On The Optimal Paper Moebius Band

TL;DR

This paper investigates conjectures about the geometric properties of smooth embedded paper Moebius bands, focusing on aspect ratio bounds and convergence behavior, by translating these conjectures into algebraic positivity conditions.

Contribution

The paper reduces two major conjectures on paper Moebius bands to 10 algebraic statements about explicit piecewise expressions, providing a new approach to these geometric problems.

Findings

Reduction of conjectures to algebraic positivity conditions

Identification of explicit algebraic expressions related to the problem

Establishment of a framework for verifying geometric conjectures through algebraic analysis

Abstract

Let $A=\sqrt 3$. There are two main conjectures about paper Moebius bands. First, a smooth embedded paper Moebius band must have aspect ratio at least A. Second, any sequence of smooth embedded paper Moebius bands having aspect ratio converging to A converges, in the Hausdorff topology and up to isometries, to an equilateral triangle of semiperimeter A. We will reduce these conjectures to 10 statements about the positivity of certain explicit piecewise algebraic expressions.