The Hopf fibration, drawn with a pen
In 1931 Heinz Hopf described a way to map the 3-sphere, the sphere that lives in four-dimensional space, onto an ordinary sphere. Each point on the ordinary sphere corresponds to a whole circle, called a fibre, and any two fibres are linked exactly once. Project the fibres onto a page and the ones from a single latitude wrap into a torus.
The Hopf Fibration generator draws each fibre as one closed pen stroke. Set Fibres from 8 to 120 and Latitudes from 1 to 5, then turn the shape with Theta, Yaw and Pitch. The back half can stay solid, turn dashed or disappear. Colour By ring gives each latitude its own pen, so a three-latitude piece plots as three nested tori.
The recipe
Enter these settings and the same drawing comes back, line for line.
- Generator
- Hopf Fibration
- Seed
- 1
- Page
- AxiDraw V3 · A4, portrait
- Pens
- Nine Inks
- Fibres
- 80
- Latitudes
- 3
- Theta
- 88
- Yaw
- 150
- Pitch
- 60
- Samples
- 200
- Back half
- Solid
- Colour By
- Ring
Search Forgeplot on the App Store.
