Rectangular Unfoldings of Polycubes

Abstract

In this paper, we investigate the problem that asks if there exists a net of a polycube that is exactly a rectangle with slits. For this nontrivial question, we show affirmative solutions. First, we show some concrete examples: (1) no rectangle with slits with fewer than 24 squares can fold to any polycube, (2) a 4 × 7 rectangle with slits can fold to a heptacube (nonmanifold), (3) both of a 3 × 8 rectangle and a 4 × 6 rectangle can fold to a hexacube (nonmanifold), and (4) a 5 × 6 rectangle can fold to a heptacube (manifold). Second, we show a construction of an infinite family of polycubes folded from a rectangle with slits. The smallest one given by this construction is a 6 × 20 rectangle with slits that can fold to a polycube of genus 5. This construction gives us a polycube for any positive genus. Moreover, by this construction, we can show that there exists a rectangle with slits that can fold to k different polycubes for any given positive integer k.