Sketch · 2026-07-20

Modular times tables on a circle.

Space N points around a circle, numbered 0 to N minus one. For every point i, draw a straight chord to point k times i, wrapped modulo N. One rule, two integers. At k equals 2 the chords bunch into a cardioid. At k equals 3 they build a nephroid. Every whole k above that adds another cusp. Between the whole numbers, the envelope drifts.

N points 300

Multiplier k 2.00

Chords drawn 300

N — points on the circle 300

k — multiplier 2.00

Why the cardioid shows up

The k equals 2 pattern is the caustic you see on the inside of a coffee cup on a sunny morning. Parallel rays reflect off the wall and bunch along a curve — that curve is a cardioid. The chord construction here is the discrete twin: each chord is tangent to the same envelope. Modular arithmetic gives you evenly spaced points on the circle, the multiplier k picks the mapping, and the envelope emerges without anyone drawing it.

Higher k, more cusps

At k equals 3 you get a nephroid, the two-cusped kidney. Each integer beyond that adds a cusp: k equals 4 gives three cusps, k equals 5 gives four, in general the count is k minus one. These are epicycloids traced by a small circle rolling inside a larger one. Between integers, the envelope isn't a named curve; it drifts smoothly through shapes that only exist because you're averaging over N points.

Primes, near-primes and stars

When k is close to a rational p over q with small q, the chords fall into q-fold star patterns. At k equals 51 and N equals 300, the ratio 51 over 300 reduces to 17 over 100, and a seventeen-fold star drops out. When k has no small rational neighbour, the pattern looks like tangled lace — no obvious symmetry, but still confined to a single caustic. Sliding k slowly walks you through this alternation between order and mess.

Sources

Mathologer (2015). Times Tables, Mandelbrot and the Heart of Mathematics. YouTube. The video that put the k=2 cardioid in front of a general audience.
Simmons, S. (1974). Cardioids and Modulo Multiplication on a Circle. Mathematics Magazine, 47(3), 147–148. An early write-up of the construction.
Lawrence, J. D. (1972). A Catalog of Special Plane Curves. Dover. Reference for the epicycloid family.