The ratio is the form

Place N points on a ring and draw a chord from every point k to the point m·k, wrapping past N. That is the entire rule — it is the times table of m, plotted. The envelope that appears at m = 2 is a cardioid, the same curve sunlight draws on the bottom of a cup, because reflection off a circle performs the identical doubling. Every multiplication table is a different mandala, and the fractional values in between morph one into the next.

Two pendulums, one driving x and the other y, both slowly dying of friction. When their frequencies form a musical interval, the trace is a closed figure that precesses as it decays. This is the honest version of what cymatics videos reach for: a perfect fifth genuinely is 3:2, and this is what 3:2 looks like when friction is allowed to tell the truth about time. Victorians kept these machines in their parlors and called the figures harmonograms.

Start with three circles, each touching the other two, all inside a fourth. Descartes sent Princess Elisabeth of Bohemia the relation their curvatures must satisfy; from it, every gap between three tangent circles admits exactly one more circle. Fill the gaps forever. For this seed the curvatures are not approximately whole numbers — they are exactly the integers printed on them, all the way down.

Seven-sided tiles cannot tile a flat plane — three of them crowd a corner. Hyperbolic space has more room: angles shrink as polygons grow, and at the right size three heptagons close a corner perfectly. The Poincaré disk shows the whole infinite plane at once. Every cell here is exactly the same size in the space's own metric; the shrinking toward the rim is the price of the map, not a fact about the territory. Escher saw this figure in a paper of Coxeter's and spent years learning to draw it — his Circle Limit prints are this construction.

Each dawn in Tamil Nadu, kolam are drawn in rice flour at the threshold: a grid of dots, then a thread that weaves around every dot and closes on itself. The rule is the oldest in either folio — a line at 45 degrees, bouncing off the frame. Whether the drawing needs one thread or several is pure number theory: the count is gcd(m, n). A 7×5 grid closes in a single unbroken line; 6×6 shatters into six. The weave alternates over and under without ever failing — mirror curves are alternating knots, and the tradition held that theorem in its hands long before anyone wrote it down. The folio ends here on purpose: this is the one figure still drawn by hand every morning, and erased by night.