Harmonic Curves · 2D
Spirograph
Hypotrochoid: a circle rolling inside a circle
Roll a small toothed wheel inside a ring and trace a point on it. The ratio of teeth decides how many loops before the curve closes.
- Loops
- R ÷ gcd(R, r)
Watch it drawn with compass and straightedge, hear the chord of its lines, and turn the Dimension dial to raise its form in space.
Facts
- The pattern repeats after r ÷ gcd(R, r) trips around the ring.
- With a coprime ratio of teeth the pen visits every tooth before closing.
- Planets seen from Earth trace similar looping paths (epicycles).
Across the dimensions
- Shadow of Spirograph on the TorusThe curve runs around a ring, and a torus spans that ring: lifted onto it, the loops become a knot whose shadow is the curve.
