In mathematics, a 3sphere, glome or hypersphere is a higherdimensional analogue of a sphere. It may be embedded in 4dimensional 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 2sphere, a twodimensional surface), the boundary of a ball in four dimensions is a 3sphere (an object with three dimensions). A 3sphere is an example of a 3manifold and an nsphere.
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Transcription
Definition
In coordinates, a 3sphere 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, 4dimensional space (R^{4}) such that
The 3sphere centered at the origin with radius 1 is called the unit 3sphere 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 3sphere is then given by
or
This description as the quaternions of norm one identifies the 3sphere with the versors in the quaternion division ring. Just as the unit circle is important for planar polar coordinates, so the 3sphere is important in the polar view of 4space involved in quaternion multiplication. See polar decomposition of a quaternion for details of this development of the threesphere. This view of the 3sphere is the basis for the study of elliptic space as developed by Georges Lemaître.^{[1]}
Properties
Elementary properties
The 3dimensional surface volume of a 3sphere of radius r is
while the 4dimensional hypervolume (the content of the 4dimensional region bounded by the 3sphere) is
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 that reaches its maximal size when the hyperplane cuts right through the "equator" of the 3sphere. Then the 2sphere shrinks again down to a single point as the 3sphere leaves the hyperplane.
In a given threedimensional hyperplane, a 3sphere can rotate about an "equatorial plane" (analogous to a 2sphere rotating about a central axis), in which case it appears to be a 2sphere whose size is constant.
Topological properties
A 3sphere is a compact, connected, 3dimensional manifold without boundary. It is also simply connected. What this means, in the broad sense, is that any loop, or circular path, on the 3sphere can be continuously shrunk to a point without leaving the 3sphere. The Poincaré conjecture, proved in 2003 by Grigori Perelman, provides that the 3sphere is the only threedimensional manifold (up to homeomorphism) with these properties.
The 3sphere is homeomorphic to the onepoint compactification of R^{3}. In general, any topological space that is homeomorphic to the 3sphere is called a topological 3sphere.
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) = {} 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 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 3sphere gives a homology sphere; typically these are not homeomorphic to the 3sphere.
As to the homotopy groups, we have π_{1}(S^{3}) = π_{2}(S^{3}) = {} and π_{3}(S^{3}) is infinite cyclic. The higherhomotopy 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} 
Geometric properties
The 3sphere 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 3sphere giving it the structure of a Riemannian manifold. As with all spheres, the 3sphere has constant positive sectional curvature equal to 1/r^{2} where r is the radius.
Much of the interesting geometry of the 3sphere stems from the fact that the 3sphere 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 0sphere and the 1sphere (see circle group).
Unlike the 2sphere, the 3sphere 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 leftinvariant vector fields forming a basis for the Lie algebra of the 3sphere. This implies that the 3sphere is parallelizable. It follows that the tangent bundle of the 3sphere is trivial. For a general discussion of the number of linear independent vector fields on a nsphere, see the article vector fields on spheres.
There is an interesting action of the circle group T on S^{3} giving the 3sphere 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 twosphere S^{2}. Since S^{3} is not homeomorphic to S^{2} × S^{1}, the Hopf bundle is nontrivial.
Topological construction
There are several wellknown constructions of the threesphere. Here we describe gluing a pair of threeballs and then the onepoint compactification.
Gluing
A 3sphere can be constructed topologically by "gluing" 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. In analogy with the case of the 2sphere (see below), the gluing surface is called an equatorial sphere.
Note that the interiors of the 3balls are not glued to each other. One way to think of the fourth dimension is as a continuous realvalued function of the 3dimensional coordinates of the 3ball, perhaps considered to be "temperature". We take the "temperature" to be zero along the gluing 2sphere and let one of the 3balls be "hot" and let the other 3ball be "cold". The "hot" 3ball could be thought of as the "upper hemisphere" and the "cold" 3ball could be thought of as the "lower hemisphere". The temperature is highest/lowest at the centers of the two 3balls.
This construction is analogous to a construction of a 2sphere, performed by gluing 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 and glue corresponding points on their boundaries. Again one may think of the third dimension as temperature. Likewise, we may inflate the 2sphere, moving the pair of disks to become the northern and southern hemispheres.
Onepoint compactification
After removing a single point from the 2sphere, what remains is homeomorphic to the Euclidean plane. In the same way, removing a single point from the 3sphere yields threedimensional space. An extremely useful way to see this is via stereographic projection. We first describe the lowerdimensional version.
Rest the south pole of a unit 2sphere on the xyplane in threespace. 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 3sphere (again removing the north pole) maps to threespace 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 onepoint compactification is via the exponential map. Returning to our picture of the unit twosphere sitting on the Euclidean plane: Consider a geodesic in the plane, based at the origin, and map this to a geodesic in the twosphere 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 onepoint compactification.
The exponential map for 3sphere is similarly constructed; it may also be discussed using the fact that the 3sphere is the Lie group of unit quaternions.
Coordinate systems on the 3sphere
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 3dimensional manifold one should be able to parameterize S^{3} by three coordinates, just as one can parameterize the 2sphere 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 2sphere, 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 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 2sphere 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 3sphere 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 2sphere 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 2ψ.
Hopf coordinates
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 3sphere as the Hopf bundle
For any fixed value of η between 0 and π/2, the coordinates (ξ_{1}, ξ_{2}) parameterize a 2dimensional 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 3sphere 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^{[2]}
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.
Stereographic coordinates
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.
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 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 0dimensional 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 exponential of a linear combination of the Pauli matrices. It is seen that an arbitrary element U ∈ SU(2) can be written as
 ^{[3]}
The condition that the determinant of U is +1 implies that the coefficients α_{1} are constrained to lie on a 3sphere.
In literature
In Edwin Abbott Abbott's Flatland, published in 1884, and in Sphereland, a 1965 sequel to Flatland by Dionys Burger, the 3sphere is referred to as an oversphere, and a 4sphere is referred to as a hypersphere.
Writing in the American Journal of Physics,^{[4]} Mark A. Peterson describes three different ways of visualizing 3spheres and points out language in The Divine Comedy that suggests Dante viewed the Universe in the same way; Carlo Rovelli supports the same idea.^{[5]}
In Art Meets Mathematics in the Fourth Dimension,^{[6]} Stephen L. Lipscomb develops the concept of the hypersphere dimensions as it relates to art, architecture, and mathematics.
See also
 1sphere, 2sphere, nsphere
 tesseract, polychoron, simplex
 Pauli matrices
 Hopf bundle, Riemann sphere
 Poincaré sphere
 Reeb foliation
 Clifford torus
References
 ^ Georges Lemaître (1948) "Quaternions et espace elliptique", Acta Pontifical Academy of Sciences 12:57–78
 ^ Banchoff, Thomas. "The Flat Torus in the ThreeSphere".
 ^ Schwichtenberg, Jakob (2015). Physics from symmetry. Cham. ISBN 9783319192017. OCLC 910917227.
{{cite book}}
: CS1 maint: location missing publisher (link)  ^ Peterson, Mark A. (1979). "Dante and the 3sphere". American Journal of Physics. 47 (12): 1031–1035. Bibcode:1979AmJPh..47.1031P. doi:10.1119/1.11968. Archived from the original on 23 February 2013.
 ^ Rovelli, Carlo (9 September 2021). General Relativity: The Essentials. Cambridge: Cambridge University Press. ISBN 9781009013697. Retrieved 13 September 2021.
 ^ Lipscomb, Stephen (2014). Art meets mathematics in the fourth dimension (2 ed.). Berlin. ISBN 9783319062549. OCLC 893872366.
{{cite book}}
: CS1 maint: location missing publisher (link)
 David W. Henderson, Experiencing Geometry: In Euclidean, Spherical, and Hyperbolic Spaces, second edition, 2001, [1] (Chapter 20: 3spheres and hyperbolic 3spaces.)
 Jeffrey R. Weeks, The Shape of Space: How to Visualize Surfaces and Threedimensional Manifolds, 1985, ([2]) (Chapter 14: The Hypersphere) (Says: A Warning on terminology: Our twosphere is defined in threedimensional space, where it is the boundary of a threedimensional ball. This terminology is standard among mathematicians, but not among physicists. So don't be surprised if you find people calling the twosphere a threesphere.)
 Zamboj, Michal (8 Jan 2021). "Synthetic construction of the Hopf fibration in a double orthogonal projection of 4space". Journal of Computational Design and Engineering. 8 (3): 836–854. arXiv:2003.09236v2. doi:10.1093/jcde/qwab018.
External links
 Weisstein, Eric W. "Hypersphere". MathWorld. Note: This article uses the alternate naming scheme for spheres in which a sphere in ndimensional space is termed an nsphere.