The Circles of Descartes

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In a circle, the curvature (or bend) is the reciprocal of the radius, . (When measured from the inside, the curvature is negative.) Four mutually tangent circles have the property . This is the Descartes circle theorem. With the centers in the complex plane (), the following formula holds:



Using both formulas, a vast Apollonian packing can be found. This Demonstration focuses on initial circle sets with integer curvature. Hover over an empty circle to see its curvature. Various wonderful results can occur:

1. If the initial curvatures are integers, so are all the curvatures.

2. If the first circle is a line, the configuration is the set of Ford circles.

3. With a large interior circle, the configuration is a Pappus chain.

4. Modulus 12, only 4 values can occur. Circles with a similar value get the same color.

Frederick Soddy, who won a Nobel prize for his study of isotopes, is better known for his research on circles and a poem he published in Nature in 1936.

The Kiss Precise by Frederick Soddy For pairs of lips to kiss maybe Involves no trigonometry. 'Tis not so when four circles kiss Each one the other three. To bring this off the four must be As three in one or one in three. If one in three, beyond a doubt Each gets three kisses from without. If three in one, then is that one Thrice kissed internally.


Contributed by: Ed Pegg Jr (December 2008)
Open content licensed under CC BY-NC-SA




[1] J. C. Lagarias, C. L. Mallows, A. R. Wilks, "Beyond the Descartes Circle Theorem,", Jan. 9, 2001.

[2] D. Austin, "When Kissing Involves Trigonometry,", March 2006.

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