Gerlach & von der Mosel · 2011
Sphere-filling ropes
The longest closed curve of thickness sin(π/8) on the unit sphere. For this thickness the shape is known exactly: one piecewise-circular loop.
- Thickness Θ
- sin(π/8)
- 0.3827
- Length
- 16.42
- 2π / Θ
- Eastern turn
- π/4
- k = 1
- Semicircles
- 8
- 2 solutions
Solved thickness Θ = sin(π/8). Eastern hemisphere turned by π/4.
Drag the sphere to orbit. The centerline lies on the unit sphere; the plaster core has radius 1 − Θ, so a rope of radius Θ sits on it and neighboring strands touch. Shown a hair inside that radius so the contacts stay readable.
Amer. Math. Monthly 118 (2011), 863–876. arXiv:1005.4609