The Cardioid
One curve, three constructions. The cardioid appears as a rolling circle’s trace (epicycloid), as the envelope of a modular multiplication circle (times table), and as the main bulb boundary of the Mandelbrot set. The same heart-shaped curve keeps showing up in unrelated mathematics.
What’s happening
The epicycloid construction
A cardioid is the trace of a point on the circumference of a circle that rolls around the outside of a fixed circle of equal radius. The word comes from the Greek kardia (heart). When the two radii are equal, you get a cardioid. Other ratios produce different epicycloids — a 2:1 ratio gives a nephroid (kidney), and 3:1 gives a three-cusped epicycloid.
The times table construction
Distribute N points equally around a circle, labeled 0 to N−1. For each point i, draw a chord to point (i × m) mod N, where m is the multiplier. When m = 2, the envelope of all these chords forms a cardioid. Other values of m produce nephroids (m = 3), astroids, and other beautiful curves. This construction, popularized by Mathologer, connects modular arithmetic to curve geometry.
The Mandelbrot connection
The main cardioid of the Mandelbrot set is the set of complex numbers c for which the iteration z → z² + c has a stable fixed point. Its boundary is parametrized by c = (eiθ/2)(1 − eiθ/2), which is precisely a cardioid centered at (1/4, 0) with cusp at the origin. This is not coincidence — it falls directly out of the fixed-point stability condition |2z*| = 1.
Polar equation
r(θ) = a(1 + cos θ) (horizontal cardioid) Parametric: x = a(2cosθ - cos2θ), y = a(2sinθ - sin2θ) Area = 3πa²/2 = 6πa² (for rolling circle version where a is the rolling radius)
The cardioid has exactly one cusp where the generating point touches the fixed circle, and its area is exactly 6π times the area of the rolling circle. It appears in acoustics (microphone polar patterns), antenna theory (radiation patterns), and optics (caustic curves visible in coffee cups when light reflects off the inner wall).