©2021 Compass Learning Technologies → GXWeb Apollonian Gaskets → GXWeb Jigsaws and Quizzes → GXWeb Kissing Circles Quiz

The Kiss Precise For pairs of lips to kiss maybe Involves no trigonometry. 'Tis not so when four circles kiss Each one the other three. 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. Frederick Soddy (1936) 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. Wilks, Mallows and Laguna (2001)

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Some two thousand years after Apollonius of Perga played with gaskets, Rene Descartes developed his Kissing Circles Theorem: if four circles have curvature a, b, c and x - and hence radii $\frac{1}{a}, \frac{1}{b}, \frac{1}{c}$ and $\frac{1}{x}$ - then

$$(a+b+c+x)^2$$ $$= 2\cdot (a^2 + b^2 + c^2 + x^2)$$

Solving for x gives two solutions for the curvature of the fourth circle:

$$x = a + b + c$$ $$\pm$$ $$2\cdot\sqrt{(a\cdot b + a\cdot c + b\cdot c)}$$

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.

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It took the application of complex numbers to the problem (which were not recognised by mathematicians until some 150 years after Descartes) to compute the centres of all those circles! Perhaps most amazingly, it turned out to be Descartes Kissing Circle Theorem once again which lay at the heart of this solution! In 2001, when Allan Wilks, Colin 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 (cited above).

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©2021 Compass Learning Technologies ← GXWeb Apollonian Gaskets ← GXWeb Jigsaws and Quizzes ← GXWeb Kissing Circles Quiz