Circles: Answers

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From "The Mathematics of Oz" by Clifford A. Pickover

We may define the curvature of a circle as the reciprocal of the circle's radius. Hence, if a circle is one-half the size (here, the outermost) circle, its curvature is twice that of the larger circle. This corresponds to the two circles labelled with a "2" in the picture. The next two smaller circles that fit in the remaining space between the "2" circles each have a radius of 1/3 (compared with the outermost circle) and a curvature of 3. Once we have the values for the three largest circles, we can compute the radii of the others using a formula by philosopher and mathematician René Descartes (1596 - 1650).

Given four mutually tangent circles with curvatures a, b, c, and d, the Cartesian circle equation specifies that:

(a² + b² + c² + d²) = 1/2 (a + b + c + d)²

(The outside circle in the figure is given a curvature of -1 relative to the inside circles; the negative sign indicates the other circles touch the large from the inside rather than the outside.)

If the curvatures of the three initial circles are integers (the largest bounding circle and two inner circles that fit side by side, in this case the two "2" circles), the curvature of every smaller circle is also an integer.

In 2001, mathematician Allan R. Wilks of AT&T Laboratories discovered that the centres (x, y). This circle research is an excellent example of how scientists sometimes start with a graph or visualisation and then discover interesting results when trying to understand the patterns.

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For more information, go to Circle game.