1. Date of download: 11/12/2016 Copyright © ASME. All rights reserved. From: Edge-Compositions of 3D Surfaces J. Mech. Des. 2013;135(11):111001-111001-9. doi:10.1115/1.4025378 A fold f has a fold line and a fold angle Figure Legend:

2. Date of download: 11/12/2016 Copyright © ASME. All rights reserved. From: Edge-Compositions of 3D Surfaces J. Mech. Des. 2013;135(11):111001-111001-9. doi:10.1115/1.4025378 Q can be unfolded into P if Q is the image of a folded state φF(P) Figure Legend:

3. Date of download: 11/12/2016 Copyright © ASME. All rights reserved. From: Edge-Compositions of 3D Surfaces J. Mech. Des. 2013;135(11):111001-111001-9. doi:10.1115/1.4025378 Two convex polygons placed next to each other are guaranteed not to intersect Figure Legend:

4. Date of download: 11/12/2016 Copyright © ASME. All rights reserved. From: Edge-Compositions of 3D Surfaces J. Mech. Des. 2013;135(11):111001-111001-9. doi:10.1115/1.4025378 Left: Walking and gripper robots folded out of the patterns shown. Right: The composition, a walking-gripping robot, whose unfolding was designed manually. Our goal is to generate such an unfolding automatically. (Credit: Robots were designed by Cagdas Onal and Michael Tolley.) Figure Legend:

5. Date of download: 11/12/2016 Copyright © ASME. All rights reserved. From: Edge-Compositions of 3D Surfaces J. Mech. Des. 2013;135(11):111001-111001-9. doi:10.1115/1.4025378 Algorithm 1: Constructing pleats to follow a path. (a) The edge e P and the path p to follow. (b) Reflected edges e i (solid) and perpendicular bisectors ei ⊥ (dotted). (c) Resulting pleats. Types 1, 2, and 3 self-intersections are shaded. (d) Pleats with self- intersections corrected. (e) Folded state of pleats. All e i coincide. Figure Legend:

6. Date of download: 11/12/2016 Copyright © ASME. All rights reserved. From: Edge-Compositions of 3D Surfaces J. Mech. Des. 2013;135(11):111001-111001-9. doi:10.1115/1.4025378 (a) A bridge (shaded) that intersects with (P, F). (b) The offending region is removed. Figure Legend:

7. Date of download: 11/12/2016 Copyright © ASME. All rights reserved. From: Edge-Compositions of 3D Surfaces J. Mech. Des. 2013;135(11):111001-111001-9. doi:10.1115/1.4025378 A long pleat. Top: The unfolding with offending area (shaded). Bottom: Folded state. (a) In the folded state, the pleat protrudes outside the adjacent face. (b) It can be crimped to not interfere with other folds and (c) trimmed to avoid protrusions. Figure Legend:

8. Date of download: 11/12/2016 Copyright © ASME. All rights reserved. From: Edge-Compositions of 3D Surfaces J. Mech. Des. 2013;135(11):111001-111001-9. doi:10.1115/1.4025378 Algorithm 3: Bridging an edge on the interior of the unfolding to the boundary of the convex hull. (a) Original fold pattern. The edge to join e P is bold and the convex hull is shown in gray. (b) The edge-adjacency graph with the path from e P to e b in bold. (c) Pleats attached to e b using Algorithm 2. (d) The accordion path and interior faces reflected over the boundary of the convex hull. The convex hull is also updated. (e) Pleats attached to erefP using Algorithm 2. (f) Output unfolding with the bridge added. Figure Legend:

9. Date of download: 11/12/2016 Copyright © ASME. All rights reserved. From: Edge-Compositions of 3D Surfaces J. Mech. Des. 2013;135(11):111001-111001-9. doi:10.1115/1.4025378 Example edge-adjacency graph Figure Legend:

10. Date of download: 11/12/2016 Copyright © ASME. All rights reserved. From: Edge-Compositions of 3D Surfaces J. Mech. Des. 2013;135(11):111001-111001-9. doi:10.1115/1.4025378 Algorithm 2: Bridging an edge on the boundary of the unfolding to the boundary of the convex hull. (a) Original fold pattern. The edge to join e P is bold and the convex hull is shown in gray. (b) The region PkCH (shaded) and its straight skeleton. (c) The path p from e P to the boundary of the convex hull. (d) Pleats tiled along the path. (e) Output unfolding with the bridge added. Figure Legend:

11. Date of download: 11/12/2016 Copyright © ASME. All rights reserved. From: Edge-Compositions of 3D Surfaces J. Mech. Des. 2013;135(11):111001-111001-9. doi:10.1115/1.4025378 Fold patterns generated by this algorithm. Top: Input fold patterns for the polyhedral complexes to join. The bold edges are the edges to join. Second row: The generated composite unfolding. Bridges constructed by our algorithm are shaded. Third row: The folded state of the composite unfolding. Fourth row: Physical model of the composition with just the bridge folded. Bottom: Physical models of the input surfaces and the composition folded from poster board. Figure Legend: