The Spirograph

A pen pinned in a small gear, rolling without slipping inside a big ring, traces a hypotrochoid. The curve closes only when the gears come back into phase — after R/gcd(R,r) trips of the ring — and that same number is its petal count. Whole-number teeth, whole-number flowers. Slide the gears, or roll one from a seed.

Gears

Ring teeth R 96

Wheel teeth r 25

Pen offset d 0.80

Motion

Speed 1.0×

Roll a figure

gcd(R, r)—

petals = R/gcd—

ring trips to close—

A hypotrochoid is the sum of two rotating vectors — its Fourier-epicycle cousin with exactly two terms. Self-test proves the closure law, the petal count, no-slip rolling, and the two-vector identity.