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On Alan Schoen's I-WP Minimal Surface

Dami Lee, Matthias Weber, A. Tom Yerger

Abstract

We discuss in detail Alan Schoen's I-WP surface, an embedded triply periodic minimal surface of genus 4 with cubical symmetries. We exhibit various geometric realizations of this surface with the same conformal structure and use them to prove that the associate family of the I-WP surface contains six surfaces congruent to I-WP at Bonnet angles that are multiples of $60^\circ$.

On Alan Schoen's I-WP Minimal Surface

Abstract

We discuss in detail Alan Schoen's I-WP surface, an embedded triply periodic minimal surface of genus 4 with cubical symmetries. We exhibit various geometric realizations of this surface with the same conformal structure and use them to prove that the associate family of the I-WP surface contains six surfaces congruent to I-WP at Bonnet angles that are multiples of .
Paper Structure (5 sections, 10 theorems, 7 equations, 10 figures)

This paper contains 5 sections, 10 theorems, 7 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Steßmann's Surface
  3. Schoen's I-WP Surface
  4. $S_{147}$: A 12-fold branched cover of the thrice punctured sphere
  5. Weierstrass Representation

Key Result

Theorem 3.1

For $S$, the Gauss map $G$ is a meromorphic function of degree $3$ and represents $S$ as a 3-fold cover of the sphere branched over the vertices of a regular octahedron.

Figures (10)

  • Figure 1: Schoen's I-WP surface with its skeletal graphs
  • Figure 2: Steßmann's Surface
  • Figure 3: Steßmann's Surface --- top view
  • Figure 4: The I-WP Surface
  • Figure 5: Hyperbolic Fundamental Domain of $S_{147}$
  • ...and 5 more figures

Theorems & Definitions (20)

  • Theorem 3.1
  • proof
  • Theorem 4.1
  • proof
  • Theorem 4.2
  • proof
  • Theorem 4.3
  • proof
  • Theorem 4.4
  • proof
  • ...and 10 more