In 1643’s letter to Princess Elisabeth, Descartes dealt with four tangent circles two by two. Call the curvature of a circle the inverse of the radius, $k = \frac{1}{r}$. The theorem says that the curvatures satisfy

$$(k_1 + k_2 + k_3 + k_4)^2 = 2\left(k_1^2 + k_2^2 + k_3^2 + k_4^2\right).$$

Application: three equal coins of radius 1 touch each other two by two. What is the radius of the circle that fits in the hole between them? With $k_1 = k_2 = k_3 = 1$ and $k_4 = k$:

$$\begin{aligned} (3 + k)^2 &= 2(3 + k^2) \\ 9 + 6k + k^2 &= 6 + 2k^2 \\ k^2 - 6k - 3 &= 0 \\ k &= 3 \pm 2\sqrt{3}. \end{aligned}$$
  • With the sign $+$: $k = 3 + 2\sqrt{3} \approx 6{,}46$, that is, radius $\approx 0{,}155$. It’s the small circle in the middle.
  • With the sign $-$: $k = 3 - 2\sqrt{3} \approx -0{,}46$. Negative curvature indicates a circle that involves the other three, with radius $\approx 2{,}155$.
Three coins, the middle circle and the one that surrounds everything. Applying Descartes’ theorem again to each hole, Apollonius’ gasket appears.

Check the geometry: the centers of the coins form a equilateral triangle side 2, whose center is $\frac{2}{\sqrt{3}} \approx 1{,}155$ from each vertex. The middle circle has radius $1{,}155 - 1 = 0{,}155$, and what surrounds everything has radius $1{,}155 + 1 = 2{,}155$. The bills add up.

All topics by René Descartes · ← The angles of polyhedra and Euler’s formula · Why does the rainbow appear at 42° →

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