The Optimal Paper Moebius Band

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.