The number in each circle is its curvature: one over its radius, the outer circle's radius being one over the first number. The address of the page keeps the gasket, to share it.
Four kissing circles
Around 200 BC, Apollonius of Perga asked: given three circles, which circles touch all three? There are eight answers in general. When the three circles already touch each other, two of the answers are circles nestled in the gaps between them, and each of those makes new, smaller gaps. Fill every gap, forever, and you get an Apollonian gasket: a fractal of circles that leaves no room unfilled, but whose circles add up to no area at all.
In 1643 René Descartes found the rule that sizes the fourth circle. Write each circle's curvature, one over its radius (negative for a circle that holds the others): for four circles that touch each other, the square of their sum is twice the sum of their squares. In 1936 the chemist Frederick Soddy rediscovered it and published it as a poem in Nature, "The Kiss Precise".
The rule has a surprise. Start with four curvatures that are whole numbers, such as −1, 2, 2 and 3, and every circle in the gasket, down to the smallest, has a whole-number curvature too. Which numbers appear, and how often, is still studied by number theorists today. Tick Show the curvatures to read them.