3sphere
From Academic Kids

In mathematics, a 3sphere is a higherdimensional analogue of a sphere. A regular sphere, or 2sphere, consists of all points equidistant from a single point in ordinary 3dimensional Euclidean space, R^{3}. A 3sphere consists of all points equidistant from a single point in R^{4}. Whereas a 2sphere is a smooth 2dimensional surface, a 3sphere is an object with three dimensions, also known as 3manifold.
In an entirely analogous manner one can define higherdimensional spheres called hyperspheres or nspheres. Such objects are ndimensional manifolds.
Some people refer to a 3sphere as a glome from the Latin word glomus meaning ball. Roughly speaking, a glome is to a sphere as a sphere is to a circle.
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Definition
In coordinates, a 3sphere with center (x_{0}, y_{0}, z_{0}, w_{0}) and radius r is the set of all points (x,y,z,w) in R^{4} such that
 <math>( x  x_0 )^2 + ( y  y_0 )^2 + ( z  z_0 )^2 + ( w  w_0 )^2 = r^2. \,<math>
The 3sphere centered at the origin with radius 1 is called the unit 3sphere and is usually denoted S^{3}. It can be described as a subset of either R^{4}, C^{2}, or H (the quaternions):
 <math>S^3 = \left\{(x_1,x_2,x_3,x_4)\in\mathbb{R}^4\mid x_1^2 + x_2^2 + x_3^2 + x_4^2 = 1\right\}<math>
 <math>S^3 = \left\{(z_1,z_2)\in\mathbb{C}^2\mid z_1^2 + z_2^2 = 1\right\}<math>
 <math>S^3 = \left\{q\in\mathbb{H}\mid q = 1\right\}.<math>
The last description is often the most useful. It describes the 3sphere as the set of all unit quaternions—quaternions with absolute value equal to one. Just as the set of all unit complex numbers is important in complex geometry, the set of all unit quaternions is important to the geometry of the quaternions.
Elementary properties
The 3dimensional volume (or hyperarea) of a 3sphere of radius r is
 <math>2\pi^2 r^3 \,<math>
while the 4dimensional hypervolume (the volume of the 4dimensional region bounded by the 3sphere) is
 <math>\begin{matrix} \frac{1}{2} \end{matrix} \pi^2 r^4.<math>
Every nonempty intersection of a 3sphere with a threedimensional hyperplane is a 2sphere (unless the hyperplane is tangent to the 3sphere, in which case the intersection is a single point). As a 3sphere moves through a given threedimensional hyperplane, the intersection starts out as a point, then becomes a growing 2sphere which reaches its maximal size when the hyperplane cuts right through the "middle" of the 3sphere. Then the 2sphere shrinks again down to a single point as the 3sphere leaves the hyperplane.
Topological construction
A 3sphere can be constructed topologically by "glueing" together the boundaries of a pair of 3balls. The boundary of a 3ball is a 2sphere, and these two 2spheres are to be identified. That is, imagine a pair of 3balls of the same size, then superpose them so that their 2spherical boundaries match, and let matching pairs of points on the pair of 2spheres be identically equivalent to each other.
The interiors of the 3balls do not match: only their boundaries. In fact, the 4th dimension can be thought of as a continuous scalar field, a function of the 3dimensional coordinates of the 3ball, similar to "temperature". Let this "temperature" be zero at the 2spherical boundary, but let one of the 3balls be "hot" (have positive values of its scalar field) and let the other 3ball be "cold" (have negative values of its scalar field). The "hot" 3ball could be thought of as the "hot hemi3sphere" and the "cold" 3ball could be thought of as the "cold hemi3sphere". The temperature is highest at the hot 3ball's very center and lowest at the cold 3ball's center.
This construction is analogous to a construction of a 2sphere, performed by joining the boundaries of a pair of disks. A disk is a 2ball, and the boundary of a disk is a circle (a 1sphere). 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 travelling on the 3sphere to move from one hemi3sphere to the other hemiglome by crossing the 2spherical boundary, which could be thought of as a "3quator" — analogous to an equator on a 2sphere. The point would seem to be bouncing off the 3quator and reversing direction of motion in 3D, but also its "temperature" would become reversed, e.g. from positive on the "hot hemiglome" to zero on the 3quator to negative on the "cold hemiglome".
Topological properties
A 3sphere is a compact, 3dimensional manifold without boundary. It is also simplyconnected. What this means, loosely speaking, is that any loop, or circular path, on the 3sphere can be continuously shrunk to a point without leaving the 3sphere. There is a longstanding, unproven conjecture, known as the Poincaré conjecture, stating that the 3sphere is the only three dimensional manifold with these properties (up to homeomorphism).
The 3sphere is also homeomorphic to the onepoint compactification of R^{3}.
The homology groups of the 3sphere 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 3sphere. Initially Poincaré conjectured that all homology 3spheres are homeomorphic to S^{3}, but then he himself constructed a nonhomeomorphic one, now known as the Poincaré sphere. Infinitely many homology spheres are now known to exist. For example, a Dehn filling with slope 1/n on any knot in the threesphere gives a homology sphere; typically these are not homeomorphic to the threesphere.
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 discernable 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 
<math>\pi_{k}(S^3)<math>  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} 
There is an interesting group action of S^{1} (thought of as the group of complex numbers of absolute value 1) on S^{3} (thought of as a subset of C^{2}): λ.(z_{1},z_{2}) = (λz_{1},λz_{2}). The orbit space of this action is naturally homeomorphic to the twosphere S^{2}. The resulting map from the 3sphere to the 2sphere is known as the Hopf bundle. It is the generator of the homotopy group π_{3}(S^{2}).
Coordinates on S^{3}
Hyperspherical coordinates
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
 <math>x_0 = \cos\psi\, <math>
 <math>x_1 = \cos\phi\,\sin\theta\,\sin\psi <math>
 <math>x_2 = \sin\phi\,\sin\theta\,\sin\psi <math>
 <math>x_3 = \cos\theta\,\sin\psi <math>
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 2sphere 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 3sphere in these coordinates is given by
 <math>ds^2 = d\psi^2 + \sin^2\psi\left(d\theta^2 + \sin^2\theta\, d\phi^2\right)<math>
and the volume form by
 <math>\left(\sin^2\psi\,\sin\theta\right)\;d\psi\wedge d\theta\wedge d\phi.<math>
These coordinates have a nice description in terms of quaternions. Any unit quaternion q can be written in the form:
 q = e^{τψ} = cos ψ + τ sin ψ
where τ is a unit imaginary quaternion—that is, any quaternion which satisfies τ^{2} = −1. This is the quaternionic analogue of Euler's formula. Now the unit imaginary quaternions all lie on the unit 2sphere 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 = e^{τψ} = x_{0} + x_{1} i + x_{2} j + x_{3} 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.
Alternative hyperspherical system
Another choice of hyperspherical coordinates is (θ, ξ_{1}, ξ_{2}), where in terms of complex coordinates (z_{1}, z_{1}) ∈ C^{2} we have
 <math>z_1 = e^{i\,\xi_1}\sin\frac{\theta}{2} <math>
 <math>z_2 = e^{i\,\xi_2}\cos\frac{\theta}{2}. <math>
Here θ runs over the range 0 to π, and ξ_{1} and ξ_{2} can take any values between 0 and 2π. These coordinates are useful in the description of the 3sphere as the Hopf bundle S^{1} → S^{3} → S^{2}.
Note that for any fixed value of θ, (ξ_{1}, ξ_{2}) parameterize a 2dimensional torus, except for the degenerate cases, when θ equals 0 or π, in which case they describe a circle.
The round metric on the 3sphere in these coordinates is given by
 <math>ds^2 = \frac{1}{4}d\theta^2 + \sin^2\left(\frac{\theta}{2}\right)d\xi_1^2 + \cos^2\left(\frac{\theta}{2}\right)d\xi_2^2<math>
and the volume form by
 <math>\left(\frac{1}{4}\sin\theta\right)\;d\theta\wedge d\xi_1\wedge d\xi_2.<math>
Stereographic coordinates
Another convenient set of coordinates can be obtained via stereographic projection of S^{3} onto a tangent R^{3} hyperplane. For example, if we project onto the plane tangent to the point (1, 0, 0, 0) we can write a point p in S^{3} as
 <math>p = \left(\frac{1\u\^2}{1+\u\^2}, \frac{2\mathbf{u}}{1+\u\^2}\right) = \frac{1+\mathbf{u}}{1\mathbf{u}}<math>
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 division here is welldefined 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
 <math>\mathbf{u} = \frac{1}{1+x_0}\left(x_1, x_2, x_3\right).<math>
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
 <math>p = \left(\frac{1+\v\^2}{1+\v\^2}, \frac{2\mathbf{v}}{1+\v\^2}\right) = \frac{1+\mathbf{v}}{1+\mathbf{v}}<math>
where v = (v_{1}, v_{2}, v_{3}) is a vector in the second R^{3}. The inverse of this map takes p to
 <math>\mathbf{v} = \frac{1}{1x_0}\left(x_1,x_2,x_3\right).<math>
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 S^{3}. This defines an atlas on S^{3} consisting of two coordinate charts. Note that the transition function between these two charts on their overlap is given by
 <math>\mathbf{v} = \frac{1}{\u\^2}\mathbf{u}<math>
and viceversa.
Group structure
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 which 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. 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 which 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
 <math>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_2\end{pmatrix}.<math>
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. It turns out that this group is precisely the special unitary group SU(2). Thus, S^{3} as a Lie group is isomorphic to SU(2).
Using our hyperspherical coordinates (θ, ξ_{1}, ξ_{2}) we can then write any element of SU(2) in the form
 <math>\begin{pmatrix}e^{i\,\xi_1}\sin\frac{\theta}{2} & e^{i\,\xi_2}\cos\frac{\theta}{2} \\ e^{i\,\xi_2}\cos\frac{\theta}{2} & e^{i\,\xi_1}\sin\frac{\theta}{2}\end{pmatrix}.<math>
Related topics
 1sphere, 2sphere, nsphere
 tesseract, polychoron, simplex
 Pauli matrices
 SO(3), charts on SO(3), quaternions and spatial rotations
 Hopf bundle, Riemann sphere
 Poincaré sphere
External link
 Mathworld website (http://mathworld.wolfram.com/Hypersphere.html)