Curvatures that stay whole

Take three circles that touch one another. Eight circles touch three given ones, but when the three already touch each other, six of the eight have collapsed and exactly two are left: one nestled in the gap between them and one wrapped around all three. Choose one, and now there are more triples that touch one another, each with its own two.

Repeat forever and the circles fill every gap. That is the Apollonian gasket, and it has a property that has nothing to do with how it was drawn: if the four curvatures one starts from are whole numbers, every curvature in the infinite packing is a whole number.

The relation #

Curvature is the reciprocal of radius, taken negative for a circle that contains the others rather than sitting beside them. For four mutually tangent circles with curvatures $k_1, k_2, k_3, k_4$,

$$(k_1 + k_2 + k_3 + k_4)^2 = 2\,(k_1^2 + k_2^2 + k_3^2 + k_4^2).$$

That is Descartes’s circle theorem, from a 1643 letter to Princess Elisabeth of Bohemia, and it was rediscovered several times before Frederick Soddy put it into verse in 1936. Its higher-dimensional analogue — $n+2$ mutually tangent spheres in $n$ dimensions, with the same shape of relation — was found by Soddy at the same time, and the sphere version is why the four-circle case looks like a coincidence rather than a special fact.

Two things about it are worth noticing before anything is done with it.

It mentions no positions. Four mutually tangent circles are rigid up to the transformations that preserve tangency, so their radii determine the picture — which is exactly the kind of fact a relation between radii alone can express. That rigidity is worth pausing on: a configuration of four touching circles has no shape parameters left once the radii are chosen, which is why the count of tangent circles in the rung below collapsed from eight to two. A triangle is determined by three lengths for the same reason and with the same consequence — the constraints have used up the freedom.

It is quadratic in each curvature. Read as an equation for $k_4$ with the others given, it says

$$k_4^2 - 2(k_1+k_2+k_3)k_4 + \big[(k_1+k_2+k_3)^2 - 2(k_1^2+k_2^2+k_3^2)\big] = 0,$$

a quadratic with two roots. Those two roots are the two circles tangent to the first three, which is the count from the rung below with six of the eight collapsed.

The subtraction that generates everything #

Here is the step that makes the whole subject work, and it is a fact about quadratics rather than about circles.

The two roots of a quadratic sum to minus the coefficient of the linear term. So the two circles tangent to three given ones have curvatures $k_4$ and $k_4'$ with

$$k_4 + k_4' = 2(k_1 + k_2 + k_3).$$

Given three mutually tangent circles and one circle that touches all three, the other one is

$$k_4' = 2(k_1+k_2+k_3) - k_4,$$

with no square root anywhere. And the same relation holds for the products of curvature and centre — writing the centres as complex numbers, $k_4' z_4' = 2(k_1z_1 + k_2z_2 + k_3z_3) - k_4z_4$ — so the new circle’s position comes out of the same subtraction.

The consequence is immediate. Start with four circles whose curvatures are whole numbers; every circle generated afterwards has curvature an integer combination of earlier ones. So every curvature in the gasket is whole, forever, and no computation more elaborate than addition ever occurs.

Where inversion comes in #

The derivation above is algebra and it is complete. What it is not is a reason, and the reason is the map this ladder is about.

The subtraction can be derived from the quadratic, and it can be seen. The second view is the one worth having.

Given three mutually tangent circles and a fourth touching all three, invert the whole picture in a circle centred at the point where two of the three touch. Those two become parallel lines — each passes through the centre of inversion, so each becomes a line, and they cannot meet because their preimages met only at the centre. The third circle becomes a circle squeezed between the two lines, and the two solutions become two circles of the same size squeezed between the same two lines, one on each side of it.

In that picture the symmetry is a reflection, and the two solutions are obviously a pair. Carrying the reflection back through the inversion gives the pair of circles the theorem describes — and the fact that the two solutions are related by a reflection is what makes their curvatures related by a subtraction rather than by anything worse.

That reduction is the same one the rung below used for the general tangency problem: put the configuration where the answer is obvious. What is new here is that the position is reached by inverting at a tangency point, which the general problem could not do because its circles were not touching. The centre of inversion is the one point a reader gets to choose, and choosing it well is the whole of the technique — the same choice that turns a ring of tangent circles into a ring of equal ones.

The seed, and what makes one #

Not every quadruple of whole numbers is the curvature set of four mutually tangent circles: the Descartes relation is a constraint, and it is a strong one.

Checking $(-1, 2, 2, 3)$: the sum is 6 and its square is 36; the sum of squares is $1 + 4 + 4 + 9 = 18$ and twice that is 36. It works. Checking $(-1, 2, 2, 4)$: the sum is 7, its square 49; twice the sum of squares is $2(1+4+4+16) = 50$. It does not, and no four circles have those curvatures.

So a seed is a solution of a quadratic form’s equation in whole numbers, and finding all of them is a question in the theory of quadratic forms rather than in geometry. Rewriting the relation as $2(k_1^2+k_2^2+k_3^2+k_4^2) - (k_1+k_2+k_3+k_4)^2 = 0$ makes it a quadratic form in four variables set equal to zero, which is where the arithmetic in the sections below comes from. The picture and the number theory are the same equation read two ways — the same coincidence that makes the whole-number right triangles a question about a conic.

What the integer curvatures mean #

An infinite family of circles all of whose curvatures are whole numbers is an arithmetic object, and the arithmetic turns out to be substantial.

Every gasket has a root quadruple — the four smallest curvatures, from which everything else is generated — and gaskets are classified by it. The one drawn here is $(-1, 2, 2, 3)$; others are $(-2, 3, 6, 7)$, $(-3, 4, 12, 13)$, and so on. Two gaskets with different root quadruples are genuinely different packings, not the same one drawn differently.

The question of which integers appear in a given gasket is much harder than it looks and was open until recently. Each gasket has a set of forbidden residues modulo 24, and the conjecture — now largely a theorem — is that every sufficiently large integer not in a forbidden class does appear. Proving it needed methods from analytic number theory rather than from geometry, which is a good indication of how far the arithmetic is from the picture.

There is also a group behind the packing. The four generating reflections — one for each circle, each replacing it by its Descartes partner — generate a group acting on quadruples of curvatures, and that group is an arithmetic subgroup of a matrix group. The gasket is its orbit. That is the modern reading, and it is why a question about touching circles ends up in the same file as questions about quadratic forms — and why a packing’s arithmetic is decided by which residues a form can represent rather than by anything visible in the drawing.

Counting the circles #

How fast does the packing grow, and how fast do the circles shrink? Both are worth a number, because they are what make the picture drawable at all.

Each generation replaces every triple of mutually tangent circles by three new triples, so the number of new circles grows by a factor of three at every generation. Four generations is a few hundred circles and eight would be tens of thousands; the drawing stops where the strokes stop being distinguishable rather than where the recursion stops being interesting.

The radii fall much faster than the count rises, which is why the total area converges. There is a clean statement of it: the number of circles in a gasket with curvature at most $x$ grows like $x^{\delta}$ where $\delta \approx 1.3057$ is the dimension of the residual set — so the exponent that governs the counting and the exponent that governs the geometry are the same number, which is the reason the constant is hard.

How much of the plane is left #

The circles fill the gaps and never finish, and what remains is worth measuring.

The residual set — the points of the disc lying on no circle of the packing — has area zero. So the circles account for all of the area, and yet the residual set is uncountable and dense in the gasket.

Its dimension is not zero and not one. It is about $1.3057$, established numerically and known not to be any nicer number; the packing is a fractal, and this is one of the few natural ones whose dimension is a genuinely hard constant rather than a ratio of logarithms. A set built by removing middles has a dimension one can write down; this one does not. The contrast with a shape that reproduces itself at a fixed scale is the point: a self-similar object’s dimension is a ratio of two logarithms, and the gasket is self-similar under a group rather than under a single map, so no such ratio is available.

What the whole numbers are not #

A caution, because the integrality invites an over-reading.

The curvatures being whole numbers is a property of the seed, not of gaskets. Start from four mutually tangent circles whose curvatures are not whole — which is the general case, since the Descartes relation has plenty of irrational solutions — and the packing generated is a perfectly good Apollonian gasket in which no curvature is an integer. Nothing about the geometry singles out the integral ones.

What singles them out is that they exist at all. A quadratic relation in four variables with a solution in whole numbers is not guaranteed one, and the fact that $(-1,2,2,3)$ works is an arithmetic accident of exactly the kind that starts a subject. Every question in the sections above — which integers appear, which residues are forbidden, what the group is — is a question about that accident and not about circles.

The pattern is one this collection meets often. A construction produces an object; the object turns out to satisfy an equation; the equation has a life of its own that the construction knew nothing about. Pythagorean triples parameterised by a circle are the standard example, and this is the same thing happening to a picture rather than to a formula.

What the pictures cannot show #

Nothing here is the gasket. Every figure draws a finite number of generations, and the object with the properties described is the limit of all of them. The integer curvatures are a fact about the limit and about every stage; the dimension is a fact about the limit alone.

The curvatures are checked and the integrality is argued. Each drawn circle’s curvature is asserted to be a whole number, at the sizes drawn. That every curvature is whole follows from the subtraction, which is prose.

The generations are not the packing’s own structure. A circle’s generation is a fact about the order the recursion reached it in, and two circles of the same curvature can arrive at different depths. The tinting and the depth parameter are about the drawing; the packing has no generations.

Small circles carry no labels. Beyond curvature fifteen the circles are too small to hold a number at a readable size, and the list of curvatures under the figure is the honest substitute.

Five generations is not many. At depth five the packing has a few hundred circles and the true object has infinitely many in every gap; the figure at that depth is chosen to be the largest that still resolves, not the largest that is interesting.

And the dimension is quoted. Nothing on this page measures it. It is a number from the literature, obtained by methods with nothing in common with the drawing, and it is here because a reader looking at the black gaps deserves to know they are not empty.

Where the ladder goes next #

Two inversions compose to a map that preserves orientation, and the group they generate is the group of Möbius transformations. The rung above is about that group and the quantity it preserves: the cross-ratio of four points, which is unchanged by every such map, and which is real exactly when the four points lie on a circle.

That invariant is what makes the next rung’s theorem sharp. A circle packing of a planar graph is unique — but only up to the group two inversions generate, which is exactly the freedom this ladder has been exploiting all along to move a configuration somewhere convenient.

What is worth carrying away #

When a relation is quadratic in a quantity, its two roots are related by their sum, and that is often more useful than either root.

Descartes’s theorem is a quadratic in the fourth curvature. Solving it needs a square root and produces a number that is generally irrational. Using the sum of the roots instead produces the other solution from the one already known by subtraction alone — and it is that step, not the theorem, that makes the gasket’s curvatures whole numbers. The second root of a quadratic is nearly always cheaper than the first, and a recursion that only ever needs the second is a recursion with no arithmetic in it.