The sphere that complex numbers live on

The previous rung established that stereographic projection preserves circles and angles. This one identifies the plane with the complex numbers and finds that those two properties are not merely convenient — they say that the sphere and the plane-plus-one-point are the same object in a much stronger sense than a bijection.

The object is the Riemann sphere: the complex numbers with a single extra point called $\infty$, given the topology in which large numbers are near it. Stereographic projection is the reason that is a sphere rather than a formal device, and the reason is that the north pole is where the large numbers go.

The rotations become divisions #

A rotation of the sphere is a rigid motion — the simplest thing that can be done to a ball. Read through the projection, it becomes a map of the plane, and the map is

$$z \mapsto \frac{az + b}{cz + d}$$

for complex $a$, $b$, $c$, $d$ with $ad - bc \neq 0$. Those are the Möbius transformations.

That the rotations land in this class is a computation, and the hero figure carries it out numerically. What is worth understanding is why the class is the right one to expect.

Möbius transformations are exactly the conformal bijections of the sphere to itself. Any map preserving angles and defined on the whole sphere must be one, by a theorem of complex analysis. Rotations preserve angles, so they must be Möbius transformations. The previous rung’s conformality is therefore not merely a property of the projection; it is what forces the correspondence.

Not every Möbius transformation is a rotation, though. The rotations are the ones whose matrix is unitary — the subgroup $\mathrm{SU}(2)$ up to sign — while the whole group is $\mathrm{PSL}(2,\mathbb{C})$, which is six-dimensional against the rotations’ three. The extra three dimensions are the maps that move the sphere without keeping it rigid: dilations, and the loxodromic spirals below.

Infinity as an ordinary place #

The single most useful consequence has nothing to do with rotations.

Consider $f(z) = 1/z$. On the plane it is undefined at zero and unbounded near it, and the phrase “tends to infinity” is a statement about a limit that does not exist. On the sphere it is the map sending $0$ to $\infty$ and $\infty$ to $0$, which is a rotation by half a turn about the axis through $1$ and $-1$ — a perfectly ordinary rigid motion with two fixed points and no singularity anywhere.

That is the same repair the punctured sphere makes to a bijection, applied to functions rather than to points. A pole is not a failure of a function; it is a point where the function takes the value $\infty$, and once $\infty$ is a point of the target that sentence is literal. A rational function of degree $n$ becomes a map of the sphere to itself that is exactly $n$-to-one everywhere, counted properly, with no exceptions to state.

That reformulation removes a whole category of special case. The fundamental theorem of algebra says a polynomial of degree $n$ has $n$ roots; on the sphere it says the map has degree $n$ and therefore hits every value $n$ times, including $\infty$, which it hits $n$ times at the single point $\infty$. One statement instead of a theorem and a convention.

Three points decide everything #

One fact about the group is worth stating on its own, because it is the reason Möbius transformations are as easy to work with as they are.

Given any three distinct points and any three distinct targets, there is exactly one Möbius transformation carrying the first triple to the second. Three points, three conditions, and the group has three complex dimensions — so the count is right and the theorem confirms it.

The construction is explicit. The map

$$z \mapsto \frac{(z - z_1)(z_2 - z_3)}{(z - z_3)(z_2 - z_1)}$$

sends $z_1$, $z_2$, $z_3$ to $0$, $1$, $\infty$. Composing one such map with the inverse of another carries any triple to any other, and uniqueness follows because a map fixing $0$, $1$ and $\infty$ has $b = 0$, $c = 0$ and $a = d$, hence is the identity.

That expression is the cross ratio, and its invariance is the group’s fundamental invariant: four points have a cross ratio, and a Möbius transformation does not change it. Since four points lie on a circle exactly when their cross ratio is real, the circle-preserving property of the last rung falls out of one equation.

The hero figure is built on this theorem. It fits a Möbius transformation through three points and then tests it at twenty more; the fit is legitimate precisely because three points determine the map, and the twenty tests are what turn a fit into a verification.

Four ways to move the sphere #

The Möbius transformations sort into four kinds, and the sorting is done by a single number.

Write the map as a matrix $\begin{pmatrix} a & b \\ c & d\end{pmatrix}$ scaled so that $ad - bc = 1$, and take its trace $a + d$. Then:

• trace real and strictly between $-2$ and $2$: elliptic, a rotation about two fixed points, every orbit a closed curve;
• trace real and outside that range: hyperbolic, a flow from one fixed point to the other along arcs;
• trace exactly $\pm 2$: parabolic, the two fixed points having merged into one;
• trace not real: loxodromic, a spiral that both turns and travels.

The classification is by the trace because the fixed points solve $cz^2 + (d-a)z - b = 0$, whose discriminant is $(a+d)^2 - 4$ after normalising. So how many fixed points and what kind of motion are the same question, which is why one number answers both.

It is worth checking the four cases against the maps everyone already knows. Multiplication by a complex number of modulus one is $z \mapsto e^{i\theta}z$, whose matrix has trace $2\cos(\theta/2)$ — real and inside the interval, so elliptic, and indeed it is a rotation about $0$ and $\infty$. Multiplication by a positive real $\lambda$ has trace $\sqrt\lambda + 1/\sqrt\lambda$, real and outside, so hyperbolic — and it flows from $0$ to $\infty$, which is what scaling does. Translation $z \mapsto z + b$ has trace exactly $2$, so parabolic, with its single fixed point at $\infty$; and a general complex multiplication is loxodromic, being a rotation and a scaling at once. The four kinds are the four things a complex number can do, transported to the sphere.

The elliptic maps are exactly the rotations, up to conjugation, and the loxodromic ones are what a rotation composed with a dilation looks like. The word comes from navigation: a loxodrome is a course of constant bearing, which spirals to the pole, and the spiral in the figure is the same curve for the same reason.

What the group is used for #

Three uses, and each is a different subject.

Hyperbolic geometry. The Möbius transformations preserving the upper half-plane are the isometries of the hyperbolic plane, and the classification above becomes the classification of hyperbolic isometries: elliptic ones rotate about an interior point, hyperbolic ones translate along a geodesic, parabolic ones fix a single point on the boundary. Two worlds that both obey the rules is that subject; this rung is where its transformations come from.

Special relativity. The set of directions an observer can look in is a sphere — the celestial sphere — and a Lorentz boost acts on it by changing where each star appears. That action turns out to be a Möbius transformation, so the group of Lorentz transformations is the same group as $\mathrm{PSL}(2,\mathbb{C})$. A boost along a fixed direction is hyperbolic, with fixed points fore and aft: it magnifies the sky ahead and shrinks the sky behind, which is stellar aberration. Combine a boost with a rotation and the result is loxodromic, spiralling — which is the geometric statement of the fact that two boosts in different directions compose into a boost and a rotation, the Thomas precession, and the trace makes it visible rather than surprising.

Complex dynamics. Iterating a rational map on the sphere is the setting for Julia sets and the Mandelbrot set, and the reason the sphere is the right stage is that it is compact — orbits cannot escape, they can only converge to $\infty$, which is a point like any other. Where Newton’s method goes instead is an instance, and its basins are drawn on a plane that is really this sphere — which is why the basin containing very large numbers is a basin rather than an edge effect, and why the picture would be a partition of a sphere if it could be drawn on one.

Putting the conformality figure among the dynamics is worth doing once, because the two are easy to conflate. A loxodromic map spirals, and a spiral looks like a distortion; it is not. Every one of the four kinds preserves every angle at every point. What the four kinds differ in is where points go, not what shapes become, and a figure of the angles is the reminder.

What is being given up #

The sphere has no addition. The complex numbers form a field and the Riemann sphere does not, because $\infty + \infty$ and $0 \times \infty$ have no defensible values. What survives is the multiplicative and Möbius structure, and any argument using addition has to stay in the plane.

The rotations are only half the story of rigidity. A rotation preserves distances on the sphere; a general Möbius transformation does not, and the round metric of the sphere is not preserved by anything outside $\mathrm{SU}(2)$. So the plane picture is conformally faithful and metrically misleading everywhere, which is the previous rung’s area distortion arriving in a new costume.

Orientation is fixed and reflections are excluded. Möbius transformations preserve orientation. The angle-preserving maps that reverse it — $z \mapsto \bar z$ and its relatives — are not in the group, and adding them gives a larger group with two components. Every statement above about “the conformal maps” quietly means the orientation-preserving ones.

And the point at infinity is one point, not many. In the plane there are infinitely many directions in which to go off to infinity, and stereographic projection identifies them all. That is the right choice for complex analysis and the wrong one for real projective geometry, where the different directions become distinct points on a line at infinity. Two compactifications of the same plane, chosen for different purposes, and neither is more correct.

How the two groups fit together #

The claim that the rotations sit inside the Möbius transformations deserves to be made precise, because the way they sit is the source of a good deal of mathematics.

A Möbius transformation is a two-by-two complex matrix up to scale, so the group is $\mathrm{PSL}(2,\mathbb{C})$ — six real dimensions. The rotations of the sphere form $\mathrm{SO}(3)$, three real dimensions. The map between them is not an inclusion of the obvious kind: it is a two-to-one covering $\mathrm{SU}(2) \to \mathrm{SO}(3)$, with the matrix and its negative giving the same rotation.

That two-to-one covering is the reason spin exists. A rotation by a full turn corresponds to the matrix $-I$ rather than to $I$, so an object described by the matrix rather than by the rotation comes back changed after one turn and unchanged after two. The physical statement is strange and the mathematical statement is not: $\mathrm{SU}(2)$ is the universal cover of $\mathrm{SO}(3)$, and the two-to-one is the index of the trivial subgroup in the fundamental group $\mathbb{Z}/2$.

So the same picture — a sphere, a projection, a matrix — carries the classification of conformal maps, the isometries of hyperbolic space, the Lorentz group, and the reason a half-integer spin is possible. That is an unusual amount of load for one construction, and it is worth being clear about what carries it: the correspondence between a rigid motion of a ball and a ratio of two linear expressions.

What the pictures cannot show #

Every drawing is of a bounded piece of an unbounded plane. The interesting behaviour of a Möbius transformation is often at $\infty$ — a parabolic map fixes it, a hyperbolic map flows into it — and a frame cuts exactly there. The panels show what happens near the origin and leave the fixed point at infinity outside the picture.

The orbits are drawn as trails and the map is not a flow. Iterating a Möbius transformation gives a sequence of points, not a curve. The curves in the figure are the sequences joined up, which is a legitimate way of showing where a point goes and is not the object: the map has no intermediate positions.

The trace is printed and the normalisation that makes it meaningful is not. A matrix and twice that matrix give the same Möbius transformation and different traces, so the number under each panel is only a classification after the determinant has been set to one. The figure does that scaling and asserts it; the panel shows a number whose meaning depends on a step taken off the page.

And a rotation of the sphere is drawn as two flat pictures. The claim is that a rigid motion in space induces the map on the right, and the figure shows the before and after in the plane, because drawing the sphere turning is a sequence of frames rather than a picture. The verification is numerical.

Where the ladder goes next #

Named here as debts, both raised above. The astrolabe — this projection built as an instrument, whose rotating overlay is a Möbius transformation of the sky and whose engraved circles are the circle property in brass. And the general one-point compactification, which does to any locally compact space what this rung does to the plane, and whose result is a sphere only in this case.

Sideways, the multiplication that becomes a rotation is complex multiplication read as turning, the classification by fixed points is the same trace argument as for a linear map’s eigenvalues, and the compactness that makes iteration well behaved is what a basin picture relies on without saying so.

Riemann’s reason for it #

The sphere is named for Riemann, and his purpose in introducing it was not the one this essay has been describing.

He was studying multi-valued functions — the square root, the logarithm — which cannot be defined consistently on the whole plane because going round the origin returns them to a different value. His answer was to build a surface on which the function is single-valued: several sheets of the plane, joined along cuts, so that going round the origin moves onto the next sheet.

The Riemann sphere is the simplest such surface — one sheet, one extra point — and it appears in his 1851 dissertation as the base case rather than as the point. What the general construction gives is that every algebraic function has a surface on which it becomes single-valued, and that the surface’s genus is an invariant of the function.

So the object this rung is about was a by-product of a much larger idea, and the larger idea is the one the mathematics went on to use: a function with awkward behaviour is a well-behaved function on a different space, and finding that space is the work. Adding one point to make $1/z$ respectable is that programme’s smallest possible instance, and the multi-sheeted covers of the circle are the same construction in a topological rather than an analytic setting.

What is worth carrying away #

Adding a single point to a space can turn a list of exceptions into a theorem.

Before the point at infinity, rational functions have poles, some circles are lines, division by zero is undefined, and every statement needs a clause. After it, rational functions are maps of a sphere with a degree, circles and lines are one kind of curve, and $1/z$ is a half-turn. Nothing was proved by adding the point; the statements became sayable.

The habit worth taking is to ask what a theory’s exceptions have in common. If they can all be removed by adjoining one object, the object is probably already implicit in the theory, and finding it is usually easier than working around its absence.