The 3-sphere and the Hopf fibration

A circle is a one-dimensional world: an ant on it has one direction, forwards and back, and if it keeps going forwards it comes back to where it started. A sphere is a two-dimensional world of finite area with no edge. The next step is the 3-sphere $\S^3$: a world where you can move in three directions, as we do, but which is finite and closes up on itself. It lives in four-dimensional space, and the vertices of all the regular polytopes lie on it. In this chapter we will learn to picture it, compute its volume, and take apart the most beautiful thing on it — the Hopf fibration.

What the 3-sphere is

The circle of radius 1 is the set of points of the plane at distance 1 from the origin: $x^2 + y^2 = 1$. The ordinary sphere is the set of points of space at that distance: $x^2 + y^2 + z^2 = 1$. The 3-sphere is the set of points of four-dimensional space at distance 1: $$\S^3 = \{(x, y, z, w) : x^2 + y^2 + z^2 + w^2 = 1\}.$$ The number in the name is the dimension of the sphere itself, not of the space around it. The circle $\S^1$ is one-dimensional: a place on it is given by one number, an angle. The sphere $\S^2$ is two-dimensional: latitude and longitude. A point of $\S^3$ takes three numbers (three angles, say), so it is three-dimensional. Such objects are called manifolds: near each of its points $\S^3$ looks like a piece of ordinary three-dimensional space — just as a small patch of the Earth’s surface is indistinguishable from a plane.

The easiest way to picture it is by slices, as in the chapter on cross-sections. Cut the ordinary sphere with the planes $z = h$ and you get the parallels, circles of radius $\sqrt{1 - h^2}$: a point at the north pole, growing circles, the equator of radius 1, shrinking circles again, a point at the south pole. Cut $\S^3$ with the three-dimensional “planes” $w = h$ and you get spheres of the same radius $\sqrt{1 - h^2}$. If a 3-sphere passed through our world, we would see a point swell into a ball, grow to radius 1 and shrink back to a point. These spheres are the “parallels” of the 3-sphere; on the showcase an animation runs through them.

Two balls glued along their edge

There is another way, and it conveys better what living in it would be like. The sphere $\S^2$ can be glued from two discs — the northern and southern hemispheres, flattened out: each point on the rim of one disc is identified with the same point on the rim of the other. An ant that reaches the rim of the northern disc does not hit a wall; it crosses onto the rim of the southern one and carries on — now from the rim towards the centre.

The 3-sphere is glued in the same way from two ordinary balls. Its half $w \ge 0$ projects onto our space as the ball $x^2 + y^2 + z^2 \le 1$, and so does the half $w \le 0$. Glue the two balls along their boundary spheres, point to point. An astronaut in such a world flies from the centre of the first ball to its edge, crosses the boundary sphere — and finds herself on the surface of the second ball, from where she flies towards its centre. Flying straight on, she crosses the second ball and comes back into the first from the opposite side. There are no walls anywhere, and the volume of the whole world is finite.

The third way is stereographic projection. Put a lamp at the pole $(0, 0, 0, 1)$ and project every other point of $\S^3$ from it onto the three-dimensional space $w = 0$. The point $(x, y, z, w)$ goes to $$\left(\frac{x}{1 - w},\ \frac{y}{1 - w},\ \frac{z}{1 - w}\right).$$ The opposite pole goes to the origin, the sphere $w = 0$ stays where it is, and points near the lamp fly off arbitrarily far. So $\S^3$ minus one point is the whole of our space $\R^3$, and the missing point is “infinity”, where all directions meet. The projection preserves angles and takes circles to circles (or to lines, if the circle passes through the lamp), which is why almost every picture in this chapter is drawn with it.

Volume

The four-dimensional ball of radius $r$ has volume $\tfrac{\pi^2}{2} r^4$, and its boundary, the 3-sphere, has “area” (strictly, three-dimensional volume) $2\pi^2 r^3$. The second number comes from the same trick Archimedes used for the area of the ordinary sphere: cut it into thin layers.

For the ordinary sphere the same computation gives a layer of $2\pi\sqrt{1 - h^2} \cdot dh/\sqrt{1 - h^2} = 2\pi\,dh$: the area of a zone depends only on its height (Archimedes’ theorem), and the whole sphere is $4\pi$. The 3-sphere no longer has that evenness. The ball’s volume is built up from nested spheres, like an onion: $\int_0^r 2\pi^2\rho^3\,d\rho = \tfrac{\pi^2}{2}r^4$.

We can go straight to a formula for a ball of any dimension. Write $V_n(r)$ for the volume of the $n$-dimensional ball; $V_0 = 1$ (a point) and $V_1(r) = 2r$ (a segment).

The area of the boundary sphere is the derivative of the volume with respect to the radius: $S_{n-1}(r) = V_n'(r) = \tfrac{n}{r}V_n(r)$. Here are the first values for $r = 1$ (the last column is the share of the enclosing cube $[-1, 1]^n$ that the ball fills):

The volume of the unit ball grows up to $n = 5$ and then falls to zero: for $n \ge 7$ the factor $2\pi/n$ is less than one, and $V_{20}$ is down to about 0.026. The sphere’s area peaks a little later, at $n = 7$. But do not read deep meaning into the five: it depends on the choice of unit length. The volume of an $n$-dimensional ball is measured in units of length to the $n$-th power, and comparing $V_5$ with $V_6$ is like comparing an area with a volume. The recursion shows that the volume grows (every other dimension) as long as $n < 2\pi r^2$: for $r = 1$ that boundary is about 6.28, hence the peak at five, while for $r = 1.5$ the peak moves to $n = 13$.

What does not depend on the units is the share of the enclosing cube $[-r, r]^n$ that the ball fills: $V_n(r)/(2r)^n$. In the plane the disc fills 78.5% of the square, in space the ball fills 52.4% of the cube, in four dimensions 30.8%, in ten a quarter of a percent, in twenty $2.5 \cdot 10^{-8}$. Almost all the volume of a high-dimensional cube sits in its corners. More on this and other oddities of high dimensions in the next chapter.

A sphere you can multiply

The points of the circle $\S^1$ can be thought of as complex numbers $e^{i\varphi}$ of modulus 1. They can be multiplied: the modulus of a product is the product of the moduli, so the product lies on the circle again. The circle is a group, and multiplying by $e^{i\varphi}$ rotates the plane by the angle $\varphi$.

The 3-sphere does the same, only it needs four-dimensional numbers — quaternions $q = a + bi + cj + dk$ with $i^2 = j^2 = k^2 = ijk = -1$. The modulus of a quaternion is $|q| = \sqrt{a^2 + b^2 + c^2 + d^2}$, and again $|pq| = |p|\,|q|$. So the quaternions of modulus 1 are exactly the points of $\S^3$, and they form a group under multiplication. Of all spheres, only $\S^0$ (the two numbers $\pm1$), $\S^1$ and $\S^3$ carry such a structure (a smooth group, a Lie group). The unit octonions give a multiplication on $\S^7$, but it is not associative, and it does not make a group.

Unit quaternions describe rotations of our space: write a vector $(x, y, z)$ as $v = xi + yj + zk$, and $v \mapsto q\,v\,\bar q$ is a rotation, with $q$ and $-q$ giving the same one. The 3-sphere covers the set of all rotations twice. That is why quaternions are used for the orientation of satellites, robots and game cameras (more in the chapter on rotation). And the vertices of the 16-, 24- and 600-cell are finite subgroups of this very group.

The Clifford torus

Write a point of $\R^4$ as two complex numbers: $z_1 = x + iy$, $z_2 = z + iw$. The equation of the 3-sphere becomes $|z_1|^2 + |z_2|^2 = 1$. Look at the points where the moduli are equal: $$|z_1| = |z_2| = \tfrac{1}{\sqrt2}.$$ These are $\bigl(\tfrac{1}{\sqrt2}e^{i\alpha},\ \tfrac{1}{\sqrt2}e^{i\beta}\bigr)$ with two independent angles — a torus, the product of two circles of radius $1/\sqrt2$. It is called the Clifford torus, after the English mathematician William Clifford. It is flat: it can be glued from a square with no stretching at all. In three dimensions this cannot be done smoothly — you cannot make a paper doughnut without creases — but in four it works.

The Clifford torus divides $\S^3$ into two equal parts: $|z_1| \le |z_2|$ and $|z_1| \ge |z_2|$. Each part is a solid torus, a doughnut with its dough. The core circle of the first is $z_1 = 0$, of the second $z_2 = 0$. So the 3-sphere is also two solid tori glued along their surfaces, with the meridian of one glued to the longitude of the other. In stereographic projection the circle $z_2 = 0$ becomes the unit circle in the $xy$ plane, and the circle $z_1 = 0$ becomes the $z$ axis, which closes up through infinity. A line threaded through a circle — they are linked like two links of a chain. The Clifford torus in this projection is on the showcase.

The Hopf fibration

Now for the main event. Send a point $(z_1, z_2)$ of the 3-sphere to the ratio $$h(z_1, z_2) = \frac{z_1}{z_2} \in \mathbb{C} \cup \{\infty\}.$$ The complex plane with a point at infinity added is the ordinary sphere $\S^2$ (the Riemann sphere: the same stereographic projection, one level down). The result is a map $h\colon \S^3 \to \S^2$ — the Hopf map; Heinz Hopf published it in 1931.

Which points go to the same place? The ratio $z_1/z_2$ does not change if both numbers are multiplied by the same factor $\lambda$, and to stay on the sphere we need $|\lambda| = 1$, that is, $\lambda = e^{i\theta}$. So the preimage of each point of $\S^2$ is a circle $\{(e^{i\theta}z_1, e^{i\theta}z_2)\}$. It is called a fibre, and the whole picture is the Hopf fibration: the 3-sphere is split into circles, one over each point of the 2-sphere. In the language of quaternions the same thing looks like this: writing $q = z_1 + j z_2$, the fibre is $\{q\,e^{i\theta}\}$, and the map becomes $q \mapsto q\,i\,\bar q$ (which is a unit vector, that is, a point of $\S^2$).

What does it look like? In our projection the fibre over one pole of the 2-sphere is the unit circle, and over the other a vertical line. The fibres over a parallel fill a torus. The closer the parallel is to the first pole, the thinner the torus around the unit circle; the closer to the second, the fatter it gets and the more tightly its hole hugs the line. The equator gives the Clifford torus. All of space is packed with nested tori, and each torus is woven from circles that go around it once the long way and once the short way. Such circles on a torus were described in 1848 by the French astronomer Yvon Villarceau: they are cut out by a plane that touches the torus at two points. They are called Villarceau circles.

Hopf’s real discovery was not the picture itself. The map $h$ cannot be continuously deformed into a constant map — the 3-sphere cannot be “pulled” onto a single point of the 2-sphere — and the obstruction is precisely the linking of the fibres. The linking number (now called the Hopf invariant) does not change under deformation, and for a constant map it would be zero. It was the first example of a sphere wrapped non-trivially around a sphere of lower dimension. Hopf’s paper opened up a large field, the homotopy groups of spheres, which are still far from fully computed.

Where it shows up

• The qubit. The state of a quantum bit is a pair of complex numbers $(\alpha, \beta)$ with $|\alpha|^2 + |\beta|^2 = 1$ — a point of $\S^3$. But an overall phase factor $e^{i\theta}$ cannot be observed: $(\alpha, \beta)$ and $(e^{i\theta}\alpha, e^{i\theta}\beta)$ are the same physical state. That is exactly a Hopf fibre, and the space of distinguishable states is a 2-sphere, the Bloch sphere. Passing from the state vector to its point on that sphere is the Hopf map. R. Mosseri and R. Dandoloff (2001) showed that for a pair of qubits a similar role is played by the next Hopf fibration, $\S^7 \to \S^4$.
• The magnetic monopole. In the same year, 1931, and independently of Hopf, Paul Dirac showed that if a magnetic charge existed, electric charge would have to be quantised. The phase of an electron’s wave function cannot be defined consistently all over a sphere around the monopole. In 1977 Andrzej Trautman, apparently the first to do so, noticed that the field of a monopole of the smallest charge is built exactly like the Hopf fibration.
• Orientation in space. A rotation of a rigid body is a direction — where a chosen axis points, a point of $\S^2$ — plus an angle of turn about it, a point on the fibre circle. This is how the evenly spaced grids of rotations for robot motion planning built by A. Yershova, S. Jain, S. LaValle and J. Mitchell (2010) work.
• Liquid crystals and magnets. Hopfions are stable “knots” in a field of directions, in which the points sharing a direction form linked circles, like Hopf fibres. They are created in chiral liquid crystals (P. Ackerman and I. Smalyukh, 2017), and in 2023 rings of hopfions were seen for the first time inside a magnetic crystal, FeGe.

The Poincaré conjecture

The 3-sphere is the simplest closed (finite and edgeless) three-dimensional manifold. How could you recognise it from the inside, without stepping into the fourth dimension? On an ordinary sphere every loop can be shrunk to a point — a rubber band slides off a ball — but on a torus it cannot: a loop around the hole gets stuck. Manifolds on which every loop shrinks are called simply connected. In 1904 Henri Poincaré asked: is every simply connected closed three-dimensional manifold, up to continuous deformation, the 3-sphere?

The conjecture was one of the seven Millennium Prize Problems of the Clay Mathematics Institute. Perelman posted three preprints in November 2002 and in March and July 2003, declined the Fields Medal in 2006 and the Clay Institute’s million-dollar prize in 2010.

Summary

• $\S^3 = \{x^2 + y^2 + z^2 + w^2 = 1\}$ is a three-dimensional manifold: finite, edgeless, and near each point like our space. It can be pictured by its slices, as two balls glued along their edge, or through stereographic projection as $\R^3$ plus a point at infinity.
• The four-dimensional ball has volume $\tfrac{\pi^2}{2}r^4$, the 3-sphere $2\pi^2 r^3$; in general $V_n = \tfrac{2\pi r^2}{n}V_{n-2}$ and $V_n = \pi^{n/2}r^n/\Gamma(\tfrac n2 + 1)$. The unit ball is largest at $n = 5$, and the share of the enclosing cube it fills tends to zero.
• The points of $\S^3$ are the unit quaternions; they form a group and describe rotations of three-dimensional space.
• The Clifford torus $|z_1| = |z_2| = 1/\sqrt2$ is flat and splits $\S^3$ into two solid tori.
• The Hopf fibration splits $\S^3$ into great circles, one over each point of $\S^2$; any two are linked once. It describes the qubit, Dirac’s monopole, the orientation of bodies and hopfions.
• The Poincaré conjecture, proved by Perelman: a simply connected closed three-dimensional manifold is $\S^3$.

Next comes the strange geometry of many dimensions: what else happens to balls and cubes when the dimensions pile up.

Sources

• H. Hopf. Über die Abbildungen der dreidimensionalen Sphäre auf die Kugelfläche. Mathematische Annalen 104 (1931), 637–665. doi:10.1007/BF01457962.
• M. A. Peterson. Dante and the 3-sphere. American Journal of Physics 47 (1979), 1031–1035. doi:10.1119/1.11968.
• A. Yvon Villarceau. Théorème sur le tore. Nouvelles Annales de Mathématiques 7 (1848), 345–347.
• P. A. M. Dirac. Quantised singularities in the electromagnetic field. Proceedings of the Royal Society A 133 (1931), 60–72.
• A. Trautman. Solutions of the Maxwell and Yang–Mills equations associated with Hopf fibrings. International Journal of Theoretical Physics 16 (1977), 561–565.
• R. Mosseri, R. Dandoloff. Geometry of entangled states, Bloch spheres and Hopf fibrations. Journal of Physics A: Mathematical and General 34 (2001), 10243–10252. arXiv:quant-ph/0108137.
• A. Yershova, S. Jain, S. M. LaValle, J. C. Mitchell. Generating uniform incremental grids on SO(3) using the Hopf fibration. International Journal of Robotics Research 29 (2010), 801–812.
• P. J. Ackerman, I. I. Smalyukh. Diversity of knot solitons in liquid crystals manifested by linking of preimages in torons and hopfions. Physical Review X 7 (2017), 011006.
• F. Zheng et al. Hopfion rings in a cubic chiral magnet. Nature 623 (2023), 718–723.
• H. Poincaré. Cinquième complément à l’analysis situs. Rendiconti del Circolo Matematico di Palermo 18 (1904), 45–110.
• G. Perelman. arXiv:math/0211159, math/0303109, math/0307245.
• Wikipedia: 3-sphere, Hopf fibration, Clifford torus, Volume of an n-ball, Poincaré conjecture.