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Circle Game

Science News, April 21, 2001 by Ivars Peterson

Packing circles within a circle turns a mathematical surprise

The geometric realm of circles seems an unlikely setting for a startling mathematical discovery. A staple of geometry textbooks, the circle is already the subject of myriad theorems and countless exercises.

Nonetheless, researchers motivated by curiosity about a striking pattern of smaller circles packed within a larger circle have now found an extraordinary new formula that mathematicians had previously overlooked. The formula, which has to do with the curvatures of touching circles, has also yielded elegant generalizations that encompass spheres and analogs of circles in higher dimensions.

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"No one had noticed it before," says statistician Allan R. Wilks of AT&T Laboratories in Florham Park, N.J., who initiated the research. "It's amazing to find something new in something so old and so classical."

The route to the discovery began with a high school geometry problem. In 1998, during a lull in a conference in Germany, Wilks started talking with a colleague who was pondering his daughter's homework assignment. The question concerned a pattern made up of two identical circles that fit side by side inside a third circle. The assignment was to find the radius of a fourth circle nestled between the outer circle and the two inner circles.

After Wilks had returned to the United States, he found the three-circle pattern still on his mind. In drawing it, he could easily insert smaller circles into the empty spaces among the larger circles. Each new circle would be as large as possible without overlapping the circles already present.

Wilks wondered about the relative sizes of the touching circles. He found his answer in a geometry textbook. In 1643, French mathematician Rene Descartes had developed a formula relating the curvatures of four circles that all touch, or are tangent to, each other.

He defined the curvature of a circle as the reciprocal of its radius. Hence, if the radius of a circle is one-fifth (1/5) that of another, its curvature is 5 times that of the larger circle.

Given four mutually tangent circles with curvatures a, b, c, and d, the Descartes circle equation specifies that ([a.sup.2] + [b2.sup.] + [c.sup.2] + [d.sup.2]) = 1/2 [(a + b+ c + d).sup.2].

The same formula holds for three touching circles nested within a fourth circle, which is the configuration that Wilks was examining. In this case, however, the curvature of the outer circle would be a negative number, because the other circles touch it from the inside rather than the outside. The formula also applies to configurations in which one or two of the touching circles are replaced by straight lines. A line counts as a circle with zero curvature.

Using the Descartes circle equation, Wilks could readily calculate the curvature and radius of each new circle he placed in his drawing. For example, if the initial, outer circle has a radius of 1, its curvature relative to circles inside it would be -1. Two smaller circles of the same size drawn inside the initial circle would each have a radius of 1/2 and, hence, a curvature of 2. The next largest circles that would fit snugly in the remaining space between circles would each have a radius of 1/3 and a curvature of 3.

Once an initial configuration of three circles is set, the Descartes circle equation specifies the size of every smaller circle that fits into the pattern. Moreover, if the curvatures of the three initial circles are integers, the curvature of every smaller circle is also an integer.

Wilks decided to use a computer to plot a large diagram of the circle pattern, labeling each circle with its curvature. "It makes a pleasing picture," Wilks says. "It's symmetric, and the numbers get bigger as the circles get smaller."

For convenience, he decided to put the origin of his plot at the center of the outer circle and to orient the x axis so that it passes through the centers of the two inner circles. It was a fortuitous choice. When Wilks checked the coordinates of the centers of the circles he was plotting, he noticed that the pairs were all rational numbers (fractions). To his surprise, he found that multiplying each coordinate of a circle's center by the circle's curvature always produced an integer. In other words, if a circle has curvature b and its center at point (x, y), bx and by are integers.

Intrigued by the result, Wilks showed his work to his coworker Colin L. Mallows, also a statistician at AT&T Labs. Mallows came up with a mathematical proof of the relationship that Wilks had found. He then generalized the result to produce a formula that relates the curvatures and coordinates of the centers of four mutually tangent circles--just as Descartes had done for curvatures alone.

There had been no clue at all that such a should exist. "It was completely unexpected," says mathematician Ronald L. Graham of the University of California, San Diego.

The new formula looks like the original Descartes equation for four mutually tangent circles, provided the coordinates of the centers are expressed as so-called complex numbers.

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Circle Game

Science News, April 21, 2001 by Ivars Peterson

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