĀN means "possession."
Female
Spanish
Spanish name ENCARNACIÓN means "incarnation."
Male
Hungarian
Hungarian name, possibly ZOLTÃN means "sultan."Â
Female
Spanish
Spanish name ASCENCIÓN means "ascension."
Surname or Lastname
Spanish (Truán)
Spanish (Truán) : nickname from truhán ‘knave’, ‘joker’.English (Cornwall) : unexplained; possibly a variant spelling of Trewin.
Male
Gaelic
Gaelic byname DUIBHÃN means "little black one."
Male
Irish
Variant spelling of Irish Gaelic Tighearnán, TIGERNÃN means "little lord."
Male
Irish
Irish name ABBÃN means "little abbot."
Male
Irish
Variant spelling of Irish Cathán, CADÃN means "little battle."
Male
Irish
Irish Gaelic name ULTÃN means "of Ulster."
Male
Vietnamese
Vietnamese name THUÃN means "tamed."
Male
Irish
Variant spelling of Irish Lorccán, LORCÃN means "little fierce one."
Female
Spanish
Spanish religious name VISITACIÓN means "visitation."
Male
Vietnamese
Vietnamese name VÄ‚N means "cloud" or "male."
Male
Spanish
Spanish form of Latin Salomon, SALOMÓN means "peaceable."
N SPHERE
N SPHERE
Boy/Male
Muslim
Support of the religion (Islam)
Boy/Male
Hindu, Indian
Lord Vishnu; Husband of Tulsi (Plant)
Girl/Female
Tamil
The energy
Female
Egyptian
, rightly guided.
Girl/Female
Hindu, Indian, Tamil
Future
Girl/Female
Indian
Non duality, One without second
Boy/Male
Tamil
Sentiment of Love and affection
Boy/Male
Hindu, Indian
Peaceful
Girl/Female
Tamil
Jyotishmati | ஜà¯à®¯à¯‹à®¤à®¿à®·à®®à®¤à®¿
Luminous, Lustrous
Girl/Female
Tamil
Charukeshi | சாரà¯à®•à¯‡à®·à¯€
Name of a Raga
N SPHERE
N SPHERE
N SPHERE
N SPHERE
N SPHERE
n.
See Daw, n.
n.
A measure of space equal to half an M (or em); an en.
n.
See Merrymake, n.
n.
See Kilt, n.
n.
See Keeve, n.
n.
See Lecher, n.
n.
See Keeve, n.
n.
See Elective, n.
n.
See Stour, n.
n.
See Jetty, n.
n.
See Nomad, n.
n.
See Mad, n.
n.
Offset, n., 4.
n.
See Intendant, n.
n.
See Hyp, n.
n.
See Solar, n.
n.
See Vanquish, n.
n.
See Invalid, n.
n.
See Lodge, n.