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A Tisket, a Tasket, an Apollonian Gasket

Fractals made of circles do funny things to mathematicians

Dana Mackenzie

Getting the Bends

Even the first step of drawing an Apollonian gasket is far from straightforward. Given three circles, how do you draw a fourth circle that is exactly tangent to all three?

Apparently the first mathematician to seriously consider this question was Apollonius of Perga, a Greek geometer who lived in the third century B.C. He has been somewhat overshadowed by his predecessor Euclid, in part because most of his books have been lost. However, Apollonius’s surviving book Conic Sections was the first to systematically study ellipses, hyperbolas and parabolas—curves that have remained central to mathematics ever since.

One of Apollonius’s lost manuscripts was called Tangencies. According to later commentators, Apollonius apparently solved the problem of drawing circles that are simultaneously tangent to three lines, or two lines and a circle, or two circles and a line, or three circles. The hardest case of all was the case where the three circles are tangent.

No one knows, of course, what Apollonius’ solution was, or whether it was correct. After many of the writings of the ancient Greeks became available again to European scholars of the Renaissance, the unsolved “problem of Apollonius” became a great challenge. In 1643, in a letter to Princess Elizabeth of Bohemia, the French philosopher and mathematician René Descartes correctly stated (but incorrectly proved) a beautiful formula concerning the radii of four mutually touching circles. If the radii are r, s, t and u, then Descartes’s formula looks like this:

1/r2+1/s2+1/t2+1/u2= 1/2 (1/r+1/s+1/t+1/u)2.

All of these reciprocals look a little bit extravagant, so the formula is usually simplified by writing it in terms of the curvatures or the bends of the circles. The curvature is simply defined as the reciprocal of the radius. Thus, if the curvatures are denoted by a, b, c and d, then Descartes’s formula reads as follows:


Formula relating any four mutually tangent circle radiiClick to Enlarge ImageAs the third figure shows, Descartes’s formula greatly simplifies the task of finding the size of the fourth circle, assuming the sizes of the first three are known. It is much less obvious that the very same equation can be used to compute the location of the fourth circle as well, and thus completely solve the drawing problem. This fact was discovered in the late 1990s by Allan Wilks and Colin Mallows of AT&T Labs, and Wilks used it to write a very efficient computer program for plotting Apollonian gaskets. One such plot went on his office door and eventually got made into the aforementioned T-shirt.

Descartes himself could not have discovered this procedure, because it involves treating the coordinates of the circle centers as complex numbers. Imaginary and complex numbers were not widely accepted by mathematicians until a century and a half after Descartes died.

In spite of its relative simplicity, Descartes’s formula has never become widely known, even among mathematicians. Thus, it has been rediscovered over and over through the years. In Japan, during the Edo period, a delightful tradition arose of posting beautiful mathematics problems on tablets that were hung in Buddhist or Shinto temples, perhaps as an offering to the gods. One of these “Japanese temple problems,” or sangaku, is to find the radius of a circle that just touches two circles and a line, which are themselves mutually tangent. This is a restricted version of the Apollonian problem, where one circle has infinite radius (or zero bend). The anonymous author shows that, in this case, √a–+√b–=√c–, a sort of demented version of the Pythagorean theorem. This formula, by the way, explains the pattern I saw in the screensaver. If the first two circles have bends 1 and 1, then the circle between them will have bend 4, because √1–+√1–=√4–. The next circle will have bend 9, because √1–+√4–=√9–. Needless to say, the pattern continues forever. (This also explains what the numbers in the first figure mean. Each circle is labeled with its own bend.)

Apollonian circles experienced perhaps their most glorious rediscovery in 1936, when the Nobel laureate (in chemistry, not mathematics) Frederick Soddy became mesmerized by their charm. He published in Nature a poetic version of Descartes’ theorem, which he called “The Kiss Precise”:

Four circles to the kissing come
The smaller are the benter.
The bend is just the inverse of
The distance from the center.
Though their intrigue left Euclid dumb,
There’s now no need for rule of thumb.
Since zero bend’s a dead straight line,
And concave bends have minus sign,
The sum of the squares of all four bends
Is half the square of their sum.

Soddy went on to state a version for three-dimensional spheres (which he was also not the first to discover) in the final stanza of his poem.

Ever since Soddy’s prosodic effort, it has become something of a tradition to publish any extension of his theorem in poetic form as well. The following year, Thorold Gosset published an n-dimensional version, also in Nature. In 2002, when Wilks, Mallows and Jeff Lagarias published a long article in the American Mathematical Monthly, they ended it with a continuation of Soddy’s poem entitled “The Complex Kiss Precise”:

Yet more is true: if all four discs
Are sited in the complex plane,
Then centers over radii
Obey the self-same rule again.

(The authors note that the poem is to be pronounced in the Queen’s English.)

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