## Zachary Abel, Erik Demaine, Martin Demaine, Hiroaki Matsui, Günter
Rote, and Ryuhei Uehara:

# 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.