3-sphere |
In mathematics, a 3-sphere is a higher-dimensional analogue of a sphere. An ordinary sphere, or 2-sphere, consists of all points equidistant from a single point in ordinary 3-dimensional Euclidean space, R3. A 3-sphere consists of all points equidistant from a single point in R4. Whereas a 2-sphere is a smooth surface, a 3-sphere is an object with three Dimensions, also known as 3-manifold.
In an entirely analogous manner one can define higher-dimensional spheres called hyperspheres or n -spheres. Such objects are n -dimensional manifolds.
Some people refer to a 3-sphere 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. In Edwin Abbott Abbott s Flatland , published in 1884, the 3-sphere is referred to as an oversphere.
= Definition =
In coordinates, a 3-sphere with center ( x 0, y 0, z 0, w 0) and radius r is the set of all points ( x , y , z ,w) in R4 such that :( x - x_0 )^2 + ( y - y_0 )^2 + ( z - z_0 )^2 + ( w - w_0 )^2 = r^2. , The 3-sphere centered at the origin with radius 1 is called the unit 3-sphere and is usually denoted S 3. It can be described as a subset of either R4, C2, or H (the quaternions):
:S^3 = left{(x_1,x_2,x_3,x_4)inmathbb{R}^4mid x_1^2 + x_2^2 + x_3^2 + x_4^2 = 1 ight}
:S^3 = left{(z_1,z_2)inmathbb{C}^2mid |z_1|^2 + |z_2|^2 = 1 ight}
:S^3 = left{qinmathbb{H}mid |q| = 1 ight}.
The last description is often the most useful. It describes the 3-sphere 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 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 :egin{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 middle of the 3-sphere. Then the 2-sphere shrinks again down to a single point as the 3-sphere leaves the hyperplane.
= Topological construction =
Two convenient constructions for the topologist are the reverse of slicing in half and puncturing .
== Unslicing ==
A 3-sphere can be constructed topology by quotient space 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 hemiglome 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 hemiglome to zero on the 3-quator to negative on the cold hemiglome .
== 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.
= Topological properties =
A 3-sphere is a compact, 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. There is a long-standing, unproven conjecture, known as the Poincaré conjecture, stating that the 3-sphere is the only three dimensional manifold with these properties (up to homeomorphism).
The 3-sphere is also homeomorphic to the one-point compactification of R3.
The in the three-sphere gives a homology sphere; typically these are not homeomorphic to the three-sphere.
As to the homotopy groups, we have 1(S3) = 2(S3) = {0} and 3(S3) 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.
There is an interesting . It is the generator of the homotopy group 3(S2).
= Coordinate systems on the 3-sphere =
== 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 (, , ) where :x_0 = cospsi, :x_1 = cosphi,sin heta,sinpsi :x_2 = sinphi,sin heta,sinpsi :x_3 = cos heta,sinpsi where and runs over the range 0 to , and runs over 0 to 2. Note that for any fixed value of , and parameterize a 2-sphere of radius sin(), except for the degenerate cases, when equals 0 or , in which case they describe a point.
The metric tensor on the 3-sphere in these coordinates is given by :ds^2 = dpsi^2 + sin^2psileft(d heta^2 + sin^2 heta, dphi^2 ight) and the volume form by :dV = left(sin^2psi,sin heta ight),dpsiwedge d hetawedge dphi.
These coordinates have a nice description in terms of quaternions. Any unit quaternion q can be written in the form: : q = e = cos + sin where is a unit imaginary quaternion—that is, any quaternion which satisfies 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 can be written: : = cos sin i + sin sin j + cos k With in this form, the unit quaternion q is given by : q = e = 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 through an angle of 2.
== Hopf coordinates ==
Another choice of hyperspherical coordinates, (, 1, 2), makes use of the embedding of S 3 in C2. In complex coordinates ( z 1, z 2) ∈ C2 we write :z_1 = e^{i,xi_1}sineta :z_2 = e^{i,xi_2}coseta. Here runs over the range 0 to /2, and 1 and 2 can take any values between 0 and 2. These coordinates are useful in the description of the 3-sphere as the Hopf bundle :S^1 o S^3 o S^2.,
For any fixed value of between 0 and /2, the coordinates (1, 2) parameterize a 2-dimensional torus. In the degenerate cases, when equals 0 or /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 S 3 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 S 3 as :p = left(frac{1-|u|^2}{1+|u|^2}, frac{2mathbf{u}}{1+|u|^2} ight) = frac{1+mathbf{u}}{1-mathbf{u}} where u = ( u 1, u 2, u 3) is a vector in R3 and || u ||2 = u 12 + u 22 + u 32. 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 well-defined 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 :mathbf{u} = frac{1}{1+x_0}left(x_1, x_2, x_3 ight).
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} ight) = frac{-1+mathbf{v}}{1+mathbf{v}} where v = ( v 1, v 2, v 3) 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_3 ight).
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 (topology) on S 3 consisting of two chart (topology). 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, 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 (mathematics). Moreover, since quaternionic multiplication is smooth function, 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 symplectic group or U(1, H).
It turns out that the only . It turns out that the only spheres which are parallelizable are S 1, S 3, and S 7.
By using a matrix (math) representation of the quaternions, H, one obtains a matrix representation of S 3. One convenient choice is :x_1+ x_2 i + x_3 j + x_4 k mapsto egin{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 2Ã?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 (, 1, 2) we can then write any element of SU(2) in the form :egin{pmatrix}e^{i,xi_1}sineta & e^{i,xi_2}coseta \ -e^{-i,xi_2}coseta & e^{-i,xi_1}sinetaend{pmatrix}.
= Tangents =
A unit 3-sphere embedded in 4-space has a 3-space of tangent vectors, T p S 3, at every point p . If ( x 0, x 1, x 2, x 3) are the coordinates of p , then the vector with coordinates ( x 1, x 0, x 3, x 2) is in T p S 3, and the collection of all these vectors forms a continuous unit vector field on S 3. (This is a Fiber bundle#Sections of the tangent bundle, T S 3.) Such a construction is clearly possible for spheres in all even-dimensional spaces, S 2 n 1; but an implication of the Atiyah-Singer index theorem is that it is impossible for S 2 n (for positive n ).
=Related topics=
*unit circle, sphere, hypersphere *Tesseract, Polychoron, simplex *Pauli matrices *rotation group SO(3) **charts on SO(3) **quaternions and spatial rotations *Hopf bundle, Riemann sphere *Poincaré sphere *Reeb foliation *Clifford torus
=External link=
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