3-sphere

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"Glome" redirects here. 3-sphere_sentence_0

For the fictitious kingdom in the C.S. Lewis novel, see Till We Have Faces. 3-sphere_sentence_1

In mathematics, a 3-sphere, or glome, is a higher-dimensional analogue of a sphere. 3-sphere_sentence_2

It may be embedded in 4-dimensional Euclidean space as the set of points equidistant from a fixed central point. 3-sphere_sentence_3

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). 3-sphere_sentence_4

A 3-sphere is an example of a 3-manifold and an n-sphere. 3-sphere_sentence_5

Definition 3-sphere_section_0

In coordinates, a 3-sphere with center (C0, C1, C2, C3) and radius r is the set of all points (x0, x1, x2, x3) in real, 4-dimensional space (R) such that 3-sphere_sentence_6

The 3-sphere centered at the origin with radius 1 is called the unit 3-sphere and is usually denoted S: 3-sphere_sentence_7

It is often convenient to regard R as the space with 2 complex dimensions (C) or the quaternions (H). 3-sphere_sentence_8

The unit 3-sphere is then given by 3-sphere_sentence_9

or 3-sphere_sentence_10

This description as the quaternions of norm one identifies the 3-sphere with the versors in the quaternion division ring. 3-sphere_sentence_11

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. 3-sphere_sentence_12

See polar decomposition of a quaternion for details of this development of the three-sphere. 3-sphere_sentence_13

This view of the 3-sphere is the basis for the study of elliptic space as developed by Georges Lemaître. 3-sphere_sentence_14

Properties 3-sphere_section_1

Elementary properties 3-sphere_section_2

The 3-dimensional cubic hyperarea of a 3-sphere of radius r is 3-sphere_sentence_15

while the 4-dimensional quartic hypervolume (the volume of the 4-dimensional region bounded by the 3-sphere) is 3-sphere_sentence_16

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). 3-sphere_sentence_17

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. 3-sphere_sentence_18

Then the 2-sphere shrinks again down to a single point as the 3-sphere leaves the hyperplane. 3-sphere_sentence_19

Topological properties 3-sphere_section_3

A 3-sphere is a compact, connected, 3-dimensional manifold without boundary. 3-sphere_sentence_20

It is also simply connected. 3-sphere_sentence_21

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. 3-sphere_sentence_22

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. 3-sphere_sentence_23

The 3-sphere is homeomorphic to the one-point compactification of R. In general, any topological space that is homeomorphic to the 3-sphere is called a topological 3-sphere. 3-sphere_sentence_24

The homology groups of the 3-sphere are as follows: H0(S,Z) and H3(S,Z) are both infinite cyclic, while Hi(S,Z) = {0} for all other indices i. 3-sphere_sentence_25

Any topological space with these homology groups is known as a homology 3-sphere. 3-sphere_sentence_26

Initially Poincaré conjectured that all homology 3-spheres are homeomorphic to S, but then he himself constructed a non-homeomorphic one, now known as the Poincaré homology sphere. 3-sphere_sentence_27

Infinitely many homology spheres are now known to exist. 3-sphere_sentence_28

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. 3-sphere_sentence_29

As to the homotopy groups, we have π1(S) = π2(S) = {0} and π3(S) is infinite cyclic. 3-sphere_sentence_30

The higher-homotopy groups (k ≥ 4) are all finite abelian but otherwise follow no discernible pattern. 3-sphere_sentence_31

For more discussion see homotopy groups of spheres. 3-sphere_sentence_32

3-sphere_table_general_0

Homotopy groups of S3-sphere_table_caption_0
k3-sphere_cell_0_0_0 03-sphere_cell_0_0_1 13-sphere_cell_0_0_2 23-sphere_cell_0_0_3 33-sphere_cell_0_0_4 43-sphere_cell_0_0_5 53-sphere_cell_0_0_6 63-sphere_cell_0_0_7 73-sphere_cell_0_0_8 83-sphere_cell_0_0_9 93-sphere_cell_0_0_10 103-sphere_cell_0_0_11 113-sphere_cell_0_0_12 123-sphere_cell_0_0_13 133-sphere_cell_0_0_14 143-sphere_cell_0_0_15 153-sphere_cell_0_0_16 163-sphere_cell_0_0_17
πk(S)3-sphere_cell_0_1_0 03-sphere_cell_0_1_1 03-sphere_cell_0_1_2 03-sphere_cell_0_1_3 Z3-sphere_cell_0_1_4 Z23-sphere_cell_0_1_5 Z23-sphere_cell_0_1_6 Z123-sphere_cell_0_1_7 Z23-sphere_cell_0_1_8 Z23-sphere_cell_0_1_9 Z33-sphere_cell_0_1_10 Z153-sphere_cell_0_1_11 Z23-sphere_cell_0_1_12 Z2⊕Z23-sphere_cell_0_1_13 Z12⊕Z23-sphere_cell_0_1_14 Z84⊕Z2⊕Z23-sphere_cell_0_1_15 Z2⊕Z23-sphere_cell_0_1_16 Z63-sphere_cell_0_1_17

Geometric properties 3-sphere_section_4

The 3-sphere is naturally a smooth manifold, in fact, a closed embedded submanifold of R. The Euclidean metric on R induces a metric on the 3-sphere giving it the structure of a Riemannian manifold. 3-sphere_sentence_33

As with all spheres, the 3-sphere has constant positive sectional curvature equal to 1/r where r is the radius. 3-sphere_sentence_34

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). 3-sphere_sentence_35

The only other spheres with such a structure are the 0-sphere and the 1-sphere (see circle group). 3-sphere_sentence_36

Unlike the 2-sphere, the 3-sphere admits nonvanishing vector fields (sections of its tangent bundle). 3-sphere_sentence_37

One can even find three linearly independent and nonvanishing vector fields. 3-sphere_sentence_38

These may be taken to be any left-invariant vector fields forming a basis for the Lie algebra of the 3-sphere. 3-sphere_sentence_39

This implies that the 3-sphere is parallelizable. 3-sphere_sentence_40

It follows that the tangent bundle of the 3-sphere is trivial. 3-sphere_sentence_41

For a general discussion of the number of linear independent vector fields on a n-sphere, see the article vector fields on spheres. 3-sphere_sentence_42

There is an interesting action of the circle group T on S giving the 3-sphere the structure of a principal circle bundle known as the Hopf bundle. 3-sphere_sentence_43

If one thinks of S as a subset of C, the action is given by 3-sphere_sentence_44

The orbit space of this action is homeomorphic to the two-sphere S. Since S is not homeomorphic to S × S, the Hopf bundle is nontrivial. 3-sphere_sentence_45

Topological construction 3-sphere_section_5

There are several well-known constructions of the three-sphere. 3-sphere_sentence_46

Here we describe gluing a pair of three-balls and then the one-point compactification. 3-sphere_sentence_47

Gluing 3-sphere_section_6

A 3-sphere can be constructed topologically by "gluing" together the boundaries of a pair of 3-balls. 3-sphere_sentence_48

The boundary of a 3-ball is a 2-sphere, and these two 2-spheres are to be identified. 3-sphere_sentence_49

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. 3-sphere_sentence_50

In analogy with the case of the 2-sphere (see below), the gluing surface is called an equatorial sphere. 3-sphere_sentence_51

Note that the interiors of the 3-balls are not glued to each other. 3-sphere_sentence_52

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". 3-sphere_sentence_53

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". 3-sphere_sentence_54

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". 3-sphere_sentence_55

The temperature is highest/lowest at the centers of the two 3-balls. 3-sphere_sentence_56

This construction is analogous to a construction of a 2-sphere, performed by gluing the boundaries of a pair of disks. 3-sphere_sentence_57

A disk is a 2-ball, and the boundary of a disk is a circle (a 1-sphere). 3-sphere_sentence_58

Let a pair of disks be of the same diameter. 3-sphere_sentence_59

Superpose them and glue corresponding points on their boundaries. 3-sphere_sentence_60

Again one may think of the third dimension as temperature. 3-sphere_sentence_61

Likewise, we may inflate the 2-sphere, moving the pair of disks to become the northern and southern hemispheres. 3-sphere_sentence_62

One-point compactification 3-sphere_section_7

After removing a single point from the 2-sphere, what remains is homeomorphic to the Euclidean plane. 3-sphere_sentence_63

In the same way, removing a single point from the 3-sphere yields three-dimensional space. 3-sphere_sentence_64

An extremely useful way to see this is via stereographic projection. 3-sphere_sentence_65

We first describe the lower-dimensional version. 3-sphere_sentence_66

Rest the south pole of a unit 2-sphere on the xy-plane in three-space. 3-sphere_sentence_67

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. 3-sphere_sentence_68

Stereographic projection of a 3-sphere (again removing the north pole) maps to three-space in the same manner. 3-sphere_sentence_69

(Notice that, since stereographic projection is conformal, round spheres are sent to round spheres or to planes.) 3-sphere_sentence_70

A somewhat different way to think of the one-point compactification is via the exponential map. 3-sphere_sentence_71

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. 3-sphere_sentence_72

Under this map all points of the circle of radius π are sent to the north pole. 3-sphere_sentence_73

Since the open unit disk is homeomorphic to the Euclidean plane, this is again a one-point compactification. 3-sphere_sentence_74

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. 3-sphere_sentence_75

Coordinate systems on the 3-sphere 3-sphere_section_8

The four Euclidean coordinates for S are redundant since they are subject to the condition that x0 + x1 + x2 + x3 = 1. 3-sphere_sentence_76

As a 3-dimensional manifold one should be able to parameterize S by three coordinates, just as one can parameterize the 2-sphere using two coordinates (such as latitude and longitude). 3-sphere_sentence_77

Due to the nontrivial topology of S it is impossible to find a single set of coordinates that cover the entire space. 3-sphere_sentence_78

Just as on the 2-sphere, one must use at least two coordinate charts. 3-sphere_sentence_79

Some different choices of coordinates are given below. 3-sphere_sentence_80

Hyperspherical coordinates 3-sphere_section_9

It is convenient to have some sort of hyperspherical coordinates on S in analogy to the usual spherical coordinates on S. One such choice — by no means unique — is to use (ψ, θ, φ), where 3-sphere_sentence_81

where ψ and θ run over the range 0 to π, and φ runs over 0 to 2π. 3-sphere_sentence_82

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. 3-sphere_sentence_83

The round metric on the 3-sphere in these coordinates is given by 3-sphere_sentence_84

and the volume form by 3-sphere_sentence_85

These coordinates have an elegant description in terms of quaternions. 3-sphere_sentence_86

Any unit quaternion q can be written as a versor: 3-sphere_sentence_87

where τ is a unit imaginary quaternion; that is, a quaternion that satisfies τ = −1. 3-sphere_sentence_88

This is the quaternionic analogue of Euler's formula. 3-sphere_sentence_89

Now the unit imaginary quaternions all lie on the unit 2-sphere in Im H so any such τ can be written: 3-sphere_sentence_90

With τ in this form, the unit quaternion q is given by 3-sphere_sentence_91

where x0,1,2,3 are as above. 3-sphere_sentence_92

When q is used to describe spatial rotations (cf. 3-sphere_sentence_93

quaternions and spatial rotations), it describes a rotation about τ through an angle of 2ψ. 3-sphere_sentence_94

Hopf coordinates 3-sphere_section_10

For unit radius another choice of hyperspherical coordinates, (η, ξ1, ξ2), makes use of the embedding of S in C. In complex coordinates (z1, z2) ∈ C we write 3-sphere_sentence_95

This could also be expressed in R as 3-sphere_sentence_96

Here η runs over the range 0 to π/2, and ξ1 and ξ2 can take any values between 0 and 2π. 3-sphere_sentence_97

These coordinates are useful in the description of the 3-sphere as the Hopf bundle 3-sphere_sentence_98

For any fixed value of η between 0 and π/2, the coordinates (ξ1, ξ2) parameterize a 2-dimensional torus. 3-sphere_sentence_99

Rings of constant ξ1 and ξ2 above form simple orthogonal grids on the tori. 3-sphere_sentence_100

See image to right. 3-sphere_sentence_101

In the degenerate cases, when η equals 0 or π/2, these coordinates describe a circle. 3-sphere_sentence_102

The round metric on the 3-sphere in these coordinates is given by 3-sphere_sentence_103

and the volume form by 3-sphere_sentence_104

To get the interlocking circles of the Hopf fibration, make a simple substitution in the equations above 3-sphere_sentence_105

In this case η, and ξ1 specify which circle, and ξ2 specifies the position along each circle. 3-sphere_sentence_106

One round trip (0 to 2π) of ξ1 or ξ2 equates to a round trip of the torus in the 2 respective directions. 3-sphere_sentence_107

Stereographic coordinates 3-sphere_section_11

Another convenient set of coordinates can be obtained via stereographic projection of S from a pole onto the corresponding equatorial R hyperplane. 3-sphere_sentence_108

For example, if we project from the point (−1, 0, 0, 0) we can write a point p in S as 3-sphere_sentence_109

where u = (u1, u2, u3) is a vector in R and ||u|| = u1 + u2 + u3. 3-sphere_sentence_110

In the second equality above, we have identified p with a unit quaternion and u = u1i + u2j + u3k with a pure quaternion. 3-sphere_sentence_111

(Note that the numerator and denominator commute here even though quaternionic multiplication is generally noncommutative). 3-sphere_sentence_112

The inverse of this map takes p = (x0, x1, x2, x3) in S to 3-sphere_sentence_113

We could just as well have projected from the point (1, 0, 0, 0), in which case the point p is given by 3-sphere_sentence_114

where v = (v1, v2, v3) is another vector in R. The inverse of this map takes p to 3-sphere_sentence_115

Note that the u coordinates are defined everywhere but (−1, 0, 0, 0) and the v coordinates everywhere but (1, 0, 0, 0). 3-sphere_sentence_116

This defines an atlas on S consisting of two coordinate charts or "patches", which together cover all of S. Note that the transition function between these two charts on their overlap is given by 3-sphere_sentence_117

and vice versa. 3-sphere_sentence_118

Group structure 3-sphere_section_12

When considered as the set of unit quaternions, S inherits an important structure, namely that of quaternionic multiplication. 3-sphere_sentence_119

Because the set of unit quaternions is closed under multiplication, S takes on the structure of a group. 3-sphere_sentence_120

Moreover, since quaternionic multiplication is smooth, S can be regarded as a real Lie group. 3-sphere_sentence_121

It is a nonabelian, compact Lie group of dimension 3. 3-sphere_sentence_122

When thought of as a Lie group S is often denoted Sp(1) or U(1, H). 3-sphere_sentence_123

It turns out that the only spheres that admit a Lie group structure are S, thought of as the set of unit complex numbers, and S, the set of unit quaternions. 3-sphere_sentence_124

One might think that S, the set of unit octonions, would form a Lie group, but this fails since octonion multiplication is nonassociative. 3-sphere_sentence_125

The octonionic structure does give S one important property: parallelizability. 3-sphere_sentence_126

It turns out that the only spheres that are parallelizable are S, S, and S. 3-sphere_sentence_127

By using a matrix representation of the quaternions, H, one obtains a matrix representation of S. One convenient choice is given by the Pauli matrices: 3-sphere_sentence_128

This map gives an injective algebra homomorphism from H to the set of 2 × 2 complex matrices. 3-sphere_sentence_129

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. 3-sphere_sentence_130

The set of unit quaternions is then given by matrices of the above form with unit determinant. 3-sphere_sentence_131

This matrix subgroup is precisely the special unitary group SU(2). 3-sphere_sentence_132

Thus, S as a Lie group is isomorphic to SU(2). 3-sphere_sentence_133

Using our Hopf coordinates (η, ξ1, ξ2) we can then write any element of SU(2) in the form 3-sphere_sentence_134

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. 3-sphere_sentence_135

It is seen that an arbitrary element U ∈ SU(2) can be written as 3-sphere_sentence_136

The condition that the determinant of U is +1 implies that the coefficients α1 are constrained to lie on a 3-sphere. 3-sphere_sentence_137

In literature 3-sphere_section_13

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. 3-sphere_sentence_138

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. 3-sphere_sentence_139

See also 3-sphere_section_14

3-sphere_unordered_list_0


Credits to the contents of this page go to the authors of the corresponding Wikipedia page: en.wikipedia.org/wiki/3-sphere.