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 congruent copy of every rectifiable 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 in a narrow interval. This project pursues a computer-checked lower bound for the area of a Moser set by discretising the space of candidate convex hulls and the space of planar isometries, and then verifying that no sufficiently small polygon can contain every unit worm.

The strategy is iterative pruning of a working set of convex candidate polygons: polygons are dropped when they are too large, when another candidate is a subset, or when a new worm fails to embed inside them. At termination the working set is empty, which (modulo the discretisation error bookkeeping) rules out all Moser sets below the area threshold. We currently target the threshold \(0.232240\).