Math Art
Hackman Screw-Motion 1-Noid

Hackman Screw-Motion 1-Noid

Hackman Screw-Motion 1-Noid is a singly periodic minimal surface, given by a Weierstrass representation, immersed, with a rod group.

Open Hackman Screw-Motion 1-Noid in the interactive viewer →

def-weierstrass immersed implemented minimal minimal-periodic singly-periodic tradition-classical

Properties

Family
minimal-periodic
Given by
a Weierstrass representation
Curvature
zero mean curvature
Periodicity
singly periodic
Ends
0
Embedding
immersed
Symmetry kind
rod group
Fidelity
exact
Exactness
numerical-integral

Definition

Toroidal 1-noids: quotients of screw-motion invariant singly periodic minimal surfaces with a single catenoidal end in the quotient (the maximum principle forbids 1-noids in R3 proper). Hackman proved one exists on every conformal type of torus; the alternating fence of half-catenoids is the simplest member of the wider picture. THE SHIPPED ROW is the k = 1/3 member on tau = t + 2i with t = 0.333316172865672, re-solved from the notebook's own period condition WITH its verbatim closed-form Bonnet phase (sigma/zeta/theta with Conjugate[]s on the sheared torus). The batch-5 hypothesis phi = -arg(sigma(-k/2) sigma(k/2)) was tested against that closed form FIRST and is WRONG -- off by pi (a silent dh sign flip that would render perfectly) plus a t-dependent 6e-6 drift. Gated: period re-solve, pure phase, end loop (0,0,0), deck z+1 purely vertical h = 1.095693, screw rise exactly k h with base-point-independent offset.

Sources

Derived from the Math Art generator registries and cross-referenced to the literature; not transcribed from any gallery's compilation.