A New Slant on Lebesgue's Universal Covering Problem

TL;DR

This paper uses computational methods to improve bounds on Lebesgue's universal covering problem, proposing conjectures and a novel geometric approach that suggests the minimal cover area is around 0.84408.

Contribution

It introduces a new geometric modification and computational approach that improves the upper bound for the minimal universal cover area.

Findings

New upper bound of 0.844113 for the cover area

Modified hexagon by removing corners at a small slant angle

Simulations suggest minimal area around 0.84408

Abstract

Lebesgue's universal covering problem is re-examined using computational methods. This leads to conjectures about the nature of the solution which if correct could provide a blueprint for a complete solution. Empirical lower bounds for the minimal area are computed using different hypotheses based on the conjectures. A new upper bound of 0.844113 for the area of the minimal cover is derived improving previous results. This method for determining the bound is suggested by the conjectures and computational observations but is proved independently of them. The key innovation is to modify previous best results by removing corners from a regular hexagon at a small slant angle to the edges of the dodecahedron used before. Simulations indicate that the minimum area for a convex universal cover is likely to be around 0.84408.