Taffy on the Sphere

Taffy on the sphere

Puller

preset

matrix M (

)

lattice τ

Taffy

shape

loop offset ε0.200

fill lines1

Rods are the four 2-torsion points of T²: 0=(0,0) at ∞, 1=(½,0), 2=(½,½), 3=(0,½). A loop on the sphere enclosing rods i and j lifts to a closed geodesic of T² parallel to the segment p_ip_j, offset by ε (the disk around the two branch points lifts to an annulus); pulling it n times gives the closed geodesic of slope Mⁿ(p_j−p_i). "Fill lines" draws nested loops at offsets ε·k/K so the taffy reads as a band. The arc itself is the ε = 0 limit. Rod 0 sits at ∞ (the north pole) and is fixed by every M: for the 3-rod pullers it is the extra puncture of S_0,4, for Thibodeau's 4-rod puller it is a rod.

Pulling

time s0.000

max stage8

speed0.25/s

Look

line width2.5 px

detail500/u

chart zoom×1.00

sphere72%

show the sphere (uncheck for strands only) rods (branch points of ℘) rod orbits over one period flat torus inset colour strands along their length slowly spin the sphere

Readout

What you are seeing

The involution ι(z) = −z on the torus T² = ℂ/Λ has four fixed points, and the Weierstrass function ℘ : T² → ℂ∪{∞} is exactly the quotient map T² → T²/ι, a sphere branched over ℘(0) = ∞ and the three half-period values e₁, e₂, e₃. A linear map φ(x) = Mx on T² commutes with ι, so it descends through ℘ to a map of the sphere that permutes the four branch points — the rods of a taffy puller. The taffy is a loop around two rods; its lift is a closed geodesic of the torus, and the puller just changes its slope (Thiffeault, The mathematics of taffy pullers).

To make the pulling continuous we run the one-parameter group M^s: at time s the strand is M^sγ in the plane, the lattice is Λ_s = M^sΛ, and the picture is ℘_{Λ_s}(M^sz). At integer s the lattice is back to Λ and the picture is the s-th iterate of the puller; in between, the branch points e_i(Λ_s) move and come back permuted — that is the rod motion. Strand length grows like λ^s, λ the dilatation.

Every vertex is computed on the GPU from the theta-series for ℘. space play · ←→ scrub · drag to orbit · wheel to zoom.