Times tables on a circle
Put n points around a circle, numbered 0 to n−1 like a giant clock. Draw a line from each point k to the point k × m, wrapping around the clock. With n = 10 and m = 2: 1 goes to 2, 3 goes to 6, and 7 goes to 14, which is 4 on a 10-hour clock. Drag n down to 10 to see those lines, then back up. Then press Animate.
What's going on
"Wrapping around the clock" is clock arithmetic: the line from k goes to (k × m) mod n. With m = 2 the lines wrap into a heart shape called a cardioid. Light reflecting inside a coffee cup draws curves like these: a cardioid when the lamp is at the rim, and the m = 3 shape when it is far away. With m = 3 you get two lobes, with m = 4 three lobes, and so on.
Multiplication by m = n − 1 is the same as multiplication by −1: it sends k to −k on the clock. With n = 100 and m = 99, each endpoint is reflected across the vertical diameter, so the lines go horizontally across the circle. Try m = 51 with n = 100: even-numbered points stay put, while odd-numbered points connect to the point directly opposite them, making 25 diameters.
The slider allows fractional m so the animation flows, but the real pattern lives at the whole numbers.
Sheet: Clock arithmetic.