Scherk Saddle Tower

Scherk Saddle Tower is a singly periodic minimal surface, given by a Weierstrass representation, genus 0, embedded, with a rod group.

Open Scherk Saddle Tower in the interactive viewer →

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

Formula

Gauss map g

z^{-1 + n}

Height differential dh

\frac{z^{-1 + n}}{1 + z^{2 n}}

Properties

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

Definition

Extracted from the shipped row's source by AST (complete multi-line lambdas, phi triple (dh = phi3, g = phi3/(phi1 - i*phi2), a Weierstrass-representation identity)) and reproduced numerically against the shipped callables over the complex plane, exactly -- no scalar slack.

Sources

Weierstrass data reproduced numerically against the shipped implementation (math_art/minsurf/zoo.py, row SCHERK_TOWER) by sampling g and dh over rings in the complex plane at the row's default parameters: g: matches over 144 complex samples (worst 2.48e-16); dh: matches over 144 complex samples (worst 0).

H. F. Scherk, J. reine angew. Math. 13 (1835) -- the singly periodic surface.
H. Karcher, 'Embedded minimal surfaces derived from Scherk's examples', Manuscripta Math. 62 (1988) -- the saddle towers.
H. Karcher, 'Embedded minimal surfaces derived from Scherk's examples', Manuscripta Math. 62 (1988) 83-114 -- 'translation invariant minimal surface with genus 0 in the quotient' (as quoted at minsurf ch140).
J. Perez, M. Traizet, 'The classification of singly periodic minimal surfaces with genus zero and Scherk type ends', Trans. Amer. Math. Soc. 359 (2007) 965-990.