Modular Mandala

A circle with N equally spaced points. For each point i, a single line is drawn from i to the point at position (i times the multiplier) modulo N. As the multiplier evolves continuously, the entire pattern morphs through famous classical curves: a heart-shape (cardioid) at 2, a two-lobed nephroid at 3, four-lobed trefoil at 4, and infinite logarithmic spirals near the golden ratio.

Modular arithmetic is clock arithmetic. On a 12-hour clock, 9 plus 5 wraps around to 2. The same idea applies to multiplication: 9 times 5 mod 12 equals 45 mod 12, which is 9. If you place 12 points around a circle and connect i to (i times 5) mod 12, you get a specific star pattern. With more points, the discrete chords blur into smooth curved envelopes.

When the multiplier is 2, point i connects to point 2i mod N. As i sweeps around the circle, the chord from i to 2i traces all the tangent lines of a single curve, and that curve happens to be a cardioid: the heart-shape produced when a circle rolls around the outside of another circle of equal size, tracing a point on its edge. The connection between modular arithmetic and rolling circles is one of the most beautiful surprises in elementary mathematics.

Multiplier 3 produces a nephroid: the kidney-shape traced by a circle rolling around another circle twice its size. More generally, multiplier k produces an epicycloid with (k minus 1) cusps. Multiplier 4 gives a three-cusped trefoil, multiplier 5 a four-cusped curve, and so on. Between integer values you see the curves morph smoothly through complex transient envelopes.

Near phi (1.618...) and at multipliers like 1+phi, 2+phi, the lines do not converge to any closed envelope because phi cannot be approximated well by any rational number. Instead they trace logarithmic spirals, the same curve that appears in nautilus shells, sunflower seed heads, and galaxy arms. This is the same mathematical principle that makes the golden angle the optimal packing angle in phyllotaxis.

An envelope is a curve that is tangent to every member of a family of curves. The lines you see in this mandala are not the cardioid itself; the cardioid is the curve they all touch from the outside. Your eye constructs the envelope from the lines, which is why moving lines feel like they are tracing a hidden shape. Envelopes are central to optics: the bright caustic on the bottom of a coffee cup is an envelope of reflected light rays.

The mathematics of envelopes and epicycloids goes back to ancient Greek astronomers and was developed in detail from Roemer and Bernoulli onward. The specific connection between modular multiplication on a circle and these classical curves was popularized by Australian mathematician Burkard Polster (Mathologer on YouTube) in his 2015 video Times Tables, Mandelbrot and the Heart of Mathematics. The visualization spread widely from there into generative art and recreational math.

It combines two things the brain finds deeply pleasant. First, it is highly regular without being repetitive: every line follows the same rule, but the visual outcome constantly transforms. Second, the additive line overlap creates soft luminance gradients the eye reads as natural light. Together they pull attention into pure form, away from internal narrative. Many people find watching it has a similar effect to watching fire, water, or clouds.

Yes. String art on a circular pegboard uses essentially this construction: pegs are equally spaced points, and a thread connecting them in modular order produces the same curves. Spirograph traces hypotrochoids and epitrochoids, close relatives of the curves shown here (the cardioid and nephroid are special epicycloids). All three are different ways of drawing curves through repeated arithmetic on a circle.

Sacred geometry is the family of geometric forms that have appeared across cultures as religious or contemplative symbols: the Flower of Life, mandalas, the Sri Yantra. They share clean ratios, symmetry, and recursive construction. The modular mandala is a literal mathematical mandala: full radial symmetry, recursive structure, and the same hypnotic quality. It is what sacred geometry looks like when constructed by the cleanest possible arithmetic rule.

Watch the multiplier evolve continuously, or drag the slider to scrub manually. Click Cardioid to snap to 2, Nephroid for 3, Trefoil for 4, or Phi for the golden ratio. Press the play/pause button to freeze or resume animation. Drag the canvas to rotate. Open the info panel for the full explanation.