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In mathematics, a **3-sphere**, or **glome**,^{ [1] } is a higher-dimensional analogue of a sphere. It may be embedded in 4-dimensional Euclidean space as the set of points equidistant from a fixed central point. Analogous to how the boundary of a ball in three dimensions is an ordinary sphere (or 2-sphere, a two-dimensional surface), the boundary of a ball in four dimensions is a 3-sphere (an object with three dimensions). A 3-sphere is an example of a 3-manifold and an n-sphere.

- Definition
- Properties
- Elementary properties
- Topological properties
- Geometric properties
- Topological construction
- Gluing
- One-point compactification
- Coordinate systems on the 3-sphere
- Hyperspherical coordinates
- Hopf coordinates
- Stereographic coordinates
- Group structure
- In literature
- See also
- References
- External links

In coordinates, a 3-sphere with center (*C*_{0}, *C*_{1}, *C*_{2}, *C*_{3}) and radius r is the set of all points (*x*_{0}, *x*_{1}, *x*_{2}, *x*_{3}) in real, 4-dimensional space (**R**^{4}) such that

The 3-sphere centered at the origin with radius 1 is called the **unit 3-sphere** and is usually denoted *S*^{3}:

It is often convenient to regard **R**^{4} as the space with 2 complex dimensions (**C**^{2}) or the quaternions (**H**). The unit 3-sphere is then given by

or

This description as the quaternions of norm one identifies the 3-sphere with the versors in the quaternion division ring. Just as the unit circle is important for planar polar coordinates, so the 3-sphere is important in the polar view of 4-space involved in quaternion multiplication. See polar decomposition of a quaternion for details of this development of the three-sphere. This view of the 3-sphere is the basis for the study of elliptic space as developed by Georges Lemaître.^{ [2] }

The 3-dimensional surface volume of a 3-sphere of radius r is

while the 4-dimensional hypervolume (the volume of the 4-dimensional region bounded by the 3-sphere) is

Every non-empty intersection of a 3-sphere with a three-dimensional hyperplane is a 2-sphere (unless the hyperplane is tangent to the 3-sphere, in which case the intersection is a single point). As a 3-sphere moves through a given three-dimensional hyperplane, the intersection starts out as a point, then becomes a growing 2-sphere that reaches its maximal size when the hyperplane cuts right through the "equator" of the 3-sphere. Then the 2-sphere shrinks again down to a single point as the 3-sphere leaves the hyperplane.

A 3-sphere is a compact, connected, 3-dimensional manifold without boundary. It is also simply connected. What this means, in the broad sense, is that any loop, or circular path, on the 3-sphere can be continuously shrunk to a point without leaving the 3-sphere. The Poincaré conjecture, proved in 2003 by Grigori Perelman, provides that the 3-sphere is the only three-dimensional manifold (up to homeomorphism) with these properties.

The 3-sphere is homeomorphic to the one-point compactification of **R**^{3}. In general, any topological space that is homeomorphic to the 3-sphere is called a **topological 3-sphere**.

The homology groups of the 3-sphere are as follows: H_{0}(S^{3},**Z**) and H_{3}(S^{3},**Z**) are both infinite cyclic, while H_{i}(S^{3},**Z**) = {0} for all other indices i. Any topological space with these homology groups is known as a homology 3-sphere. Initially Poincaré conjectured that all homology 3-spheres are homeomorphic to *S*^{3}, but then he himself constructed a non-homeomorphic one, now known as the Poincaré homology sphere. Infinitely many homology spheres are now known to exist. For example, a Dehn filling with slope 1/*n* on any knot in the 3-sphere gives a homology sphere; typically these are not homeomorphic to the 3-sphere.

As to the homotopy groups, we have π_{1}(S^{3}) = π_{2}(S^{3}) = {0} and π_{3}(S^{3}) is infinite cyclic. The higher-homotopy groups (*k* ≥ 4) are all finite abelian but otherwise follow no discernible pattern. For more discussion see homotopy groups of spheres.

k | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 |

π_{k}(S^{3}) | 0 | 0 | 0 | Z | Z_{2} | Z_{2} | Z_{12} | Z_{2} | Z_{2} | Z_{3} | Z_{15} | Z_{2} | Z_{2}⊕Z_{2} | Z_{12}⊕Z_{2} | Z_{84}⊕Z_{2}⊕Z_{2} | Z_{2}⊕Z_{2} | Z_{6} |

The 3-sphere is naturally a smooth manifold, in fact, a closed embedded submanifold of **R**^{4}. The Euclidean metric on **R**^{4} induces a metric on the 3-sphere giving it the structure of a Riemannian manifold. As with all spheres, the 3-sphere has constant positive sectional curvature equal to 1/*r*^{2} where r is the radius.

Much of the interesting geometry of the 3-sphere stems from the fact that the 3-sphere has a natural Lie group structure given by quaternion multiplication (see the section below on group structure). The only other spheres with such a structure are the 0-sphere and the 1-sphere (see circle group).

Unlike the 2-sphere, the 3-sphere admits nonvanishing vector fields (sections of its tangent bundle). One can even find three linearly independent and nonvanishing vector fields. These may be taken to be any left-invariant vector fields forming a basis for the Lie algebra of the 3-sphere. This implies that the 3-sphere is parallelizable. It follows that the tangent bundle of the 3-sphere is trivial. For a general discussion of the number of linear independent vector fields on a n-sphere, see the article vector fields on spheres.

There is an interesting action of the circle group **T** on *S*^{3} giving the 3-sphere the structure of a principal circle bundle known as the Hopf bundle. If one thinks of *S*^{3} as a subset of **C**^{2}, the action is given by

- .

The orbit space of this action is homeomorphic to the two-sphere *S*^{2}. Since *S*^{3} is not homeomorphic to *S*^{2} × *S*^{1}, the Hopf bundle is nontrivial.

There are several well-known constructions of the three-sphere. Here we describe gluing a pair of three-balls and then the one-point compactification.

A 3-sphere can be constructed topologically by "gluing" together the boundaries of a pair of 3-balls. The boundary of a 3-ball is a 2-sphere, and these two 2-spheres are to be identified. That is, imagine a pair of 3-balls of the same size, then superpose them so that their 2-spherical boundaries match, and let matching pairs of points on the pair of 2-spheres be identically equivalent to each other. In analogy with the case of the 2-sphere (see below), the gluing surface is called an equatorial sphere.

Note that the interiors of the 3-balls are not glued to each other. One way to think of the fourth dimension is as a continuous real-valued function of the 3-dimensional coordinates of the 3-ball, perhaps considered to be "temperature". We take the "temperature" to be zero along the gluing 2-sphere and let one of the 3-balls be "hot" and let the other 3-ball be "cold". The "hot" 3-ball could be thought of as the "upper hemisphere" and the "cold" 3-ball could be thought of as the "lower hemisphere". The temperature is highest/lowest at the centers of the two 3-balls.

This construction is analogous to a construction of a 2-sphere, performed by gluing the boundaries of a pair of disks. A disk is a 2-ball, and the boundary of a disk is a circle (a 1-sphere). Let a pair of disks be of the same diameter. Superpose them and glue corresponding points on their boundaries. Again one may think of the third dimension as temperature. Likewise, we may inflate the 2-sphere, moving the pair of disks to become the northern and southern hemispheres.

After removing a single point from the 2-sphere, what remains is homeomorphic to the Euclidean plane. In the same way, removing a single point from the 3-sphere yields three-dimensional space. An extremely useful way to see this is via stereographic projection. We first describe the lower-dimensional version.

Rest the south pole of a unit 2-sphere on the xy-plane in three-space. We map a point P of the sphere (minus the north pole N) to the plane by sending P to the intersection of the line NP with the plane. Stereographic projection of a 3-sphere (again removing the north pole) maps to three-space in the same manner. (Notice that, since stereographic projection is conformal, round spheres are sent to round spheres or to planes.)

A somewhat different way to think of the one-point compactification is via the exponential map. Returning to our picture of the unit two-sphere sitting on the Euclidean plane: Consider a geodesic in the plane, based at the origin, and map this to a geodesic in the two-sphere of the same length, based at the south pole. Under this map all points of the circle of radius π are sent to the north pole. Since the open unit disk is homeomorphic to the Euclidean plane, this is again a one-point compactification.

The exponential map for 3-sphere is similarly constructed; it may also be discussed using the fact that the 3-sphere is the Lie group of unit quaternions.

The four Euclidean coordinates for *S*^{3} are redundant since they are subject to the condition that *x*_{0}^{2} + *x*_{1}^{2} + *x*_{2}^{2} + *x*_{3}^{2} = 1. As a 3-dimensional manifold one should be able to parameterize *S*^{3} by three coordinates, just as one can parameterize the 2-sphere using two coordinates (such as latitude and longitude). Due to the nontrivial topology of *S*^{3} it is impossible to find a single set of coordinates that cover the entire space. Just as on the 2-sphere, one must use *at least* two coordinate charts. Some different choices of coordinates are given below.

It is convenient to have some sort of hyperspherical coordinates on *S*^{3} in analogy to the usual spherical coordinates on *S*^{2}. One such choice — by no means unique — is to use (*ψ*, *θ*, *φ*), where

where ψ and θ run over the range 0 to π, and φ runs over 0 to 2π. Note that, for any fixed value of ψ, θ and φ parameterize a 2-sphere of radius r sin *ψ*, except for the degenerate cases, when ψ equals 0 or π, in which case they describe a point.

The round metric on the 3-sphere in these coordinates is given by^{[ citation needed ]}

and the volume form by

These coordinates have an elegant description in terms of quaternions. Any unit quaternion q can be written as a versor:

where τ is a unit imaginary quaternion; that is, a quaternion that satisfies *τ*^{2} = −1. This is the quaternionic analogue of Euler's formula. Now the unit imaginary quaternions all lie on the unit 2-sphere in Im **H** so any such τ can be written:

With τ in this form, the unit quaternion q is given by

where *x*_{0,1,2,3} are as above.

When q is used to describe spatial rotations (cf. quaternions and spatial rotations), it describes a rotation about τ through an angle of 2*ψ*.

For unit radius another choice of hyperspherical coordinates, (*η*, *ξ*_{1}, *ξ*_{2}), makes use of the embedding of *S*^{3} in **C**^{2}. In complex coordinates (*z*_{1}, *z*_{2}) ∈ **C**^{2} we write

This could also be expressed in **R**^{4} as

Here η runs over the range 0 to π/2, and *ξ*_{1} and *ξ*_{2} can take any values between 0 and 2π. These coordinates are useful in the description of the 3-sphere as the Hopf bundle

For any fixed value of η between 0 and π/2, the coordinates (*ξ*_{1}, *ξ*_{2}) parameterize a 2-dimensional torus. Rings of constant *ξ*_{1} and *ξ*_{2} above form simple orthogonal grids on the tori. See image to right. In the degenerate cases, when η equals 0 or π/2, these coordinates describe a circle.

The round metric on the 3-sphere in these coordinates is given by

and the volume form by

To get the interlocking circles of the Hopf fibration, make a simple substitution in the equations above^{ [3] }

In this case η, and *ξ*_{1} specify which circle, and *ξ*_{2} specifies the position along each circle. One round trip (0 to 2π) of *ξ*_{1} or *ξ*_{2} equates to a round trip of the torus in the 2 respective directions.

Another convenient set of coordinates can be obtained via stereographic projection of *S*^{3} from a pole onto the corresponding equatorial **R**^{3} hyperplane. For example, if we project from the point (−1, 0, 0, 0) we can write a point p in *S*^{3} as

where **u** = (*u*_{1}, *u*_{2}, *u*_{3}) is a vector in **R**^{3} and ||*u*||^{2} = *u*_{1}^{2} + *u*_{2}^{2} + *u*_{3}^{2}. In the second equality above, we have identified p with a unit quaternion and **u** = *u*_{1}*i* + *u*_{2}*j* + *u*_{3}*k* with a pure quaternion. (Note that the numerator and denominator commute here even though quaternionic multiplication is generally noncommutative). The inverse of this map takes *p* = (*x*_{0}, *x*_{1}, *x*_{2}, *x*_{3}) in *S*^{3} to

We could just as well have projected from the point (1, 0, 0, 0), in which case the point p is given by

where **v** = (*v*_{1}, *v*_{2}, *v*_{3}) is another vector in **R**^{3}. The inverse of this map takes p to

Note that the **u** coordinates are defined everywhere but (−1, 0, 0, 0) and the **v** coordinates everywhere but (1, 0, 0, 0). This defines an atlas on *S*^{3} consisting of two coordinate charts or "patches", which together cover all of *S*^{3}. Note that the transition function between these two charts on their overlap is given by

and vice versa.

When considered as the set of unit quaternions, *S*^{3} inherits an important structure, namely that of quaternionic multiplication. Because the set of unit quaternions is closed under multiplication, *S*^{3} takes on the structure of a group. Moreover, since quaternionic multiplication is smooth, *S*^{3} can be regarded as a real Lie group. It is a nonabelian, compact Lie group of dimension 3. When thought of as a Lie group *S*^{3} is often denoted Sp(1) or U(1, **H**).

It turns out that the only spheres that admit a Lie group structure are *S*^{1} , thought of as the set of unit complex numbers, and *S*^{3}, the set of unit quaternions (The degenerate case *S*^{0} which consists of the real numbers 1 and -1 is also a Lie group, albeit a 0-dimensional one). One might think that *S*^{7}, the set of unit octonions, would form a Lie group, but this fails since octonion multiplication is nonassociative. The octonionic structure does give *S*^{7} one important property: * parallelizability *. It turns out that the only spheres that are parallelizable are *S*^{1}, *S*^{3}, and *S*^{7}.

By using a matrix representation of the quaternions, **H**, one obtains a matrix representation of *S*^{3}. One convenient choice is given by the Pauli matrices:

This map gives an injective algebra homomorphism from **H** to the set of 2 × 2 complex matrices. It has the property that the absolute value of a quaternion q is equal to the square root of the determinant of the matrix image of q.

The set of unit quaternions is then given by matrices of the above form with unit determinant. This matrix subgroup is precisely the special unitary group SU(2). Thus, *S*^{3} as a Lie group is isomorphic to SU(2).

Using our Hopf coordinates (*η*, *ξ*_{1}, *ξ*_{2}) we can then write any element of SU(2) in the form

Another way to state this result is if we express the matrix representation of an element of SU(2) as a linear combination of the Pauli matrices. It is seen that an arbitrary element *U* ∈ SU(2) can be written as

The condition that the determinant of U is +1 implies that the coefficients α_{1} are constrained to lie on a 3-sphere.

In Edwin Abbott Abbott's * Flatland *, published in 1884, and in * Sphereland *, a 1965 sequel to Flatland by Dionys Burger, the 3-sphere is referred to as an **oversphere**, and a 4-sphere is referred to as a **hypersphere**.

Writing in the American Journal of Physics,^{ [4] } Mark A. Peterson describes three different ways of visualizing 3-spheres and points out language in * The Divine Comedy * that suggests Dante viewed the Universe in the same way.

In mathematics and physics, **Laplace's equation** is a second-order partial differential equation named after Pierre-Simon Laplace who first studied its properties. This is often written as

In geometry, the **stereographic projection** is a particular mapping (function) that projects a sphere onto a plane. The projection is defined on the entire sphere, except at one point: the projection point. Where it is defined, the mapping is smooth and bijective. It is conformal, meaning that it preserves angles at which curves meet. It is neither isometric nor area-preserving: that is, it preserves neither distances nor the areas of figures.

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 under the operation of composition. By definition, a rotation about the origin is a transformation that preserves the origin, Euclidean distance, and orientation. Every non-trivial rotation is determined by its axis of rotation and its angle of rotation. Composing two rotations results in another rotation; every rotation has a unique inverse rotation; and the identity map satisfies the definition of a rotation. Owing to the above properties, the set of all rotations is a group under composition. Rotations are not commutative, making it a nonabelian group. Moreover, the rotation group has a natural structure as a manifold for which the group operations are smoothly differentiable; so it is in fact a Lie group. It is compact and has dimension 3.

In physics and mathematics, the **Lorentz group** is the group of all Lorentz transformations of Minkowski spacetime, the classical and quantum setting for all (non-gravitational) physical phenomena. The Lorentz group is named for the Dutch physicist Hendrik Lorentz.

In the mathematical field of differential topology, the **Hopf fibration** describes a 3-sphere in terms of circles and an ordinary sphere. Discovered by Heinz Hopf in 1931, it is an influential early example of a fiber bundle. Technically, Hopf found a many-to-one continuous function from the 3-sphere onto the 2-sphere such that each distinct *point* of the 2-sphere is mapped from a distinct great circle of the 3-sphere. Thus the 3-sphere is composed of fibers, where each fiber is a circle — one for each point of the 2-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), named after the physicist Felix Bloch.

Spatial rotations in three dimensions can be parametrized using both Euler angles and unit quaternions. This article explains how to convert between the two representations. Actually this simple use of "quaternions" was first presented by Euler some seventy years earlier than Hamilton to solve the problem of magic squares. For this reason the dynamics community commonly refers to quaternions in this application as "Euler parameters".

In mathematics, the group of **rotations about a fixed point in four-dimensional Euclidean space** is denoted **SO(4)**. The name comes from the fact that it is the special orthogonal group of order 4.

**Conical coordinates** are a three-dimensional orthogonal coordinate system consisting of concentric spheres and by two families of perpendicular cones, aligned along the *z*- and *x*-axes, respectively.

In special functions, a topic in mathematics, **spin-weighted spherical harmonics** are generalizations of the standard spherical harmonics and—like the usual spherical harmonics—are functions on the sphere. Unlike ordinary spherical harmonics, the spin-weighted harmonics are U(1) gauge fields rather than scalar fields: mathematically, they take values in a complex line bundle. The spin-weighted harmonics are organized by degree l, just like ordinary spherical harmonics, but have an additional **spin weight**s that reflects the additional U(1) symmetry. A special basis of harmonics can be derived from the Laplace spherical harmonics *Y*_{lm}, and are typically denoted by _{s}*Y*_{lm}, where l and m are the usual parameters familiar from the standard Laplace spherical harmonics. In this special basis, the spin-weighted spherical harmonics appear as actual functions, because the choice of a polar axis fixes the U(1) gauge ambiguity. The spin-weighted spherical harmonics can be obtained from the standard spherical harmonics by application of spin raising and lowering operators. In particular, the spin-weighted spherical harmonics of spin weight *s* = 0 are simply the standard spherical harmonics:

In geometry, various **formalisms** exist to express a rotation in three dimensions as a mathematical transformation. In physics, this concept is applied to classical mechanics where rotational kinematics is the science of quantitative description of a purely rotational motion. The orientation of an object at a given instant is described with the same tools, as it is defined as an imaginary rotation from a reference placement in space, rather than an actually observed rotation from a previous placement in space.

In mathematics, the **Prolate spheroidal wave functions** (PSWF) are a set of orthogonal bandlimited functions. They are eigenfunctions of a timelimiting operation followed by a lowpassing operation. Let denote the time truncation operator, such that if and only if is timelimited within . Similarly, let denote an ideal low-pass filtering operator, such that if and only if is bandlimited within . The operator turns out to be linear, bounded and self-adjoint. For we denote with the n-th eigenfunction, defined as

In mathematics, the **spectral theory of ordinary differential equations** is the part of spectral theory concerned with the determination of the spectrum and eigenfunction expansion associated with a linear ordinary differential equation. In his dissertation Hermann Weyl generalized the classical Sturm–Liouville theory on a finite closed interval to second order differential operators with singularities at the endpoints of the interval, possibly semi-infinite or infinite. Unlike the classical case, the spectrum may no longer consist of just a countable set of eigenvalues, but may also contain a continuous part. In this case the eigenfunction expansion involves an integral over the continuous part with respect to a spectral measure, given by the Titchmarsh–Kodaira formula. The theory was put in its final simplified form for singular differential equations of even degree by Kodaira and others, using von Neumann's spectral theorem. It has had important applications in quantum mechanics, operator theory and harmonic analysis on semisimple Lie groups.

In mathematics, a **Coulomb wave function** is a solution of the **Coulomb wave equation**, named after Charles-Augustin de Coulomb. They are used to describe the behavior of charged particles in a Coulomb potential and can be written in terms of confluent hypergeometric functions or Whittaker functions of imaginary argument.

In general relativity, a point mass deflects a light ray with impact parameter by an angle approximately equal to

In fluid dynamics, the **Oseen equations** describe the flow of a viscous and incompressible fluid at small Reynolds numbers, as formulated by Carl Wilhelm Oseen in 1910. Oseen flow is an improved description of these flows, as compared to Stokes flow, with the (partial) inclusion of convective acceleration.

In fluid dynamics, a **cnoidal wave** is a nonlinear and exact periodic wave solution of the Korteweg–de Vries equation. These solutions are in terms of the Jacobi elliptic function *cn*, which is why they are coined *cn*oidal waves. They are used to describe surface gravity waves of fairly long wavelength, as compared to the water depth.

**Symmetries in quantum mechanics** describe features of spacetime and particles which are unchanged under some transformation, in the context of quantum mechanics, relativistic quantum mechanics and quantum field theory, and with applications in the mathematical formulation of the standard model and condensed matter physics. In general, symmetry in physics, invariance, and conservation laws, are fundamentally important constraints for formulating physical theories and models. In practice, they are powerful methods for solving problems and predicting what can happen. While conservation laws do not always give the answer to the problem directly, they form the correct constraints and the first steps to solving a multitude of problems.

In image analysis, the **generalized structure tensor (GST)** is an extension of the Cartesian structure tensor to curvilinear coordinates. It is mainly used to detect and to represent the "direction" parameters of curves, just as the Cartesian structure tensor detects and represents the direction in Cartesian coordinates. Curve families generated by pairs of locally orthogonal functions have been the best studied.

**Moffatt eddies** are sequences of eddies that develop in corners bounded by plane walls due to an arbitrary disturbance acting at asymptotically large distances from the corner. Although the source of motion is the arbitrary disturbance at large distances, the eddies develop quite independently and thus solution of these eddies emerges from an eigenvalue problem, a self-similar solution of the second kind.

- ↑ Weisstein, Eric W. "Glome".
*MathWorld*. Retrieved 2017-12-04. - ↑ Georges Lemaître (1948) "Quaternions et espace elliptique",
*Acta*Pontifical Academy of Sciences 12:57–78 - ↑ Banchoff, Thomas. "The Flat Torus in the Three-Sphere".
- ↑ Mark A. Peterson. "Dante and the 3-sphere" Archived 2013-02-23 at Archive.today , American Journal of Physics, vol 47, number 12, 1979, pp1031-1035

- David W. Henderson,
*Experiencing Geometry: In Euclidean, Spherical, and Hyperbolic Spaces, second edition*, 2001, (Chapter 20: 3-spheres and hyperbolic 3-spaces.) - Jeffrey R. Weeks,
*The Shape of Space: How to Visualize Surfaces and Three-dimensional Manifolds*, 1985, () (Chapter 14: The Hypersphere) (Says:*A Warning on terminology: Our two-sphere is defined in three-dimensional space, where it is the boundary of a three-dimensional ball. This terminology is standard among mathematicians, but not among physicists. So don't be surprised if you find people calling the two-sphere a three-sphere.*) - Zamboj, Michal (8 Jan 2021). "Synthetic construction of the Hopf fibration in the double orthogonal projection of the 4-space". arXiv: 2003.09236v2 [math.HO].

- Weisstein, Eric W. "Hypersphere".
*MathWorld*.*Note*: This article uses the alternate naming scheme for spheres in which a sphere in n-dimensional space is termed an n-sphere.

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