Definitions

missing point

3-sphere

In mathematics, a 3-sphere is a higher-dimensional analogue of a sphere. It consists of the set of points equidistant from a fixed central point in 4-dimensional Euclidean space. Just as an ordinary sphere (or 2-sphere) is a two dimensional surface that forms the boundary of a ball in three dimensions, a 3-sphere is an object with three dimensions that forms the boundary of a ball in four dimensions. A 3-sphere is an example of a 3-manifold.

A 3-sphere is also called a hypersphere, although the term hypersphere can in general describe any n-sphere for n ≥ 3.

Definition

In coordinates, a 3-sphere with center (C0C1C2C3) and radius r is the set of all points (x0x1x2x3) in real, 4-dimensional space (R4) such that

sum_{i=0}^3(x_i - C_i)^2 = (x_0 - C_0 )^2 + (x_1 - C_1 )^2 + (x_2 - C_2 )^2+ (x_3 - C_3 )^2 = r^2.
The 3-sphere centered at the origin with radius 1 is called the unit 3-sphere and is usually denoted S3:

S^3 = left{(x_0,x_1,x_2,x_3)inmathbb{R}^4 : x_0^2 + x_1^2 + x_2^2 + x_3^2 = 1right}.

It is often convenient to regard R4 as the space with 2 complex dimensions (C2) or the quaternions (H). The unit 3-sphere is then given by

S^3 = left{(z_1,z_2)inmathbb{C}^2 : |z_1|^2 + |z_2|^2 = 1right}
or
S^3 = left{qinmathbb{H} : |q| = 1right}.

The last description is often the most useful. It describes the 3-sphere as the set of all unit quaternionsquaternions with absolute value equal to unity. 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.

Properties

Elementary properties

The 3-dimensional volume (or hyperarea) of a 3-sphere of radius r is

2pi^2 r^3 ,
while the 4-dimensional hypervolume (the volume of the 4-dimensional region bounded by the 3-sphere) is
begin{matrix} frac{1}{2} end{matrix} pi^2 r^4.

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 which 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.

Topological properties

A 3-sphere is a compact, connected, 3-dimensional manifold without boundary. It is also simply-connected. What this means, loosely speaking, 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 proposes that the 3-sphere is the only three dimensional manifold with these properties (up to homeomorphism). This conjecture was proved in 2003 by Grigori Perelman.

The 3-sphere is homeomorphic to the one-point compactification of mathbb{R}^3. Generally, any topological space which is homeomorphic to the 3-sphere is called a topological 3-sphere.

The homology groups of the 3-sphere are as follows: H0(S3,Z) and H3(S3,Z) are both infinite cyclic, while Hi(S3,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 S3, 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 three-sphere gives a homology sphere; typically these are not homeomorphic to the three-sphere.

As to the homotopy groups, we have π1(S3) = π2(S3) = {0} and π3(S3) 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.

Homotopy groups of S3
k 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
πk(S3) 0 0 0 Z Z2 Z2 Z12 Z2 Z2 Z3 Z15 Z2 Z2Z2 Z12Z2 Z84Z2Z2 Z2Z2 Z6

Geometric properties

The 3-sphere is naturally a smooth manifold, in fact, a closed embedded submanifold of R4. The Euclidean metric on R4 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/r2 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 S3 giving the 3-sphere the structure of a principal circle bundle known as the Hopf bundle. If one thinks of S3 as a subset of C2, the action is given by

(z_1,z_2)cdotlambda = (z_1lambda,z_2lambda)quad foralllambdainmathbb T.
The orbit space of this action is homeomorphic to the two-sphere S2. Since S3 is not homeomorphic to S2×S1, the Hopf bundle is nontrivial.

Topological construction

Two convenient constructions for the topologist are the reverse of "slicing in half" and "puncturing".

Unslicing

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.

The interiors of the 3-balls do not match: only their boundaries. In fact, the fourth dimension can be thought of as a continuous scalar field, a function of the 3-dimensional coordinates of the 3-ball, similar to "temperature". Let this "temperature" be zero at the 2-spherical boundary, but let one of the 3-balls be "hot" (have positive values of its scalar field) and let the other 3-ball be "cold" (have negative values of its scalar field). The "hot" 3-ball could be thought of as the "hot hemi-3-sphere" and the "cold" 3-ball could be thought of as the "cold hemi-3-sphere". The temperature is highest at the hot 3-ball's very center and lowest at the cold 3-ball's center.

This construction is analogous to a construction of a 2-sphere, performed by joining 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 so that their circular boundaries match, then let corresponding points on the circular boundaries become equivalent identically to each other. The boundaries are now glued together. Now "inflate" the disks. One disk inflates upwards and becomes the Northern hemisphere and the other inflates downwards and becomes the Southern hemisphere.

It is possible for a point traveling on the 3-sphere to move from one hemi-3-sphere to the other hemi-3-sphere by crossing the 2-spherical boundary, which could be thought of as a "3-quator" — analogous to an equator on a 2-sphere. The point would seem to be bouncing off the 3-quator and reversing direction of motion in 3-D, but also its "temperature" would become reversed, e.g. from positive on the "hot hemi-3-sphere" to zero on the 3-quator to negative on the "cold hemi-3-sphere".

Unpuncturing

Consider a topological 2-sphere to be a seamless balloon. When punctured and flattened, the missing point becomes a circle (a 1-sphere) and the remaining balloon surface becomes a disk (a 2-ball) inside the circle. In the same way, a 3-ball is a punctured and flattened 3-sphere. To recreate the 3-sphere, merge all points on the 3-ball boundary (a 2-sphere) into a single point.

Another view of puncturing is stereographic projection. Rest the South Pole of a 2-sphere on an infinite plane, and draw lines from the North Pole through the sphere to intersect the plane. Each sphere point corresponds to a unique plane point, and vice versa, excepting the North Pole itself. The balloon has been stretched to infinity. Stereographic projection of a 3-sphere (except for the projection point) fills all of 3-space in the same manner. A benefit of this correspondence is that geometric spheres in 3-space map to geometric spheres of the 3-sphere, and planes in 3-space map to spheres containing the Pole.

Another view is a "shooting map". Place a marble at the South Pole and give it a flick of a measured strength in a chosen direction. Assuming the marble stays on the sphere and rolls without friction, its position after a fixed time interval (say, 1 second) will be some definite point of the sphere. Plotting direction in the plane and strength as radius, the North Pole is equally far away in every direction; this is the equivalent of the punctured balloon. Performing the same shooting experiment on the 3-sphere gives a map on the 3-ball. When the 3-sphere is considered a Lie group, the marble paths are one-parameter subgroups, the 3-ball is the tangent space at the identity (taken to be the South Pole), and the mapping to the 3-sphere is the exponential map.

Coordinate systems on the 3-sphere

The four Euclidean coordinates for S3 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 S3 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 S3 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.

Hyperspherical coordinates

It is convenient to have some sort of hyperspherical coordinates on S3 in analogy to the usual spherical coordinates on S2. One such choice—by no means unique—is to use (ψ, θ, φ) where

x_0 = cospsi,
x_1 = cosphi,sintheta,sinpsi
x_2 = sinphi,sintheta,sinpsi
x_3 = costheta,sinpsi
where ψ and θ runs 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 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

ds^2 = dpsi^2 + sin^2psileft(dtheta^2 + sin^2theta, dphi^2right)
and the volume form by
dV = left(sin^2psi,sinthetaright),dpsiwedge dthetawedge dphi.

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

q = eτψ = cos ψ + τ sin ψ
where τ is a unit imaginary quaternion—that is, any quaternion which satisfies τ2 = −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:
τ = cos φ sin θ i + sin φ sin θ j + cos θ k
With τ in this form, the unit quaternion q is given by
q = eτψ = x0 + x1 i + x2 j + x3 k
where the x’s 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 2ψ.

Hopf coordinates

Another choice of hyperspherical coordinates, (η, ξ1, ξ2), makes use of the embedding of S3 in C2. In complex coordinates (z1, z2) ∈ C2 we write

z_1 = e^{i,xi_1}sineta
z_2 = e^{i,xi_2}coseta.
Here η runs over the range 0 to π/2, and ξ1 and ξ2 can take any values between 0 and 2π. These coordinates are useful in the description of the 3-sphere as the Hopf bundle
S^1 to S^3 to S^2.,

For any fixed value of η between 0 and π/2, the coordinates (ξ1, ξ2) parameterize a 2-dimensional torus. 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

ds^2 = deta^2 + sin^2eta,dxi_1^2 + cos^2eta,dxi_2^2
and the volume form by
dV = sinetacoseta,detawedge dxi_1wedge dxi_2.

Stereographic coordinates

Another convenient set of coordinates can be obtained via stereographic projection of S3 onto a tangent R3 hyperplane. For example, if we project onto the plane tangent to the point (1, 0, 0, 0) we can write a point p in S3 as

p = left(frac{1-|u|^2}{1+|u|^2}, frac{2mathbf{u}}{1+|u|^2}right) = frac{1+mathbf{u}}{1-mathbf{u}}
where u = (u1, u2, u3) is a vector in R3 and ||u||2 = u12 + u22 + u32. In the second equality above we have identified p with a unit quaternion and u = u1 i + u2 j + u3 k with a pure quaternion. (Note that the division here is well-defined even though quaternionic multiplication is generally noncommutative). The inverse of this map takes p = (x0, x1, x2, x3) in S3 to
mathbf{u} = frac{1}{1+x_0}left(x_1, x_2, x_3right).

We could just have well have projected onto the plane tangent to the point (−1, 0, 0, 0) in which case the point p is given by

p = left(frac{-1+|v|^2}{1+|v|^2}, frac{2mathbf{v}}{1+|v|^2}right) = frac{-1+mathbf{v}}{1+mathbf{v}}
where v = (v1, v2, v3) is a vector in the second R3. The inverse of this map takes p to
mathbf{v} = frac{1}{1-x_0}left(x_1,x_2,x_3right).

Note that the u coordinates are defined everywhere but (−1, 0, 0, 0) and the v coordinates everywhere but (1, 0, 0, 0). Both patches together cover all of S3. This defines an atlas on S3 consisting of two coordinate charts. Note that the transition function between these two charts on their overlap is given by

mathbf{v} = frac{1}{|u|^2}mathbf{u}
and vice-versa.

Group structure

When considered as the set of unit quaternions, S3 inherits an important structure, namely that of quaternionic multiplication. Because the set of unit quaternions is closed under multiplication, S3 takes on the structure of a group. Moreover, since quaternionic multiplication is smooth, S3 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 S3 is often denoted Sp(1) or U(1, H).

It turns out that the only spheres which admit a Lie group structure are S1, thought of as the set of unit complex numbers, and S3, the set of unit quaternions. One might think that S7, the set of unit octonions, would form a Lie group, but this fails since octonion multiplication is nonassociative. The octonionic structure does give S7 one important property: parallelizability. It turns out that the only spheres which are parallelizable are S1, S3, and S7.

By using a matrix representation of the quaternions, H, one obtains a matrix representation of S3. One convenient choice is given by the Pauli matrices:

x_1+ x_2 i + x_3 j + x_4 k mapsto begin{pmatrix};;,x_1 + i x_2 & x_3 + i x_4 -x_3 + i x_4 & x_1 - i x_2end{pmatrix}.
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, S3 as a Lie group is isomorphic to SU(2).

Using our hyperspherical coordinates (η, ξ1, ξ2) we can then write any element of SU(2) in the form

begin{pmatrix}e^{i,xi_1}sineta & e^{i,xi_2}coseta -e^{-i,xi_2}coseta & e^{-i,xi_1}sinetaend{pmatrix}.

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 in SU(2) can be written as U=alpha_0 I + i alpha_i sigma^i. The condition that the determinant of U is +1 implies that the coefficients alpha_i are constrained to lie on a 3-sphere.

In literature

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, 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.

See also

References

  • 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.)

External links

  • 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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