Geometric Folding Algorithms: Linkages, Origami, Polhedra

Lecture 1: Overview

Geometric Objects

1. Linkage
   • rigid bars
   • no crossing
2. Paper
   • no stretch
   • no tears
   • no crossing
3. Polyhedra
   • no overlap
   • one piece

Questions

1. Foldability: what structures fold at all/ in particular way ?
2. Design: what shapes (or particular property) can be folded & how ?

Results:

1. Universality: everything can be folded + algorithm
2. Decision: efficient algorithm to decide foldability
3. Hardness: computationally intractable to decide foldability

Ideas

1. Linkages
   1. allowing intersection
      • converting linear motion to circular motion
   2. Rigidity: does a linkage given fold at all ?
   3. Linkages forbidding intersection
      • reconfiguration: fold from config A to config B
      • special linkages
        • chain
        • trees
          • locked trees
2. Paper
   1. Foldability: which crease patterns fold flat ?
   2. Design: what shapes can be folded ?
      1. Origamizer
      2. TreeMaker

Open Problems

1. No crossing
2. which 3d chains/2d trees have ”locked configurations”
3. Mountain and valley on paper crease to be able to be folded ?