The lab

Pappus Chain

Take a diameter and divide it at a point, forming two smaller semicircles inside a large one. The crescent-shaped region between them is the arbelos — the shoemaker's knife of antiquity. Fill it with a chain of mutually tangent circles. Each circle in the chain has a remarkable property: its center lies exactly n times its own diameter above the baseline, where n is its position in the chain. Drag the dividing point to reshape the arbelos and watch the chain reconfigure.

Arbelos · Pappus of Alexandria, c. 300 AD · Descartes circle theorem · proved by inversion

t = 0.40

R = —

r1 = —

r2 = —

circles = 0

drag baseline divider ←→

split point t 0.40

chain length 8

n	radius r_n	center height y_n	y_n / (2r_n)

Pappus of Alexandria recorded this theorem around 300 AD, crediting still older sources. The arbelos — from the Greek for shoemaker's knife, the shape a cobbler's blade makes — is the region between three mutually tangent semicircles all sharing the same diameter line. Pack circles into it and something unexpected happens: the n-th circle's center is exactly n times its own diameter above the baseline. Always. No matter how you divide the diameter.

The classical proof is algebraic and laborious. Pappus probably had one. Descartes independently proved it (and discovered his circle theorem along the way). But the modern proof by inversion reduces the whole thing to a triviality: invert the figure through a circle centered at the left endpoint of the diameter. Under inversion, the three bounding semicircles become two parallel lines and a circle (or three parallel lines, depending on your inversion center). The pappus chain circles, all mutually tangent to the boundaries, transform into equal circles stacked between the two parallel lines. Their centers lie on a straight line. Inverting back, the elegant theorem about heights follows immediately from this simple transformed picture.

The inversion proof is one of those moments in mathematics where the right change of coordinates makes the problem dissolve. The theorem doesn't become easier to understand; it becomes a different theorem, in a different geometry, where it is obvious. Then you invert back. I find this philosophically interesting: the difficulty was never in the mathematical content. It was in the coordinate system.

Toggle "Show inversion" to see the transformed picture: the arbelos flattens into a strip, and the Pappus chain becomes equally-spaced circles between parallel lines.

Drag the point on the baseline (or use the slider) to vary the split ratio t. The theorem holds for all t ∈ (0,1): the ratio y_n / (2r_n) = n, always.