# Common developments of several different orthogonal boxes

In: Proceedings of the 23rd Annual Canadian Conference on Computational Geometry, Vancouver, August 10–12, 2011, pp. 77–82.

### Abstract

We investigate the problem of finding common developments that fold to several different orthogonal boxes. It was shown that there are infinitely many orthogonal polygons that fold to two incongruent orthogonal boxes in 2008. In this paper, we first show that there is an orthogonal polygon that fold to three boxes of size 1×1×5, 1×2×3, and 0×1×11. Although we have to admit a box to have volume 0, this solves the open problem mentioned in literature. Moreover, once we admit that a box can be of volume 0, a long rectangular strip can be folded to an arbitrary number of boxes of volume 0. We next consider for finding common non-orthogonal developments that fold to plural incongruent orthogonal boxes. In literature, only orthogonal folding lines or with 45 degree lines were considered. In this paper, we show some polygons that can fold to two incongruent orthogonal boxes in more general directions.

Last update: August 22, 2011.