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# 3-sphere

In mathematics, a 3-sphere, or glome, is a higher-dimensional analogue of a sphere. [1]

103 relations: Abelian group, Absolute value, Alexandroff extension, Algebra homomorphism, American Journal of Physics, Associative property, Atlas (topology), Ball (mathematics), Charts on SO(3), Circle, Circle bundle, Circle group, Clifford torus, Compact space, Complex number, Conformal map, Conformal map projection, Connected space, Coordinate system, Cyclic group, Dante Alighieri, David W. Henderson, Determinant, Differentiable manifold, Dimension, Dionys Burger, Divine Comedy, Division ring, Edwin Abbott Abbott, Euclidean distance, Euclidean space, Euler's formula, Exponential map (Riemannian geometry), Fiber bundle, Flatland, Four-dimensional space, Georges Lemaître, Grigori Perelman, Group (mathematics), Group action, Henri Poincaré, Homeomorphism, Homology (mathematics), Homology sphere, Homotopy group, Homotopy groups of spheres, Hopf fibration, Hyperbolic Dehn surgery, Hyperplane, Hypersphere, ... Expand index (53 more) »

## Abelian group

In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written.

## Absolute value

In mathematics, the absolute value or modulus of a real number is the non-negative value of without regard to its sign.

## Alexandroff extension

In the mathematical field of topology, the Alexandroff extension is a way to extend a noncompact topological space by adjoining a single point in such a way that the resulting space is compact.

## Algebra homomorphism

A homomorphism between two associative algebras, A and B, over a field (or commutative ring) K, is a function F\colon A\to B such that for all k in K and x, y in A,.

## American Journal of Physics

The American Journal of Physics is a monthly, peer-reviewed scientific journal published by the American Association of Physics Teachers and the American Institute of Physics.

## Associative property

In mathematics, the associative property is a property of some binary operations.

## Atlas (topology)

In mathematics, particularly topology, one describes a manifold using an atlas.

## Ball (mathematics)

In mathematics, a ball is the space bounded by a sphere.

## Charts on SO(3)

In mathematics, the special orthogonal group in three dimensions, otherwise known as the rotation group SO(3), is a naturally occurring example of a manifold.

## Circle

A circle is a simple closed shape.

## Circle bundle

In mathematics, a circle bundle is a fiber bundle where the fiber is the circle \scriptstyle \mathbf^1.

## Circle group

In mathematics, the circle group, denoted by T, is the multiplicative group of all complex numbers with absolute value 1, that is, the unit circle in the complex plane or simply the unit complex numbers The circle group forms a subgroup of C&times;, the multiplicative group of all nonzero complex numbers.

## Clifford torus

In geometric topology, the Clifford torus is the simplest and most symmetric Euclidean space embedding of the cartesian product of two circles S1a and S1b.

## Compact space

In mathematics, and more specifically in general topology, compactness is a property that generalizes the notion of a subset of Euclidean space being closed (that is, containing all its limit points) and bounded (that is, having all its points lie within some fixed distance of each other).

## Complex number

A complex number is a number that can be expressed in the form, where and are real numbers, and is a solution of the equation.

## Conformal map

In mathematics, a conformal map is a function that preserves angles locally.

## Conformal map projection

In cartography, a conformal map projection is one in which any angle on Earth (a sphere or an ellipsoid) is preserved in the image of the projection, i.e. the projection is a conformal map in the mathematical sense.

## Connected space

In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint nonempty open subsets.

## Coordinate system

In geometry, a coordinate system is a system which uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space.

## Cyclic group

In algebra, a cyclic group or monogenous group is a group that is generated by a single element.

## Dante Alighieri

Durante degli Alighieri, commonly known as Dante Alighieri or simply Dante (c. 1265 – 1321), was a major Italian poet of the Late Middle Ages.

## David W. Henderson

David Wilson Henderson is a Professor Emeritus of Mathematics in the Department of Mathematics at Cornell University.

## Determinant

In linear algebra, the determinant is a value that can be computed from the elements of a square matrix.

## Differentiable manifold

In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a linear space to allow one to do calculus.

## Dimension

In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it.

## Dionys Burger

Dionys Burger (10 July 1892, Amsterdam - 19 April 1987) was a Dutch secondary school physics teacher and author of the novel Sphereland.

## Divine Comedy

The Divine Comedy (Divina Commedia) is a long narrative poem by Dante Alighieri, begun c. 1308 and completed in 1320, a year before his death in 1321.

## Division ring

In abstract algebra, a division ring, also called a skew field, is a ring in which division is possible.

## Edwin Abbott Abbott

Edwin Abbott Abbott (20 December 1838 – 12 October 1926) was an English schoolmaster and theologian, best known as the author of the novella Flatland (1884).

## Euclidean distance

In mathematics, the Euclidean distance or Euclidean metric is the "ordinary" straight-line distance between two points in Euclidean space.

## Euclidean space

In geometry, Euclidean space encompasses the two-dimensional Euclidean plane, the three-dimensional space of Euclidean geometry, and certain other spaces.

## Euler's formula

Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function.

## Exponential map (Riemannian geometry)

In Riemannian geometry, an exponential map is a map from a subset of a tangent space TpM of a Riemannian manifold (or pseudo-Riemannian manifold) M to M itself.

## Fiber bundle

In mathematics, and particularly topology, a fiber bundle (or, in British English, fibre bundle) is a space that is locally a product space, but globally may have a different topological structure.

## Flatland

Flatland: A Romance of Many Dimensions is a satirical novella by the English schoolmaster Edwin Abbott Abbott, first published in 1884 by Seeley & Co.

## Four-dimensional space

A four-dimensional space or 4D space is a mathematical extension of the concept of three-dimensional or 3D space.

## Georges Lemaître

Georges Henri Joseph Édouard Lemaître, RAS Associate (17 July 1894 &ndash; 20 June 1966) was a Belgian Catholic Priest, astronomer and professor of physics at the Catholic University of Leuven.

## Grigori Perelman

Grigori Yakovlevich Perelman (a; born 13 June 1966) is a Russian mathematician.

## Group (mathematics)

In mathematics, a group is an algebraic structure consisting of a set of elements equipped with an operation that combines any two elements to form a third element and that satisfies four conditions called the group axioms, namely closure, associativity, identity and invertibility.

## Group action

In mathematics, an action of a group is a formal way of interpreting the manner in which the elements of the group correspond to transformations of some space in a way that preserves the structure of that space.

## Henri Poincaré

Jules Henri Poincaré (29 April 1854 – 17 July 1912) was a French mathematician, theoretical physicist, engineer, and philosopher of science.

## Homeomorphism

In the mathematical field of topology, a homeomorphism or topological isomorphism or bi continuous function is a continuous function between topological spaces that has a continuous inverse function.

## Homology (mathematics)

In mathematics, homology is a general way of associating a sequence of algebraic objects such as abelian groups or modules to other mathematical objects such as topological spaces.

## Homology sphere

In algebraic topology, a homology sphere is an n-manifold X having the homology groups of an n-sphere, for some integer n ≥ 1.

## Homotopy group

In mathematics, homotopy groups are used in algebraic topology to classify topological spaces.

## Homotopy groups of spheres

In the mathematical field of algebraic topology, the homotopy groups of spheres describe how spheres of various dimensions can wrap around each other.

## Hopf fibration

In the mathematical field of differential topology, the Hopf fibration (also known as the Hopf bundle or Hopf map) describes a 3-sphere (a hypersphere in four-dimensional space) in terms of circles and an ordinary sphere.

## Hyperbolic Dehn surgery

In mathematics, hyperbolic Dehn surgery is an operation by which one can obtain further hyperbolic 3-manifolds from a given cusped hyperbolic 3-manifold.

## Hyperplane

In geometry, a hyperplane is a subspace whose dimension is one less than that of its ambient space.

## Hypersphere

In geometry of higher dimensions, a hypersphere is the set of points at a constant distance from a given point called its center.

## Injective function

In mathematics, an injective function or injection or one-to-one function is a function that preserves distinctness: it never maps distinct elements of its domain to the same element of its codomain.

## Isomorphism

In mathematics, an isomorphism (from the Ancient Greek: ἴσος isos "equal", and μορφή morphe "form" or "shape") is a homomorphism or morphism (i.e. a mathematical mapping) that can be reversed by an inverse morphism.

## Jeffrey Weeks (mathematician)

Jeffrey Renwick Weeks (born December 10, 1956) is an American mathematician, a geometric topologist and cosmologist.

## Knot theory

In topology, knot theory is the study of mathematical knots.

## Latitude

In geography, latitude is a geographic coordinate that specifies the north–south position of a point on the Earth's surface.

## Lie algebra

In mathematics, a Lie algebra (pronounced "Lee") is a vector space \mathfrak g together with a non-associative, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g; (x, y) \mapsto, called the Lie bracket, satisfying the Jacobi identity.

## Lie group

In mathematics, a Lie group (pronounced "Lee") is a group that is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure.

## Longitude

Longitude, is a geographic coordinate that specifies the east-west position of a point on the Earth's surface.

## Manifold

In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point.

## Mathematics

Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change.

## Matrix (mathematics)

In mathematics, a matrix (plural: matrices) is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns.

## Meridian (perimetry, visual field)

Meridian (plural: "meridians") is used in perimetry and in specifying visual fields.

## Metric tensor

In the mathematical field of differential geometry, a metric tensor is a type of function which takes as input a pair of tangent vectors and at a point of a surface (or higher dimensional differentiable manifold) and produces a real number scalar in a way that generalizes many of the familiar properties of the dot product of vectors in Euclidean space.

## N-sphere

In mathematics, the n-sphere is the generalization of the ordinary sphere to spaces of arbitrary dimension.

## Non-abelian group

In mathematics, and specifically in group theory, a non-abelian group, sometimes called a non-commutative group, is a group (G, ∗) in which there exists at least one pair of elements a and b of G, such that a ∗ b ≠ b ∗ a.

## Octonion

In mathematics, the octonions are a normed division algebra over the real numbers, usually represented by the capital letter O, using boldface O or blackboard bold \mathbb O. There are three lower-dimensional normed division algebras over the reals: the real numbers R themselves, the complex numbers C, and the quaternions H. The octonions have eight dimensions; twice the number of dimensions of the quaternions, of which they are an extension.

## Parallelizable manifold

In mathematics, a differentiable manifold M of dimension n is called parallelizable if there exist smooth vector fields on the manifold, such that at any point p of M the tangent vectors provide a basis of the tangent space at p. Equivalently, the tangent bundle is a trivial bundle, so that the associated principal bundle of linear frames has a section on M. A particular choice of such a basis of vector fields on M is called a parallelization (or an absolute parallelism) of M.

## Pauli matrices

In mathematical physics and mathematics, the Pauli matrices are a set of three complex matrices which are Hermitian and unitary.

## Poincaré conjecture

In mathematics, the Poincaré conjecture is a theorem about the characterization of the 3-sphere, which is the hypersphere that bounds the unit ball in four-dimensional space.

The Pontifical Academy of Sciences (Pontificia accademia delle scienze, Pontificia Academia Scientiarum) is a scientific academy of the Vatican City, established in 1936 by Pope Pius XI, and thriving with the blessing of the Papacy ever since.

## Quaternion

In mathematics, the quaternions are a number system that extends the complex numbers.

## Quaternions and spatial rotation

Unit quaternions, also known as versors, provide a convenient mathematical notation for representing orientations and rotations of objects in three dimensions.

## Quotient space (topology)

In topology and related areas of mathematics, a quotient space (also called an identification space) is, intuitively speaking, the result of identifying or "gluing together" certain points of a given topological space.

## Reeb foliation

In mathematics, the Reeb foliation is a particular foliation of the 3-sphere, introduced by the French mathematician Georges Reeb (1920–1992).

## Riemann sphere

In mathematics, the Riemann sphere, named after Bernhard Riemann, is a model of the extended complex plane, the complex plane plus a point at infinity.

## Riemannian manifold

In differential geometry, a (smooth) Riemannian manifold or (smooth) Riemannian space (M,g) is a real, smooth manifold M equipped with an inner product g_p on the tangent space T_pM at each point p that varies smoothly from point to point in the sense that if X and Y are differentiable vector fields on M, then p \mapsto g_p(X(p),Y(p)) is a smooth function.

## Rotation group SO(3)

In mechanics and geometry, the 3D rotation group, often denoted SO(3), is the group of all rotations about the origin of three-dimensional Euclidean space R3 under the operation of composition.

## Section (fiber bundle)

In the mathematical field of topology, a section (or cross section) of a fiber bundle E is a continuous right inverse of the projection function \pi.

## Sectional curvature

In Riemannian geometry, the sectional curvature is one of the ways to describe the curvature of Riemannian manifolds.

## Simplex

In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions.

## Simply connected space

In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every path between two points can be continuously transformed (intuitively for embedded spaces, staying within the space) into any other such path while preserving the two endpoints in question.

## Smoothness

In mathematical analysis, the smoothness of a function is a property measured by the number of derivatives it has that are continuous.

## Special unitary group

In mathematics, the special unitary group of degree, denoted, is the Lie group of unitary matrices with determinant 1.

## Sphere

A sphere (from Greek σφαῖρα — sphaira, "globe, ball") is a perfectly round geometrical object in three-dimensional space that is the surface of a completely round ball (viz., analogous to the circular objects in two dimensions, where a "circle" circumscribes its "disk").

## Sphereland

Sphereland: A Fantasy About Curved Spaces and an Expanding Universe is a 1965 novel by Dionys Burger, and is a sequel to Flatland, a novel by "A Square" (a pen name of Edwin Abbott Abbott).

## Spherical coordinate system

In mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the radial distance of that point from a fixed origin, its polar angle measured from a fixed zenith direction, and the azimuth angle of its orthogonal projection on a reference plane that passes through the origin and is orthogonal to the zenith, measured from a fixed reference direction on that plane.

## Square root

In mathematics, a square root of a number a is a number y such that; in other words, a number y whose square (the result of multiplying the number by itself, or) is a. For example, 4 and −4 are square roots of 16 because.

## Stereographic projection

In geometry, the stereographic projection is a particular mapping (function) that projects a sphere onto a plane.

## Submanifold

In mathematics, a submanifold of a manifold M is a subset S which itself has the structure of a manifold, and for which the inclusion map S → M satisfies certain properties.

## Surface (topology)

In topology and differential geometry, a surface is a two-dimensional manifold, and, as such, may be an "abstract surface" not embedded in any Euclidean space.

## Symplectic group

In mathematics, the name symplectic group can refer to two different, but closely related, collections of mathematical groups, denoted and, the latter is called the compact symplectic group.

## Tangent bundle

In differential geometry, the tangent bundle of a differentiable manifold M is a manifold TM which assembles all the tangent vectors in M. As a set, it is given by the disjoint unionThe disjoint union ensures that for any two points x1 and x2 of manifold M the tangent spaces T1 and T2 have no common vector.

## Tesseract

In geometry, the tesseract is the four-dimensional analogue of the cube; the tesseract is to the cube as the cube is to the square.

## Topological space

In topology and related branches of mathematics, a topological space may be defined as a set of points, along with a set of neighbourhoods for each point, satisfying a set of axioms relating points and neighbourhoods.

## Topology

In mathematics, topology (from the Greek τόπος, place, and λόγος, study) is concerned with the properties of space that are preserved under continuous deformations, such as stretching, crumpling and bending, but not tearing or gluing.

## Torus

In geometry, a torus (plural tori) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis coplanar with the circle.

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## Unit circle

In mathematics, a unit circle is a circle with a radius of one.

## Vector field

In vector calculus and physics, a vector field is an assignment of a vector to each point in a subset of space.

## Vector fields on spheres

In mathematics, the discussion of vector fields on spheres was a classical problem of differential topology, beginning with the hairy ball theorem, and early work on the classification of division algebras.

## Versor

In mathematics, a versor is a quaternion of norm one (a unit quaternion).

## Volume form

In mathematics, a volume form on a differentiable manifold is a top-dimensional form (i.e., a differential form of top degree).

## 3-manifold

In mathematics, a 3-manifold is a space that locally looks like Euclidean 3-dimensional space.

## 4-polytope

In geometry, a 4-polytope (sometimes also called a polychoron, polycell, or polyhedroid) is a four-dimensional polytope.

## References

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