1 Introduction

note: this blueprint was drafted by Claude

Moser’s worm problem asks for the (convex) region of smallest area in the plane that contains a rotated and/or translated copy of every curve of unit length (a worm). Any such region is called a Moser set. The exact optimum is unknown; the best published upper and lower bounds sandwich it between \(0.232239\) and \(\pi /12\).

This project pursues a computer-checked improvement to the lower bound, by showing that every Moser set must contain a convex polygon of area at least \(0.232240\).

The main idea is to discretise the problem. We create and iteratively mutate a working set of convex candidate polygons satisfying the property that which any convex Moser set of size below \(0.232240\) must contain a copy of at least one of the candidates. The working set is initialised with a single polygon, and then repeatedly mutated by replacing elements with collections of larger elements in ways that preserve the property, all the while removing any element of size greater than \(0.232240\). If we can find a sequence of mutations that ends with an empty working set, we may conclude that no convex Moser set of area below \(0.232240\) exists.