Tightest Arrangements of All Different Nets of a Cube

Abstract

Summary Since the 1950’s, polyominoes (that is, shapes consisting of edge-connected unit squares) have gained significant interest both as objects of combinatorial study and as classics of recreational mathematics. There is a very natural collection of such shapes, namely the 11 distinct polyhedral nets of the unit cube. An interesting question is: how tightly can the nets fit together? It is equivalent to search for the smallest perimeter a shape built of them can have. In this article, we provide the exact lower bound as an invitation to study analogous problems for different families of polyominoes.