Symmetrized Double Enneper

Symmetrized Double Enneper is a minimal surface, given by a Weierstrass representation, immersed.

Open Symmetrized Double Enneper in the interactive viewer →

aperiodic def-weierstrass immersed implemented minimal tradition-classical

Formula

Gauss map g

\frac{z^{n} \left(\mathit{R0} \exp\left(\frac{0.5 \pi i}{1 + n}\right)\right)^{2 + 2 n} \left(\left(\mathit{R0} \exp\left(\frac{0.5 \pi i}{1 + n}\right)\right)^{1 + n} - z^{1 + n}\right)}{\left(1 + \left(\mathit{R0} \exp\left(\frac{0.5 \pi i}{1 + n}\right)\right)^{2 + 2 n}\right) \left(-1 + z^{1 + n} \left(\mathit{R0} \exp\left(\frac{0.5 \pi i}{1 + n}\right)\right)^{1 + n}\right)}

Height differential dh

\frac{-z^{-2 - n} \left(-1 + \left(z \mathit{R0} \exp\left(\frac{0.5 \pi i}{1 + n}\right)\right)^{1 + n}\right) \left(z^{1 + n} - \left(\mathit{R0} \exp\left(\frac{0.5 \pi i}{1 + n}\right)\right)^{1 + n}\right)}{1 + \left(\mathit{R0} \exp\left(\frac{0.5 \pi i}{1 + n}\right)\right)^{2 + 2 n}}

Properties

Family: minimal
Given by: a Weierstrass representation
Curvature: zero mean curvature
Periodicity: not periodic
Ends: 0
Embedding: immersed
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 SYMM_DBLENN) by sampling g and dh over rings in the complex plane at the row's default parameters: g: matches over 144 complex samples (worst 1.9e-15); dh: matches over 144 complex samples (worst 1.83e-15).

B. Riemann (1867) and F. J. Lopez, A. Ros, J. Differential Geom. 33 (1991) for the (never embedded) finite Riemann family; the symmetrized member follows M. Weber, minimalsurfaces.blog, "Symmetrized Finite Riemann".
H. Karcher, "Construction of minimal surfaces" (1989) for the symmetrization method (double Enneper, k-noid families);
L. P. Jorge, W. H. Meeks III, Topology 22 (1983) for the k-noids; data after M. Weber, minimalsurfaces.blog ("Symmetrized Double Enneper", "k-Noids with Enneper Ends", "Antiprismatic k-Noids").
L. Henneberg (1875); R. Kusner, Bull. Amer. Math. Soc. 17 (1987) 291-295; F. J. Lopez, Duke Math. J. 71 (1993) 23-30 -- full citations in the weierstrass symtail engine block.