Crocheting Bour's $\mathcal{B}_m$ minimal surfaces

TL;DR

This paper explores how to create crochet patterns for Bour's minimal surfaces, a class of mathematically defined surfaces of revolution, by deriving arc length formulas and providing detailed instructions for specific cases.

Contribution

It introduces a method to generate crochet patterns for Bour's minimal surfaces using trigonometric identities, including explicit instructions for Enneper's surface.

Findings

Exact crochet instructions for Enneper's surface provided

Method to calculate arc lengths for Bour's surfaces established

Analysis of three special Bour's surfaces in detail

Abstract

Minimal surfaces can be though as a mathematical generalisation of surfaces formed by soap films. We consider Bour's minimal surfaces $\mathcal{B}_m$ that are intrinsically surfaces of revolution. We show how to generate crochet patterns for $\mathcal{B}_m$ surfaces using basic trigonometric identities to calculate required arc lengths. Three special cases of $\mathcal{B}_m$ surfaces are considered in more detail, namely Enneper's, Richmond's, and Bour's $\mathcal{B}_3$ surfaces, and we provide exact crochet instructions for the classical Enneper's surface.