What can you do with this space? So asks Andrew Stacey. ‘Fill it’ is the prompt reply, but fill it with what? Maybe like Andrew you want to use a single curve, but I want to use circles. If you do this in the way shown above in blue, the result is called an Apollonian packing, a variant of which can be seen on the cover of this issue.

Here we shall explore the history of this entrancing object, which spans over 2000 years, and percolates into a surprising variety of mathematical disciplines. Starting in the familiar world of Euclidean geometry, Apollonian packings extend into fractal geometry and measure theory; Möbius transformations and the hyperbolic plane; and then on into the distant reaches of geometric group theory, number theory, orbital mechanics, and even ship navigation.

At times we may wonder off into thickets of more obscure mathematics, so those readers who get lost should feel free to skip ahead to the next section.

## Apollonius of Perga

Apollonius (c 230 BC) was a Hellenistic mathematician, considered one of the greatest after Euclid and Archimedes. Perhaps his most important work was his eight book treaties *Κωνικα* on conic sections—once lost to European civilisation, but fortuitously preserved by the more enlightened Middle Eastern scholars and later reintroduced by Edmund Halley in 1710. The same unfortunately cannot be said of *Έπαφαι* (*De Tractionibus* or *Tangencies*). Although now lost, we have accounts of the work from other ancient authors, particularly in the writings of Pappus of Alexandria. In it, Apollonius posed and solved the following problem.

**Problem:**Given three geometric objects in the plane (points, lines, and/or circles), find all circles which meet all three simultaneously (ie which pass through any points, and are tangent to any lines or circles).

So for example, given three points which don’t lie on the same line, there is exactly one circle which passes through all three. The case which interests us at present is when we are given three circles, each of which is tangent to the other two. In the very special case that all three are tangent at the same point there are infinitely many circles tangent to all three. Usually, however, the circles will be pairwise tangent at three distinct points, in which case there are exactly two other circles tangent to all three simultaneously.

This is as far as Apollonius went; the next step would not be taken until 1643, when René Descartes discovered a formula for the size of the two tangent circles, which he wrote in a letter to Princess Elizabeth of the Palatinate. The same formula was later rediscovered by Frederick Soddy and published as a poem in Nature in 1936.

The size of a circle is determined by its radius $r$. If $r$ is small, the circle will be small, but it will also be very curved. We can define the *curvature* of the circle to be $k=1/r$. Descartes showed that if three given circles are mutually tangent at three distinct points, and have curvatures $k_1$, $k_2$, and $k_3$, then a fourth circle which is tangent to all three has curvature $k_4$ satisfying

\begin{equation}\label{eq:descartes} (k_1+k_2+k_3+k_4)^2=2(k_1^2+k_2^2+k_3^2+k_4^2) \tag{1} \end{equation}

For technical algebraic reasons, sometimes this equation gives negative values for the curvature $k_4$, which we can interpret as corresponding to a circle with curvature $|k|$ which contains the other circles in its interior. Notice that this equation is quadratic in the variable $k_4$, so there are two solutions; these will correspond to the two possibilities for the fourth circle found by Apollonius.

## Apollonian packings

So far we have constructed at most 5 mutually tangent circles. The step to infinity may seem obvious, but took another 63 years and some 1900 years after Apollonius. The earliest description seems to appear in a letter from Leibniz to des Bosses (11 March 1706):

Imagine a circle; in it draw three other circles that are the same size and as large as possible, and in any new circle and in the space between circles again draw the three largest circles of the same size that are possible. Imagine proceeding to infinity in this way…

What Leibniz is describing is in fact a *nested* Apollonian packing, since at each step he fills in every circle as well as the gaps between circles. This early description makes the nested Apollonian packing one of the first fractals, although it wasn’t studied properly until mathematicians like Cantor, Weierstrass, von Koch, and Sierpinski started discovering other fractals in the late nineteenth and early twentieth centuries. This may be because Leibniz was not interested in the mathematical construction, but rather was trying to draw an analogy to argue against the existence in infinitesimals in nature.

Henceforth we shall only consider the un-nested Apollonian packing. As a fractal, it has a number of interesting properties: it is a set of measure 0, which means that if you tried to make it by starting with a disc of metal, and then drilled out infinitely many ever smaller holes (and if you ignore that metal is made out of atoms), then you would finish up with a single piece of metal (you haven’t removed everything), but nevertheless with exactly 0 mass. It has fractal dimension approximately 1.30568, which means that mathematically it lives somewhere between a 1D curve and a 2D area. Finally, if you look at just the portion of an Apollonian packing which lies in the triangular region between three tangent circles, this is homeomorphic to the Sierpinski triangle, which means that one can be bent and stretched to look like the other.

There is a curious combinatorial consequence of Descartes’ formula for Apollonian packings. If we start with three mutually tangent circles with curvatures $k_1$, $k_2$, and $k_3$, we can solve \eqref{eq:descartes} to find that the curvatures $k_4^+$ and $k_4^-$ of the other two circles are

\begin{equation}\label{eq:descartes2}\tag{2} k_4^\pm=k_1+k_2+k_3\pm2\sqrt{k_1k_2+k_2k_3+k_3k_1} \end{equation}

Now suppose we start constructing an Apollonian packing by drawing four mutually tangent circles whose curvatures $k_1$, $k_2$, $k_3$, and $k_4^+$ are all integers. From equation \eqref{eq:descartes2} it follows that $2\sqrt{k_1k_2+k_2k_3+k_3k_1}$ must be an integer since $k_4^+$ is an integer, and so $k_4^-$ is also an integer. Now we can build the packing by filling in a fifth circle wherever we see four mutually tangent circles. By the observation above, if the four circles have integer curvatures, the fifth circle will also have integer curvature. Inductively therefore we will end up with an Apollonian packing consisting of infinitely many tangent circles, all of which have integer curvatures.

## Hyperbolic geometry

If you have some familiarity with non-Euclidean geometry, Apollonian packings may remind you of the Poincaré model of the hyperbolic plane. The hyperbolic plane $\mathbb{H}^2$ is a 2D surface on which we can do geometry just like we can on the flat Euclidean plane. Whereas a sphere has constant positive curvature (it curves the same way in all directions), and the Euclidean plane has constant zero curvature (it’s flat), $\mathbb{H}^2$ is an infinite surface which has constant negative curvature, which means that at every point it curves in the same way as a Pringle. This negative curvature makes the surface crinkle up on itself more and more as you move out towards infinity, which is inconvenient when we try to work with it. Usually then we represent it on a flat surface so we can draw pictures of it in magazines and the like. One way to do this is with the Poincaré model. This views the hyperbolic plane as a disc. In order to fit the whole infinity of $\mathbb{H}^2$ into a finite disc, we have to shrink distances as we move out towards the edge of the disc. Using this skewed way of measuring distances, the circular edge of the disc is *infinitely* far away from its centre.

We can think of an Apollonian packing as living in the Poincaré disc, with the outermost circle of the packing as the boundary circle of $\mathbb{H}^2$. Then the circles in the packing which are not tangent to this boundary are also circles in the strange hyperbolic way of measuring distance, that is, all points are equidistant from some other point in the plane—the circle’s hyperbolic centre. Circles in the packing which *are* tangent to the boundary are called *horocycles* (in Greek this literally means border circle), which are circles with infinite radius in the hyperbolic metric. Horocycles have no analogue in the Euclidean plane.

Something interesting happens when we see what an Apollonian packing looks like in the upper half-plane (UHP) model for $\mathbb{H}^2$. This model is similar to the Poincaré model, but instead of using a disc, we use the half-plane above the $x$-axis $\{(x,y)\in \mathbb{R}^2: y> 0\}$, where the $x$-axis behaves like the boundary circle and should be thought of as *at infinity*. There is a problem, in that in the Poincaré disc, the boundary of $\mathbb{H}^2$ was a circle, and so it closed up on itself. In the UHP, the boundary is a line which doesn’t close up on itself, but these are supposed to be models for the same thing. To fix this, we imagine there is a *point at infinity* $\infty$ which joins up the two ends of the boundary to form an infinite diameter circle.

If we start with any Apollonian packing living in the Poincaré disc, there is a map from the disc to the UHP preserving hyperbolic distances, under which the outer circle of the packing becomes the $x$-axis (together with the point at infinity), and exactly one of the horocycles (one of the circles tangent to the outer circle in the packing) becomes the horizontal line $y=1$. All other circles and horocycles in the packing are sent to circles which are tangent to each other as before, but are now sandwiched between the lines $y=0$ and $y=1$.

If we focus on just those circles which meet the $x$-axis we get what are called *Ford circles*. Remarkably each of these circles is tangent to the $x$-axis at a rational number $p/q$, and has radius $1/2q^2$. Moreover every rational number is the point of tangency of one of the circles (see below). Now some magic happens: suppose the Ford circles at $a/b$ and $c/d$ are tangent to each other, then there is a unique circle sandwiched between these two circles and the $x$-axis. The rational point at which this circle meets the $x$-axis is given by the *Farey sum* of $a/b$ and $c/d$

\[ \frac{a}{b}\oplus\frac{c}{d}=\frac{a+b}{c+d} \]

Note that for this to be well-defined, $a/b$ and $c/d$ must be written in their simplest form. This Farey sum, and the associated Farey sequences $F_n$ you get by looking at all rational numbers between 0 and 1 which can be written as a fraction with denominator at most $n$, turn up in several places across number theory. These include rational approximation of irrational numbers and the Riemann Hypothesis.

## Möbius transformations

If you haven’t seen hyperbolic geometry before, you may wonder how we can map the Poincaré disc model to the UHP model, and in such a way that the strange distance measure in the two models is preserved—for a start one is a finite region while the other is an infinite half-plane. The answer is to view both models as living inside the complex plane $\mathbb{C}$ (or more accurately the *extended* complex plane $\widehat{\mathbb{C}}=\mathbb{C}\cup\{\infty\}$): the Poincaré disc is the unit disc $\{z\in\mathbb{C}: |z|<1\}$, and the UHP is the region above the real axis $\{z\in\mathbb{C}:$ Im$(z)>0\}$. Then a function like

\begin{equation}\label{eq:discToUHP}\tag{3} z\mapsto-\mathrm{i}\frac{z+1}{z-1}=\frac{-\mathrm{i} z-\mathrm{i}}{z-1} \end{equation}

will do the trick. This function is an example of a Möbius transformation, which in general is a complex function of the form

\[ z\mapsto \frac{az+b}{bz+d} \]

were we require $ad-bc\ne0$ so that this function is invertible. The function \eqref{eq:discToUHP} sends the unit disc to the UHP, but it is not the only Möbius transformation which does this. In fact there are infinitely many such functions, all of which preserve the hyperbolic metric. In the previous section I claimed that starting with any Apollonian packing, we could choose one of these Möbius transformations such that the image had a very specific form, sandwiched between the lines Im$(z)=0$ and Im$(z)=1$.

**An exercise:**If you have seen Möbius transformations before, you may wish to try and prove that the purported mapping exists yourself. (Hint: remember that Möbius transformations send circles and lines to circles and lines, and are completely determined by their image on 3 distinct points.)

The upshot of this is that all Apollonian packings are the same in the hyperbolic plane, because they can all be mapped to the same packing by (invertible) functions which preserve hyperbolic distance. Once we have started thinking about *the* Apollonian packing living in the complex plane, the whole world of complex functions is open to us, and we can start to do crazy things. If we don’t restrict ourselves to just Möbius transformations, but see what happens when we apply holomorphic or anti-holomorphic functions to the packing (these are complex functions with a good notion of derivative in the sense of calculus, which in particular have the property that they preserve angles between intersecting curves), we can get some very pretty designs. We need not even require (anti-)holomorphicity. The patterns featured on the front and back covers were drawn in this way.

## Beyond the packing

Let us return to Apollonius of Perga. Remember that his treaties *Έπαφαι*, where he stated and solved the problem of finding tangent circles, is lost to history—how then do we know what he proved and how? The answer is that we don’t. The only record we have appears in the writings of Pappus of Alexandria, who lived some 400 years after Apollonius, but who references many of Apollonius’ works, including six which are no longer extant. All he says of *Tangencies* is the general problem which Apollonius was interested in, and that he solved it by solving many simple special cases and working up from there.

The first person to reprove Apollonius’ results in ‘modern’ times was Adriaan van Roomen in 1596. His solution, however, does not use ruler and compass constructions, so cannot have been the one Apollonius used. The result was later proved using methods available to Apollonius, and in the way described by Pappus, by van Roomen’s friend François Viéte.

The method of Viéte was later reworked and simplified by several mathematicians, including Isaac Newton in his *Principia*. Newton related the position of the centre of the fourth circle to its distance from the centres of the three circles to which it is supposed to be tangent. This is called *hyperbolic positioning* or *trilateration*. Newton used this viewpoint to describe the orbits of planets in the solar system, but it can also be used to help navigate ships, and to locate the source of a signal based on the different times the signal is received at three different locations. In the first world war this was used to locate artillery based on when shots were heard. This is also how modern GPS works (not by *triangulation* as is commonly believed).

So this 2000-year-old problem in abstract geometry turned out to have extremely useful applications in the real world. The Apollonian packing also shows up in lots of different areas of mathematics. For example, Ford circles inspired the Hardy–Littlewood circle method, an important tool in analytic number theory which was used to solve Waring’s Problem: for an integer $k$, can every integer be written as a sum of at most $n$ $k$^{th} powers for some value of $n$? This is true: for example, every integer is the sum of 4 squares, 9 cubes, 19 fourth powers, and so on. In 2013, Harald Helfgott used the circle method to prove the weak Goldbach conjecture: every odd number greater than 5 is the sum of 3 primes.

## To infinity

As a final application, I am a geometric group theorist, and I cannot help but talk about one place the Apollonian packing shows up in my field. Be warned: there is definitely some advanced maths coming up, but if you don’t mind skipping over some of the details, there are some very pretty pictures to make it worthwhile.

It turns out that the extended complex plane $\widehat{\mathbb{C}}$ can be thought of as the boundary of 3 dimensional hyperbolic space $\mathbb{H}^3$. If we model $\mathbb{H}^3$ as the upper half-*space *$\{(x,y,z)\in\mathbb{R}^3\mid z\ge 0\}\cup\{\infty\}$ then $\widehat{\mathbb{C}}$ is identified with the plane $\{(x,y,z)\in\mathbb{R}^3\mid z=0\}\cup\{\infty\}$. When Möbius transformations act on $\widehat{\mathbb{C}}$, they also act on the whole of $\mathbb{H}^3$, and preserve hyperbolic distance. If we start by choosing just a few Möbius transformations, these generate a group which acts on $\mathbb{H}^3$. In doing so, the group creates a pattern on the complex plane called its *limit set*. This is a picture of how the group acts ‘at infinity’. Choosing the Möbius transformations carefully gives a group whose limit set is precisely the Apollonian packing.

Let’s be a bit more precise; pick a point $p\in \widehat{\mathbb{C}}$ and choose $g$ pairs of circles $(C_i^+,C_i^-)_{i=1}^g$, each of which doesn’t intersect $p$. Each circle cuts $\widehat{\mathbb{C}}$ into two regions, call the region containing $p$ the *exterior* of that circle, and the complementary region the circle’s *interior*. We also want to arrange things so that no two circles have overlapping interiors (although two circles are allowed to be tangent). Next, for each pair of circles $(C_i^+,C_i^-)$ choose a Möbius transformation $m_i$ which maps $C_i^+$ to $C_i^-$ and which sends the interior of $C_i^+$ to the exterior of $C_i^-$. The group $G=\langle m_1,\dots,m_g\rangle$ generated by these transformations is called a (classical) Schottky group and it acts as a subgroup of the group of isometries of $\mathbb{H}^3$. Since we chose the circles to have non-overlapping interiors, we can use the delightfully named ‘Ping-Pong Lemma’ to prove that $G$ is abstractly isomorphic to the free group on $g$ generators.

So how do we get a Schottky group whose limit set is the Apollonian packing? We can cheat slightly by working backwards; starting off with the picture we want to create, then we will choose the pairs of circles in the right way. Remember that one way we thought about constructing the Apollonian packing was to start off with four mutually tangent circles and then inductively draw the fifth circle wherever we can. Our strategy will be to choose Möbius transformations which do the same thing. We are helped by the following curious fact which you may want to try and prove yourself (again using Möbius transformations): given any three mutually tangent circles, there is a unique circle (possibly through $\infty$) which passes through all three circles at right angles. Given the four initial circles, there are ${4\choose 3}=4$ triples of mutually tangent circles, so we let $C_1^\pm$ and $C_2^\pm$ be the four circles orthogonal to each of these triples, as shown on the left. The corresponding Möbius transformations are :

\begin{align*} m_1: z \mapsto \frac{z}{-2\mathrm{i}z+1} && m_2: z \mapsto \frac{(1-\mathrm{i})z+1}{z+(1+\mathrm{i})} \end{align*}

The limit set of $G=\langle m_1, m_2\rangle$ is indeed the Apollonian packing we started with. If we perturb the starting Möbius transformations just slightly by varying the matrix entries (while being careful to ensure that the resulting group acts nicely on $\mathbb{H}^3$), we get a group whose limit set is a twisted Apollonian packing.

Even though some of these perturbed limit sets look like they are still made up more or less of circles, they are in fact made up of a single continuous closed curve which is fractal, and does not intersect itself anywhere. They are examples of Jordan curves and illustrate why the Jordan Curve Theorem is so difficult to prove despite being ‘obvious’. Playing around more with different choices of Möbius generators we can produce even more beautiful examples of fractal limit sets; below are just a few to finish off. If you want to learn more about Schottky groups, their limit sets, and how to draw these pictures, I highly recommend the book *Indra’s pearls: the vision of Felix Klein*. It is the basis of this final section of this article, and gives details on exactly how you can draw these and many other pictures yourself.