Math Art
Enneper

Enneper

Enneper is a minimal surface, given by a parametrisation, genus 0, 1 end, self-intersecting.

Open Enneper in the interactive viewer →

aperiodic complete-finite-total-curvature def-parametric implemented minimal self-intersecting tradition-classical

About

One of the simplest minimal surfaces that can be written down in closed form, and a standard first example because its Weierstrass data is as plain as it gets. It has a single end and no boundary, and it passes through itself: minimal does not mean tidy. Grown far enough, the self-intersections dominate the picture.

Formula

x(u, v)
u−u33+u⁢v2u - \frac{u^{3}}{3} + u v^{2}
y(u, v)
v−v33+v⁢u2v - \frac{v^{3}}{3} + v u^{2}
z(u, v)
u2−v2u^{2} - v^{2}

Properties

Family
minimal
Given by
a parametrisation
Curvature
zero mean curvature
Periodicity
not periodic
Genus
0
Ends
1
Embedding
self-intersecting
Orientable
yes
Fidelity
exact
Exactness
elementary

Sources

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