Times Tables on a Circle
There are 240 points round the circle, numbered from 0 at the bottom, and each one is joined by a straight line to the point at twice its number. No curve is drawn anywhere, yet a heart shape appears where the lines crowd together: the cardioid. Change the multiplier and the heart becomes the two-lobed nephroid, then curves with 3, 4, 5 cusps and more. Every times table has a shape.
| 2 times table | cardioid, 1 cusp |
|---|---|
| 3 times table | nephroid, 2 cusps |
| m times table | epicycloid, m - 1 cusps |
| Points by default | 240 |
What you are looking at
Each point is joined to the point at its number times the multiplier, counting round the circle like a clock: past 239 the count starts again at 0, so point 150 doubled is 300, which lands on point 60. Short lines near the rim glow pale gold and lines that cut across the circle are violet.
The bright curve is not drawn. It is where the straight lines bunch up, each one just touching it, the way a curve appears in string art. Mathematicians call such a curve an envelope.
With Sweep on, the multiplier starts climbing once the lines are drawn, 0.07 every second, so the cardioid melts into the nephroid in about 14 seconds and keeps going. In between the whole numbers the multiplier is a fraction, and a line can end between two numbered points. Set a few points, such as 24, to follow the arithmetic line by line, or several hundred to see the curves sharply.
The math
Think of the circle as having radius 1 and of each point by its angle. The line from angle t to angle m times t, drawn for every t, always touches one curve: the path of a point on a small circle of radius 1/(m + 1) rolling round the outside of a fixed circle of radius (m - 1)/(m + 1). A curve made by one circle rolling round another is an epicycloid.
The rolling point stops dead, making a sharp point called a cusp, each time it touches the fixed circle, and that happens m - 1 times in one trip round. So the 2 times table gives one cusp, the cardioid, and the 3 times table two, the nephroid. The cusps sit (m - 1)/(m + 1) of the way out from the centre: the cardioid's cusp is a third of the way. The curve also touches the outer circle m - 1 times, between the cusps.
The cardioid and nephroid are close cousins of the curves made by circles turning on circles, the idea Ptolemy used to explain how the planets loop.
How it is computed
For each point i from 0 to N - 1 the piece multiplies i by the multiplier and removes as many whole turns of N as fit, which gives the place on the circle the line goes to. The lines are drawn in that order as the picture builds, grouped by length so each colour is drawn in one pass. The curve itself is never computed for the picture. Instead the tests check the mathematics: every line touches the epicycloid, the epicycloid has m - 1 cusps for every whole m from 2 to 12, and the cusps sit where the formula says. Everything runs in your browser, and the link in your address bar reproduces the picture you are looking at.
Questions
Why does a curve appear when only straight lines are drawn?
Each line touches the same curve at one point, and neighbouring lines touch it at neighbouring points. Where many lines cross close together they light up the curve they all share, which is called their envelope. With m times tables that envelope is an epicycloid.
What happens with a multiplier that is not a whole number?
The line from point i goes to i times m round the circle, which can fall between two numbered points. The pictures in between the whole numbers are where one curve melts into the next, which is what the sweep shows.
Why does it look tangled with big multipliers?
A curve with m - 1 cusps needs several points for every cusp to show cleanly. At 240 points and a multiplier of 99 there are fewer than three points per cusp, so the lines weave a pattern of their own instead. More points bring the curve back.