Apollonian circle packings come from the Apollonius Problem:
Given three objects, draw a circle mutually tangent to all three.
When the objects are circles themselves, and are already mutually tangent, there are exactly two solutions to this problem (an inner and outer solution), and these are called Soddy circles1
It is very easy to compute the two Soddy circles. If we denote bi = 1/ri as the bend of a circle of radius ri, then
2(b12 + b22 + b32 + b42) = (b1 + b2 + b3 + b4)2
This is known as the Descartes Circle Theorem, and the four bends together are called a Descarte quadruple. What is intriguing is that if we wish to determine the circle centers, we can do so as well. If the centers are represented as complex numbers zj, then Colin Mallows has shown that the numbers bi * zi also form a Descarte quadruple. An important note is that bends can be negative: if three mutually tangent circles are touched by a circle that includes all of them, its bend has a negative sign (you can think of this on the sphere to see why: the "sense" of the circle is inverted)
Now take four mutually tangent circles, and pick one. For the other three, find the "second" mutually tangent circle. Add that in and repeat. This yields a packing of the plane, called the Apollonian Gasket. Depending on the starting set of four circles, one can obtain many different kinds of gaskets.
Now comes the stunning part2. A set of two simple inversion operations allows us to move from one Descarte configuration to another. One operation we have already seen: if we consider the two solutions to the 3-circle Apollonian problem, they invert to each other with respect to a fixed circle. This gives us one new circle for a fixed set of three.
The other inversion operator works as follows. Take one of the Soddy circles, and invert its three touching circles into it. This yields another 4-circle configuration inside the Soddy circle.
There are a total of eight such operations (4 for each kind of inversion). Together, these operations generate a special group called a Coxeter group with appropriate syzygies. Among the properties of these operations are:
- Each pair of circles thus generated are tangent or disjoint
- If the starting configuration is (strongly) integral, so are all resulting configurations.
- The resulting set of configurations contains all strongly integral Apollonian packings
That last statement is what for me is so amazing. Essentially there is a single universal packing that contains all Apollonian packings (in the integral case). This is a truly mysterious structure.
Also, there are beguiling symmetries if we look at outer circle configurations (configurations where there is one outer circle and three inner circle). The mod 2 bend groups are symmetric with different axes of symmetry, and all outer circles having the same bends form symmetric patterns in the supergasket.
1. Apollonian circle problems were discussed in Sangaku (Japanese Temple Geometry) that I had talked about earlier. More on Sangaku here (via GJ).
2. When Allan was giving this talk, he said 'beautiful' so many times he had to restrain himself; now as I try to describe this in words, I understand his problem :)
This is based on work by Jeff Lagarias, Ron Graham, Colin Mallows, Allan Wilks and Catherine Yan.
Papers: (at www.arxiv.org)