Points round a circle, each joined to the point its number is multiplied to. The straight chords bend into a cardioid at times two, a nephroid at times three, and ever stranger curves between.
Modular multiplication on a circle · Burkard Polster, 2015
Put two hundred points round a circle and number them. Join point 1 to point 2, point 2 to point 4, point 3 to point 6: every point to its double, counting past the last point back round to the first. The chords are all straight, yet together they trace a heart, the cardioid. Times three gives a kidney, the nephroid, times four a curve with three lobes, and in general times m gives m − 1 of them.
Burkard Polster's 2015 Mathologer film showed how far this goes, down to the main heart of the Mandelbrot set. Here the multiplier can sit anywhere between whole numbers, where the curves break apart and re-form, and echoes draw the table again at nearby multipliers in fainter ink, so the lines pile up into soft clouds. In motion the multiplier drifts and the figure turns into its neighbours.
Behind the mathematics
Times tables on a circle, Mathologer, 2015
German-Australian mathematician and the presenter of Mathologer, whose 2015 film on times tables drawn on a circle showed cardioids and Mandelbrot's heart appearing out of school multiplication.