Distance between Folded Objects

Geometric folding problems have recently attracted much attention in both mathematics and theoretical computer science. In this paper, we study the following basic problem: given a set of folded conformations of a unit-length rope in 1D, how do we decide which are more similar or less similar to each other? We first define a distance function between flat folded states that incorporates both the geometry and the overlap order. Then we do some computational experiments clustering random folded ropes with this distance metric. Finally, we generalize our results for 1D folded ropes to flat folded papers (origami) in 2D.