Pythagorean trigonometric identity

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https://en.wikipedia.org/wiki/Pythagorean_trigonometric_identity

The Pythagorean trigonometric identity, also called simply the Pythagorean identity, is an identity expressing the Pythagorean theorem in terms of trigonometric functions. Along with the sum-of-angles formulae, it is one of the basic relations between the sine and cosine functions.

The identity is

${\displaystyle {\sin }^2\theta +{\cos }^2\theta =1.}$

As usual, ${\displaystyle {\sin }^2\theta }$ means $(\sin \theta {)}^2$.

Proofs and their relationships to the Pythagorean theorem

Similar right triangles showing sine and cosine of angle θ

Proof based on right-angle triangles

Any similar triangles have the property that if we select the same angle in all of them, the ratio of the two sides defining the angle is the same regardless of which similar triangle is selected, regardless of its actual size: the ratios depend upon the three angles, not the lengths of the sides. Thus for either of the similar right triangles in the figure, the ratio of its horizontal side to its hypotenuse is the same, namely cos θ.

The elementary definitions of the sine and cosine functions in terms of the sides of a right triangle are:

${\displaystyle \sin \theta =\frac{\mathrm{o}\mathrm{p}\mathrm{p}\mathrm{o}\mathrm{s}\mathrm{i}\mathrm{t}\mathrm{e}}{\mathrm{h}\mathrm{y}\mathrm{p}\mathrm{o}\mathrm{t}\mathrm{e}\mathrm{n}\mathrm{u}\mathrm{s}\mathrm{e}}=\frac{b}{c}}$

${\displaystyle \cos \theta =\frac{\mathrm{a}\mathrm{d}\mathrm{j}\mathrm{a}\mathrm{c}\mathrm{e}\mathrm{n}\mathrm{t}}{\mathrm{h}\mathrm{y}\mathrm{p}\mathrm{o}\mathrm{t}\mathrm{e}\mathrm{n}\mathrm{u}\mathrm{s}\mathrm{e}}=\frac{a}{c}}$

The Pythagorean identity follows by squaring both definitions above, and adding; the left-hand side of the identity then becomes

${\displaystyle \frac{{\mathrm{o}\mathrm{p}\mathrm{p}\mathrm{o}\mathrm{s}\mathrm{i}\mathrm{t}\mathrm{e}}^2+{\mathrm{a}\mathrm{d}\mathrm{j}\mathrm{a}\mathrm{c}\mathrm{e}\mathrm{n}\mathrm{t}}^2}{{\mathrm{h}\mathrm{y}\mathrm{p}\mathrm{o}\mathrm{t}\mathrm{e}\mathrm{n}\mathrm{u}\mathrm{s}\mathrm{e}}^2}}$

which by the Pythagorean theorem is equal to 1. This definition is valid for all angles, due to the definition of defining ${\displaystyle x=\cos \theta }$ and ${\displaystyle y=\sin \theta }$ for the unit circle and thus ${\displaystyle x=c\cos \theta }$ and ${\displaystyle y=c\sin \theta }$ for a circle of radius c and reflecting our triangle in the y axis and setting ${\displaystyle a=x}$ and ${\displaystyle b=y}$.

Alternatively, the identities found at Trigonometric symmetry, shifts, and periodicity may be employed. By the periodicity identities we can say if the formula is true for −π < θ ≤ π then it is true for all real θ. Next we prove the identity in the range π/2 < θ ≤ π, to do this we let t = θ − π/2, t will now be in the range 0 < t ≤ π/2. We can then make use of squared versions of some basic shift identities (squaring conveniently removes the minus signs):

${\displaystyle {\sin }^2\theta +{\cos }^2\theta ={\sin }^2\left(t+\frac{1}{2}\pi \right)+{\cos }^2\left(t+\frac{1}{2}\pi \right)={\cos }^{2}t+{\sin }^{2}t=1.}$

All that remains is to prove it for −π < θ < 0; this can be done by squaring the symmetry identities to get

${\displaystyle {\sin }^2\theta ={\sin }^2(-\theta )\text{ and }{\cos }^2\theta ={\cos }^2(-\theta ).}$

Related identities

Similar right triangles illustrating the tangent and secant trigonometric functions.

Trigonometric functions and their reciprocals on the unit circle. The Pythagorean theorem applied to the blue triangle shows the identity 1 + cot² θ = csc² θ, and applied to the red triangle shows that 1 + tan² θ = sec² θ.

The identities

${\displaystyle 1+{\tan }^2\theta ={\sec }^2\theta }$

and

${\displaystyle 1+{\cot }^2\theta ={\csc }^2\theta }$

are also called Pythagorean trigonometric identities. If one leg of a right triangle has length 1, then the tangent of the angle adjacent to that leg is the length of the other leg, and the secant of the angle is the length of the hypotenuse.

${\displaystyle \tan \theta =\frac{b}{a},}$

and:

${\displaystyle \sec \theta =\frac{c}{a}.}$

In this way, this trigonometric identity involving the tangent and the secant follows from the Pythagorean theorem. The angle opposite the leg of length 1 (this angle can be labeled φ = π/2 − θ) has cotangent equal to the length of the other leg, and cosecant equal to the length of the hypotenuse. In that way, this trigonometric identity involving the cotangent and the cosecant also follows from the Pythagorean theorem.

The following table gives the identities with the factor or divisor that relates them to the main identity.

Original Identity	Divisor	Divisor Equation	Derived Identity	Derived Identity (Alternate)
${\displaystyle {\sin }^2\theta +{\cos }^2\theta =1}$	${\displaystyle {\cos }^2\theta }$	${\displaystyle \frac{{\sin }^2\theta }{{\cos }^2\theta }+\frac{{\cos }^2\theta }{{\cos }^2\theta }=\frac{1}{{\cos }^2\theta }}$	${\displaystyle {\tan }^2\theta +1={\sec }^2\theta }$	${\displaystyle \begin{array}{r}{\sec }^2\theta -{\tan }^2\theta =1\\ (\sec \theta -\tan \theta )(\sec \theta +\tan \theta )=1\end{array}}$
	${\displaystyle {\sin }^2\theta }$	${\displaystyle \frac{{\sin }^2\theta }{{\sin }^2\theta }+\frac{{\cos }^2\theta }{{\sin }^2\theta }=\frac{1}{{\sin }^2\theta }}$	${\displaystyle 1+{\cot }^2\theta ={\csc }^2\theta }$	${\displaystyle \begin{array}{r}{\csc }^2\theta -{\cot }^2\theta =1\\ (\csc \theta -\cot \theta )(\csc \theta +\cot \theta )=1\end{array}}$

Proof using the unit circle

Main article: Unit circle

Point P(x,y) on the circle of unit radius at an obtuse angle θ > π/2

Sine function on unit circle (top) and its graph (bottom)

The unit circle centered at the origin in the Euclidean plane is defined by the equation:

${\displaystyle x^2+y^2=1.}$

Given an angle θ, there is a unique point P on the unit circle at an anticlockwise angle of θ from the x-axis, and the x- and y-coordinates of P are:

${\displaystyle x=\cos \theta \text{ and }y=\sin \theta .}$

Consequently, from the equation for the unit circle:

${\displaystyle {\cos }^2\theta +{\sin }^2\theta =1,}$

the Pythagorean identity.

In the figure, the point P has a negative x-coordinate, and is appropriately given by x = cos θ, which is a negative number: cos θ = −cos(π−θ). Point P has a positive y-coordinate, and sin θ = sin(π−θ) > 0. As θ increases from zero to the full circle θ = 2π, the sine and cosine change signs in the various quadrants to keep x and y with the correct signs. The figure shows how the sign of the sine function varies as the angle changes quadrant.

Because the x- and y-axes are perpendicular, this Pythagorean identity is equivalent to the Pythagorean theorem for triangles with hypotenuse of length 1 (which is in turn equivalent to the full Pythagorean theorem by applying a similar-triangles argument). See Unit circle for a short explanation.

Proof using power series

The trigonometric functions may also be defined using power series, namely (for x an angle measured in radians):

${\displaystyle \begin{array}{rl}\sin x& =\sum _{n=0}^{\mathrm{\infty }}\frac{(-1{)}^n}{(2n+1)!}{x}^{2n+1},\\ \cos x& =\sum _{n=0}^{\mathrm{\infty }}\frac{(-1{)}^n}{(2n)!}{x}^{2n}.\end{array}}$

Using the multiplication formula for power series at Multiplication and division of power series (suitably modified to account for the form of the series here) we obtain

${\displaystyle \begin{array}{rl}{\sin }^{2}x& =\sum _{i=0}^{\mathrm{\infty }}\sum _{j=0}^{\mathrm{\infty }}\frac{(-1{)}^i}{(2i+1)!}\frac{(-1{)}^j}{(2j+1)!}{x}^{(2i+1)+(2j+1)}\\ & =\sum _{n=1}^{\mathrm{\infty }}\left(\sum _{i=0}^{n-1}\frac{(-1{)}^{n-1}}{(2i+1)!(2(n-i-1)+1)!}\right)x^{2n}\\ & =\sum _{n=1}^{\mathrm{\infty }}\left(\sum _{i=0}^{n-1}(\genfrac{}{}{0ex}{}{2n}{2i+1})\right)\frac{(-1{)}^{n-1}}{(2n)!}{x}^{2n},\\ {\cos }^{2}x& =\sum _{i=0}^{\mathrm{\infty }}\sum _{j=0}^{\mathrm{\infty }}\frac{(-1{)}^i}{(2i)!}\frac{(-1{)}^j}{(2j)!}{x}^{(2i)+(2j)}\\ & =\sum _{n=0}^{\mathrm{\infty }}\left(\sum _{i=0}^n\frac{(-1{)}^n}{(2i)!(2(n-i))!}\right)x^{2n}\\ & =\sum _{n=0}^{\mathrm{\infty }}\left(\sum _{i=0}^n(\genfrac{}{}{0ex}{}{2n}{2i})\right)\frac{(-1{)}^n}{(2n)!}{x}^{2n}.\end{array}}$

In the expression for sin², n must be at least 1, while in the expression for cos², the constant term is equal to 1. The remaining terms of their sum are (with common factors removed)

${\displaystyle \sum _{i=0}^n(\genfrac{}{}{0ex}{}{2n}{2i})-\sum _{i=0}^{n-1}(\genfrac{}{}{0ex}{}{2n}{2i+1})=\sum _{j=0}^{2n}(-1{)}^j(\genfrac{}{}{0ex}{}{2n}{j})=(1-1{)}^{2n}=0}$

by the binomial theorem. Consequently,

${\displaystyle {\sin }^{2}x+{\cos }^{2}x=1,}$

which is the Pythagorean trigonometric identity.

When the trigonometric functions are defined in this way, the identity in combination with the Pythagorean theorem shows that these power series parameterize the unit circle, which we used in the previous section. This definition constructs the sine and cosine functions in a rigorous fashion and proves that they are differentiable, so that in fact it subsumes the previous two.

Proof using the differential equation

Sine and cosine can be defined as the two solutions to the differential equation:

${\displaystyle y^{″}+y=0}$

satisfying respectively y(0) = 0, y′(0) = 1 and y(0) = 1, y′(0) = 0. It follows from the theory of ordinary differential equations that the first solution, sine, has the second, cosine, as its derivative, and it follows from this that the derivative of cosine is the negative of the sine. The identity is equivalent to the assertion that the function

${\displaystyle z={\sin }^{2}x+{\cos }^{2}x}$

is constant and equal to 1. Differentiating using the chain rule gives:

${\displaystyle \frac{d}{dx}z=2\sin x\cos x+2\cos x(-\sin x)=0,}$

so z is constant. A calculation confirms that z(0) = 1, and z is a constant so z = 1 for all x, so the Pythagorean identity is established.

A similar proof can be completed using power series as above to establish that the sine has as its derivative the cosine, and the cosine has as its derivative the negative sine. In fact, the definitions by ordinary differential equation and by power series lead to similar derivations of most identities.

This proof of the identity has no direct connection with Euclid's demonstration of the Pythagorean theorem.

Proof using Euler's formula

Using Euler's formula ${\displaystyle e^{i\theta }=\cos \theta +i\sin \theta }$ and factoring ${\displaystyle {\cos }^2\theta +{\sin }^2\theta }$ as the complex difference of two squares,

${\displaystyle \begin{array}{rl}1& =e^{i\theta }{e}^{-i\theta }\\ & =(\cos \theta +i\sin \theta )(\cos \theta -i\sin \theta )\\ & ={\cos }^2\theta +{\sin }^2\theta .\end{array}}$

3-sphere

From Wikipedia, the free encyclopedia
https://en.wikipedia.org/wiki/3-sphere

Stereographic projection of the hypersphere's parallels (red), meridians (blue) and hypermeridians (green). Because this projection is conformal, the curves intersect each other orthogonally (in the yellow points) as in 4D. All curves are circles: the curves that intersect ⟨0,0,0,1⟩ have infinite radius (= straight line). In this picture, the whole 3D space maps the surface of the hypersphere, whereas in the next picture the 3D space contained the shadow of the bulk hypersphere.

Direct projection of 3-sphere into 3D space and covered with surface grid, showing structure as stack of 3D spheres (2-spheres)

In mathematics, a 3-sphere, glome or hypersphere is a higher-dimensional analogue of a sphere. In 4-dimensional Euclidean space, it is 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 2-sphere, a two-dimensional surface), the boundary of a ball in four dimensions (a gongyl) is a 3-sphere (a 3-dimensional "surface"). Topologically, a 3-sphere is an example of a 3-manifold, and it is also an n-sphere.

Definition

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 (R⁴) such that

${\displaystyle \sum _{i=0}^3(x_i-C_i{)}^2=(x_0-C_0{)}^2+(x_1-C_1{)}^2+(x_2-C_2{)}^2+(x_3-C_3{)}^2=r^2.}$

The 3-sphere centered at the origin with radius 1 is called the unit 3-sphere and is usually denoted S³:

${\displaystyle S^3=\left\{(x_0,x_1,x_2,x_3)\in {\mathbb{R}}^4:x_0^2+x_1^2+x_2^2+x_3^2=1\right\}.}$

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

${\displaystyle S^3=\left\{(z_1,z_2)\in {\mathbb{C}}^2:|z_1{|}^2+|z_2{|}^2=1\right\}}$

or

${\displaystyle S^3=\left\{q\in \mathbb{H}:\Vert q\Vert =1\right\}.}$

This description as the quaternions of norm one identifies the 3-sphere with the versors in the quaternion division ring. 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. See polar decomposition of a quaternion for details of this development of the three-sphere. This view of the 3-sphere is the basis for the study of elliptic space as developed by Georges Lemaître.

Properties

Elementary properties

The 3-dimensional surface volume of a 3-sphere of radius r is

${\displaystyle SV=2{\pi }^2{r}^3}$

while the 4-dimensional hypervolume (the content of the 4-dimensional region bounded by the 3-sphere) is

${\displaystyle H=\frac{1}{2}{\pi }^2{r}^4.}$

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 that 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.

In a given three-dimensional hyperplane, a 3-sphere can rotate about an "equatorial plane" (analogous to a 2-sphere rotating about a central axis), in which case it appears to be a 2-sphere whose size is constant.

Topological properties

A 3-sphere is a compact, connected, 3-dimensional manifold without boundary. It is also simply connected. 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. 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.

The 3-sphere is homeomorphic to the one-point compactification of R³. In general, any topological space that 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 H_i(S³, Z) = {} 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 3-sphere gives a homology sphere; typically these are not homeomorphic to the 3-sphere.

As to the homotopy groups, we have π₁(S³) = π₂(S³) = {} 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	Z2⊕Z2	Z12⊕Z2	Z84⊕Z2⊕Z2	Z2⊕Z2	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

${\displaystyle (z_1,z_2)\cdot \lambda =(z_1\lambda ,z_2\lambda )\mathrm{\forall }\lambda \in \mathbb{T}}$.

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

There are several well-known constructions of the three-sphere. Here we describe gluing a pair of three-balls and then the one-point compactification.

Gluing

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

Note that the interiors of the 3-balls are not glued to each other. 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". 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". 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". The temperature is highest/lowest at the centers of the two 3-balls.

This construction is analogous to a construction of a 2-sphere, performed by gluing 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 and glue corresponding points on their boundaries. Again one may think of the third dimension as temperature. Likewise, we may inflate the 2-sphere, moving the pair of disks to become the northern and southern hemispheres.

One-point compactification

After removing a single point from the 2-sphere, what remains is homeomorphic to the Euclidean plane. In the same way, removing a single point from the 3-sphere yields three-dimensional space. An extremely useful way to see this is via stereographic projection. We first describe the lower-dimensional version.

Rest the south pole of a unit 2-sphere on the xy-plane in three-space. 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 3-sphere (again removing the north pole) maps to three-space 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 one-point compactification is via the exponential map. 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. 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 one-point compactification.

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.

Coordinate systems on the 3-sphere

The four Euclidean coordinates for S³ are redundant since they are subject to the condition that x₀² + x₁² + x₂² + x₃² = 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

${\displaystyle \begin{array}{rl}{x}_0& =r\cos \psi \\ x_1& =r\sin \psi \cos \theta \\ x_2& =r\sin \psi \sin \theta \cos \phi \\ x_3& =r\sin \psi \sin \theta \sin \phi \end{array}}$

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 2-sphere of radius ${\displaystyle r\sin \psi }$, 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

${\displaystyle d{s}^2=r^2\left[d{\psi }^2+{\sin }^2\psi \left(d{\theta }^2+{\sin }^2\theta d{\phi }^2\right)\right]}$

and the volume form by

${\displaystyle dV=r^3\left({\sin }^2\psi \sin \theta \right)d\psi \wedge d\theta \wedge d\phi .}$

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

${\displaystyle q=e^{\tau \psi }=\cos \psi +\tau \sin \psi }$

where τ is a unit imaginary quaternion; that is, a quaternion that 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:

${\displaystyle \tau =(\cos \theta )i+(\sin \theta \cos \phi )j+(\sin \theta \sin \phi )k}$

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

${\displaystyle q=e^{\tau \psi }=x_0+x_{1}i+x_{2}j+x_{3}k}$

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

The Hopf fibration can be visualized using a stereographic projection of S³ to R³ and then compressing R³ to a ball. This image shows points on S² and their corresponding fibers with the same color.

For unit radius 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

${\displaystyle \begin{array}{rl}{z}_1& =e^{i{\xi }_1}\sin \eta \\ z_2& =e^{i{\xi }_2}\cos \eta .\end{array}}$

This could also be expressed in R⁴ as

${\displaystyle \begin{array}{rl}{x}_0& =\cos {\xi }_1\sin \eta \\ x_1& =\sin {\xi }_1\sin \eta \\ x_2& =\cos {\xi }_2\cos \eta \\ x_3& =\sin {\xi }_2\cos \eta .\end{array}}$

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

${\displaystyle S^1\to S^3\to S^2.}$

A diagram depicting the poloidal (ξ₁) direction, represented by the red arrow, and the toroidal (ξ₂) direction, represented by the blue arrow, although the terms poloidal and toroidal are arbitrary in this flat torus case.

For any fixed value of η between 0 and π/2, the coordinates (ξ₁, ξ₂) parameterize a 2-dimensional torus. Rings of constant ξ₁ and ξ₂ 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 3-sphere in these coordinates is given by

${\displaystyle d{s}^2=d{\eta }^2+{\sin }^2\eta d{\xi }_1^2+{\cos }^2\eta d{\xi }_2^2}$

and the volume form by

${\displaystyle dV=\sin \eta \cos \eta d\eta \wedge d{\xi }_1\wedge d{\xi }_2.}$

To get the interlocking circles of the Hopf fibration, make a simple substitution in the equations above^[2]

${\displaystyle \begin{array}{rl}{z}_1& =e^{i({\xi }_1+{\xi }_2)}\sin \eta \\ z_2& =e^{i({\xi }_2-{\xi }_1)}\cos \eta .\end{array}}$

In this case η, and ξ₁ specify which circle, and ξ₂ specifies the position along each circle. One round trip (0 to 2π) of ξ₁ or ξ₂ 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 S³ from a pole onto the corresponding equatorial R³ hyperplane. For example, if we project from the point (−1, 0, 0, 0) we can write a point p in S³ as

${\displaystyle p=\left(\frac{1-\Vert u{\Vert }^2}{1+\Vert u{\Vert }^2},\frac{2\mathbf{u}}{1+\Vert u{\Vert }^2}\right)=\frac{1+\mathbf{u}}{1-\mathbf{u}}}$

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 = u₁i + u₂j + u₃k 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 = (x₀, x₁, x₂, x₃) in S³ to

${\displaystyle \mathbf{u}=\frac{1}{1+x_0}\left(x_1,x_2,x_3\right).}$

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

${\displaystyle p=\left(\frac{-1+\Vert v{\Vert }^2}{1+\Vert v{\Vert }^2},\frac{2\mathbf{v}}{1+\Vert v{\Vert }^2}\right)=\frac{-1+\mathbf{v}}{1+\mathbf{v}}}$

where v = (v₁, v₂, v₃) is another vector in R³. The inverse of this map takes p to

${\displaystyle \mathbf{v}=\frac{1}{1-x_0}\left(x_1,x_2,x_3\right).}$

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 S³ consisting of two coordinate charts or "patches", which together cover all of S³. Note that the transition function between these two charts on their overlap is given by

${\displaystyle \mathbf{v}=\frac{1}{\Vert u{\Vert }^2}\mathbf{u}}$

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 that admit a Lie group structure are S¹, thought of as the set of unit complex numbers, and S³, the set of unit quaternions (The degenerate case S⁰ which consists of the real numbers 1 and −1 is also a Lie group, albeit a 0-dimensional one). 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 that 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:

${\displaystyle x_1+x_{2}i+x_{3}j+x_{4}k\mapsto \left(\begin{array}{cc}{x}_1+i{x}_2& x_3+i{x}_4\\ -x_3+i{x}_4& x_1-i{x}_2\end{array}\right).}$

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

${\displaystyle \left(\begin{array}{cc}{e}^{i{\xi }_1}\sin \eta & e^{i{\xi }_2}\cos \eta \\ -e^{-i{\xi }_2}\cos \eta & e^{-i{\xi }_1}\sin \eta \end{array}\right).}$

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

${\displaystyle U=\exp \left(\sum _{i=1}^3{\alpha }_i{J}_i\right).}$

The condition that the determinant of U is +1 implies that the coefficients α₁ 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; Carlo Rovelli supports the same idea.

In Art Meets Mathematics in the Fourth Dimension, Stephen L. Lipscomb develops the concept of the hypersphere dimensions as it relates to art, architecture, and mathematics.