Forms of the Plane · 3D
Spirograph on the Torus
The pen’s loops wound around a ring
A spirograph curve only ever runs between two circles, and a torus spans exactly that ring. Lift each point of the curve onto the torus and the loops become a knot winding around it: its shadow is the flat curve, point for point.
- Ring
- R − r
- Tube
- pen offset
Turn it in your hands, strike it to hear the chord of its edges, slice it with a plane.
Facts
- The torus’s middle circle has radius R − r, where the wheel’s centre rolls, and its tube is as wide as the pen’s offset.
- Each lobe of the flat curve is one trip around the tube.
- With the pen further out than the ring’s radius the torus closes its hole and the loops pass over the top instead of through it.
Across the dimensions
- Its shadow SpirographThe curve runs around a ring, and a torus spans that ring: lifted onto it, the loops become a knot whose shadow is the curve.
