The Optimal Paper Moebius Band

TL;DR

This paper establishes the minimum aspect ratio for smooth embedded paper Moebius bands as aa and characterizes the limiting shape as the triangular Moebius band, resolving longstanding geometric conjectures.

Contribution

It proves the minimum aspect ratio for smooth embedded paper Moebius bands and characterizes the shape as the aspect ratio approaches aa, confirming two historical conjectures.

Findings

Minimum aspect ratio > aa for smooth embedded paper Moebius bands

Convergence to the triangular Moebius band as aspect ratio approaches aa

Resolution of conjectures by Wunderlich (1962) and Halpern & Weaver (1977)

Abstract

In this paper we prove that a smooth embedded paper Moebius band must have aspect ratio greater than $\sqrt 3$. We also prove that any sequence of smooth embedded paper Moebius bands whose aspect ratio converges to $\sqrt 3$ must converge, up to isometry, to the famous triangular Moebius band. These results answer the minimum aspect ratio question discussed by W. Wunderlich in 1962 and prove the more specific conjecture of B Halpern and C. Weaver from 1977.