What you are looking at
Mark n points evenly around a circle and number them 0, 1, 2, and so on. Now pick a multiplier k. For every point i, draw a straight line from i to point ik — wrapping around the circle when you run past n, the way a clock wraps past twelve. That is the entire recipe. It is the two times table, or the seventeen times table, done in thread instead of chalk.
The lines are dead straight and there are only a few hundred of them, but they crowd together along certain curves and thin out elsewhere. Your eye reads the crowding as an edge. That edge is the envelope of the family of chords: the curve every chord happens to be tangent to. Nobody drew it. It is the negative space left over by a few hundred straight lines that all had somewhere else to be.
For k = 2 the envelope is a cardioid, the heart-shaped curve traced by a point on a circle rolling around another circle of the same size. For k = 3 you get a nephroid, the kidney with two cusps. In general the times-k table draws an epicycloid with k − 1 cusps, so you can read the multiplier straight off the picture by counting the sharp points and adding one.
The cardioid has an easy proof once you see the trick. Point i sits at angle θ, point 2i sits at angle 2θ, and the chord between them is a chord of the unit circle subtending those two angles. Chords of that family are exactly the tangent lines of the cardioid, which is why the curve appears without ever being plotted.
Try this
- Set the multiplier to 9 — this site's namesake — and count the cusps. There should be eight.
- Leave the multiplier at 2 and walk the fine adjustment slider all the way up to +1. The single cusp does not simply become two; it splits, and for most of the journey the figure is neither curve.
- Park the fine adjustment just off a whole number — 25 with +0.020, say. Whole multipliers give still figures; fractions give the same figure slowly rotating, because each step now falls slightly short of closing the loop.
- Turn the point count down to 60. The envelope survives at absurdly low resolution — twenty chords are already enough to suggest a cardioid.
- Try 21, 51, 99. The readout counts the chords that come out with no length at all, which is gcd(k − 1, n); when that number is large the figure is conspicuously more symmetrical.
- Tick Sweep the multiplier and press Play. Watch how many times the figure passes through an almost-empty state on its way between whole numbers.
Notes
This construction is old and keeps being rediscovered. It is the same idea as the curve-stitching Mary Everest Boole devised around the turn of the twentieth century to teach children about tangents with nothing but card and thread, and the same idea as the parabola every schoolchild has accidentally sewn between two rulers. The modular version — where the wrap-around does the work — reached a wide audience through Burkard Polster's Mathologer video on times tables and cardioids in 2015.
What makes it worth keeping around is the honesty of it. Nothing here is approximated. There is no curve-fitting, no smoothing, no shader. There are n line segments with integer endpoints, and the beautiful part is the part that was never drawn.
Further reading
Mary Everest Boole, Philosophy and Fun of Algebra (1909), for the spirit of the thing. For the geometry of epicycloids and their tangent families, J. Dennis Lawrence's A Catalog of Special Plane Curves remains the standard shelf reference.