Planar Drawings of Origami Polyhedra

This work studies the structure of origami bases via graph drawings of origami polyhedra. In particular, we propose a new class of polyhedra, called extreme-base polyhedra, that capture the essence of \extreme" origami bases. We develop a linear-time algorithm to nd the \natural" straight-line planar drawing of these polyhedra. This algorithm demonstrates a recursive structure in the polyhedra that was not apparent before, and leads to interesting fractals. Our hope is to discover fundamental properties of paper folding through graph drawing. One important problem in origami mathematics is to nd a useful characterization of when a crease pattern can be folded at (using every crease). This problem is known to be NP-hard 3]. However, this does not preclude the possibility of a simple characterization, only the tractability of such a characterization. For example, one conjecture that we have arrived at from this work is that every at-foldable crease pattern has a Hamiltonian cycle. This property is insuucient, but a similar condition that is NP-hard to verify yet easy to understand may ll the gap in understanding at origami. Origami bases are the starting point for designing origami models. An extreme origami base maps the boundary of a polygonal piece of paper (convex for our purposes) to a common plane. Two examples of extreme bases are given in Fig. 1(c). The crease pattern of an extreme base is the union of the medial axis of the polygon and \perpendicular folds." One interesting property of these extreme bases is that they can be folded while keeping the faces rigid. ! ! (a) (b) (c) (d) ! ! Fig. 1. (a{c) Folding of an extreme base for a triangle (top row) and quadrilateral (bottom row). (d) Polyhedra formed from partial foldings; the lower convex hull is drawn below.