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Indra's Net, Drawn

Chapter 3The math: mirrors in mirrors

You can't build a net of jewels that goes on forever. But you can put a handful of round mirrors on a table and do the arithmetic of what shows up in 'em. That arithmetic draws the prettiest pictures on this site, and mathematicians named it for the net26.

push the size to 1 and the gaps close into lace

A flat mirror, then a round one

Stand in front of a flat mirror and your reflection is as far behind the glass as you are in front of it. Easy. A round mirror — here, a circle you reflect through, the way mathematicians do it — flips near and far instead. Somethin' right up against the rim stays put. Somethin' far outside lands close to the middle. Somethin' way off yonder lands right in the center28.

dnew = R² ÷ d

Readin' it: take a point d away from the center of a round mirror of radius R. Its reflection sits on the same line out from the center, but at R² ÷ d. Twice as far out as the rim? Reflection's halfway in. Ten times out? A tenth of the way in.

Do that to every point on a circle and — here's the sweet part — you get another circle, only smaller. So circles in round mirrors stay circles. For a circle of radius r whose center sits d from the mirror's center:

s = R² ÷ (d² − r²)    dnew = s·d    rnew = s·r

Readin' it: work out one squeeze factor s, then shrink both the distance and the radius by it. A circle of radius 0.5 centered 2 out, in a mirror of radius 1: s = 1 ÷ (4 − 0.25) = 0.267, so the reflection sits 0.5333 out with radius 0.1333. That's the whole engine. Everything below is that, over and over.

Now put four of 'em in a ring

Look at mirror one in mirror two, and you see a little circle. Look at that in mirror three, a littler one. Keep goin'. The one rule is: don't bounce straight back into the mirror you just came out of, 'cause that undoes the bounce and you're back where you started.

circles on bounce n = k × (k − 1)n

Readin' it: with k mirrors, each circle has k − 1 fresh mirrors to show up in. Four mirrors: 4, then 12, then 36, 108, 324 — by bounce ten, 236,196. They pile up that fast and they don't overlap, so they have to get smaller and crowd in toward the edges. That crowdin' is the lace.

Three mirrors just touchin'. The gaps fill with smaller and smaller circles that also touch — near kin to the Apollonian gasket. · computed here · Nan · hongdam.net · CC BY 4.0
Six mirrors with a little air between 'em. Each bounce is a color; the dust of circles crowds up against a ring it never quite reaches. · computed here · Nan · hongdam.net · CC BY 4.0

These still pictures draw every circle bigger than a speck: 2,409 and 2,856 of 'em. The live one up top does the same arithmetic in your browser29.

What the jewels are, in this math

Each round mirror is a jewel. Each little circle is one jewel seen in another, seen in another. The dust they pile up against — mathematicians call it the limit set — is the part of the net you'd only reach after bouncin' forever. Felix Klein and his students drew the first of these by hand in the 1890s27; it took computers to see how fine they go.

Here's a thing to try: set the size to 1 so the mirrors touch, and watch the gaps fill in with circles that also touch. Then back it off to 0.8 and the lace comes apart into dust. Same rule, a hair's difference in the setup, a different world — which is the whole of chaos in one slider.