# Lecture 17

#### Thursday, March 28, 2000

Eric Demaine's talk.

## Folding and Cutting Paper

### Erik Demaine Department of Computer Science University of Waterloo

Take a sheet of paper, fold it flat, and make one complete straight cut. What shapes can the unfolded pieces make? The earliest reference to this idea is a 1721 Japanese puzzle book, in which a Japanese crest is made by this method. Houdini used folding and cutting for a magic trick before he became a famous escape artist. This talk discusses our work on the fold-and-cut problem: given a collection of line segments (i.e., a plane graph), find a flat folding of the plane so that precisely those line segments are folded to a common line, along which we can cut to make the desired shapes.

I will describe two algorithms that we have developed to solve this problem for any collection of line segments: convex and nonconvex polygons, multiple disjoint polygons, adjoining polygons, nested polygons, and even floating line segments and points. The first algorithm (joint work with Martin Demaine and Anna Lubiw) is based on the straight skeleton. The second algorithm (joint work with Marshall Bern, David Eppstein, and Barry Hayes) is based on disk packing.