6.849 | Fall 2012 | Graduate

Geometric Folding Algorithms: Linkages, Origami, Polyhedra

Calendar and Notes

Organizational Note: This course was taught with an inverted lecture format. Students first watched lecture videos from the last time the course was offered, in Fall 2010.  Class time was devoted to in-depth discussions of the topics covered in those lectures, and additional activities such as demonstrations, proofs, and folding exercises.  The materials here are presented in accordance with the way the course was organized in Fall 2012, which differed from the previous offering.  As a result, some of the videos, notes, and slides from Fall 2010 are not in sequential order, but rather one that is most relevant to the inverted format.

SES # TOPICS & DESCRIPTIONS LECTURE RESOURCES (FROM FALL 2010) CLASS RESOURCES
1 Overview of the course: Inverted lecture format, sample topics and problems considered.

Notes for Lecture 1 (PDF - 5.4MB)

Slides for Lecture 1 (PDF - 11.9MB)

Video of Lecture 1 (optional)

Notes for Class 1 (PDF - 4.4MB)

Slides for Class 1 (PDF - 2.9MB)

Video of Class 1

2

Origami intro: Origami alphabet, higher dimensions.

Universality: Terminology history, practical strip folding, pseudopolynomial bounds, seam placement, hide gadget via simple folds?

Simple folds: Metal/wood/plastic motivation, definition, examples, linear-time algorithm, extra creases.

Detailed Descriptions of Lecture 2 and Class 2 (PDF)

Notes for Lecture 2 (PDF - 6.2MB)

Slides for Lecture 2 (PDF)

Video of Lecture 2

Notes for Class 2 (PDF - 3.0MB)

Slides for Class 2 (PDF - 2.9MB)

Video of Class 2

3

Single-vertex crease patterns: Linear-time algorithm, local foldability examples, T-shirt folding, higher dimensions, why flat foldability?

Detailed Descriptions of Lecture 3 and Class 3 (PDF)

Notes for Lecture 3 (PDF - 5.4MB)

Slides for Lecture 3 (PDF)

Video of Lecture 3

Notes for Class 3 (PDF - 2.4MB)

Slides for Class 3 (PDF - 1.2MB)

Video of Class 3

4

Efficient origami design: Uniaxial, TreeMaker and Origamizer in practice, box-pleating tree method, tree method triangulation, universal molecule, gift wrapping, checkerboard gadgets, Origamizer software vs. mathematics, vertex/edge tucking molecules, Voronoi diagrams.

Detailed Descriptions of Lecture 4 and Class 4 (PDF)

Notes for Lecture 4 (PDF - 5.4MB)

Slides for Lecture 4 (PDF - 1.8MB)

Video of Lecture 4

Notes for Class 4 (PDF - 2.7MB)

Slides for Class 4 (PDF - 6.7MB)

Video of Class 4

5

Artistic origami design: Jason Ku designs, other materials (dollars, cardboard, hydro, metal, polypropylene), tessellations, Tess, connected cranes, modular origami, business card cubes.

Detailed Descriptions of Lecture 5 and Class 5 (PDF)

Slides for Lecture 5 (PDF - 2.3MB)

Video of Lecture 5

Slides and video courtesy of Jason Ku. Used with permission.

Notes for Class 5 (PDF)

Slides for Class 5 (PDF - 6.0MB)

Video of Class 5

6

Architectural Origami: Origamizer, Freeform Origami, Rigid Origami Simulator.

Detailed Descriptions of Lecture 6 and Class 6 (PDF)

Slides for Lecture 6 (PDF - 12.4MB)

Video of Lecture 6

Slides and video courtesy of Tomohiro Tachi. Used with permission.

Notes for Class 6 (PDF - 2.1MB)

Slides for Class 6 (PDF - 2.6MB)

Video of Class 6

7

Universal hinge patterns: Box-pleating history, maze-folding prints.

NP-hardness: Simple foldability, crease pattern flat foldability.

Detailed Descriptions of Lecture 7 and Class 7 (PDF)

Notes for Lecture 7 (PDF - 5.1MB)

Slides for Lecture 7 (PDF - 11.2MB)

Video of Lecture 7

Notes for Class 7 (PDF - 2.7MB)

Slides for Class 7 (PDF - 2.1MB)

Video of Class 7

8

Fold and one cut: Software, scissor vs. mathematical cuts, tree folding, density, examples, how many disks, comparison to tree method, continuous flattening.

Detailed Descriptions of Lecture 8 and Class 8 (PDF)

Notes for Lecture 8 (PDF- 7.1MB)

Slides for Lecture 8 (PDF)

Video of Lecture 8

Notes for Class 8 (PDF - 3.0MB)

Slides for Class 8 (PDF - 1.7MB)

Video of Class 8

9

Pleat folding: Triangulated hypars, smoothness, normals, mathematical vs. real paper, pleat folding algorithms, hypar folding.

Detailed Descriptions of Lecture 9 and Class 9 (PDF)

Notes for Lecture 9 (PDF - 6.3MB)

Slides for Lecture 9 (PDF - 2.4MB)

Video of Lecture 9

Notes for Class 9 (PDF - 3.5MB)

Slides for Class 9 (PDF)

Video of Class 9

10

Folding motions: Trouble with holes.

Linkages to sign your name: Sliding joints, contraparallelogram bracing, higher dimensions, semi-algebraic sets, splines.

Geometric construction: Straight edge and compass, origami axioms, angle trisection, cube doubling.

Detailed Descriptions of Lecture 10 and Class 10 (PDF)

Notes for Lecture 10 (PDF - 6.3MB)

Slides for Lecture 10 (PDF)

Video of Lecture 10

Notes for Class 10 (PDF - 2.7MB)

Slides for Class 10 (PDF - 1.1MB)

Video of Class 10

11

Rigidity theory: Pebble algorithms, rigid component decomposition, body-and-bar framework, angular rigidity, 5-connected double bananas.

Detailed Descriptions of Lecture 11 and Class 11 (PDF)

Notes for Lecture 11 (PDF - 8.0MB)

Slides for Lecture 11 (PDF - 1.3MB)

Video of Lecture 11

Notes for Class 11 (PDF - 3.2MB)

Slides for Class 11 (PDF)

Video of Class 11

12

Tensegrities: Dot products, springs, software, sculpture.

Project discussion

Detailed Descriptions of Lecture 12 and Class 12 (PDF)

Notes for Lecture 12 (PDF - 6.9MB)

Slides for Lecture 12 (PDF)

Video of Lecture 12

Notes for Class 12 (PDF)

Slides for Class 12 (PDF)

Video of Class 12

13

Locked linkages: Why expansiveness, energy algorithm correctness, pointed pseudotriangulations (combinatorics, rigidity, universality, expansiveness, extremeness), linear equilateral trees can’t lock, unfolding 4D chains.

Detailed Descriptions of Lecture 13 and Class 13 (PDF)

Notes for Lecture 13 (PDF - 6.8MB)

Slides for Lecture 13 (PDF - 17.8MB)

Video of Lecture 13

Notes for Class 13 (PDF - 3.5MB)

Slides for Class 13 (PDF)

Video of Class 13

14

Hinged dissections: Animations, polyform inductive construction, rectangle to rectangle, furniture, pseudopolynomial construction, 3D, Dehn invariant.

Detailed Descriptions of Lecture 14 and Class 14 (PDF)

Notes for Lecture 14 (PDF - 7.3MB)

Slides for Lecture 14 (PDF)

Video of Lecture 14

Notes for Class 14 (PDF - 2.6MB)

Slides for Class 14 (PDF - 2.2MB)

Video of Class 14

15

Polyhedron unfolding: Handles, holes, ridge trees; sun unfolding; zipper unfolding; more ununfoldable polyhedra; NP-completeness of edge unfolding; band unfolding; continuous blooming.

Detailed Descriptions of Lecture 15 and Class 15 (PDF)

Notes for Lecture 15 (PDF - 7.0MB)

Slides for Lecture 15 (PDF)

Video of Lecture 15

Notes for Class 15 (PDF - 3.6MB)

Slides for Class 15 (PDF - 2.8MB)

Video of Class 15

16

Polyhedron unfolding: Topologically convex vertex-ununfoldable polyhedron, unfolding orthogonal polyhedra with quadratic refinement.

Detailed Descriptions of Lecture 16 and Class 16 (PDF)

Notes for Lecture 16 (PDF - 6.8MB)

Slides for Lecture 16 (PDF - 1.4MB)

Video of Lecture 16

Notes for Class 16 (PDF - 2.4MB)

Slides for Class 16 (PDF - 1.5MB)

Video of Class 16

17

Polyhedron folding: Pita forms, D-forms, seam forms, convex hull and crease properties, rolling belts, Burago-Zalgaller folding into nonconvex polyhedra.

Detailed Descriptions of Lecture 17 and Class 17 (PDF)

Notes for Lecture 17 (PDF - 7.0MB)

Slides for Lecture 17 (PDF)

Video of Lecture 17

Notes for Class 17 (PDF - 2.6MB)

Slides for Class 17 (PDF)

Video of Class 17

18

Watch origami documentary Between The Folds.

Detailed Descriptions of Lecture 18 and Class 18 (PDF)

Notes for Lecture 18 (PDF - 5.7MB)

Slides for Lecture 18 (PDF - 1.1MB)

Video of Lecture 18

No class resources for this session
19

Polyhedron refolding: Fractal unfolding, three boxes, flat boxes.

Kinetic sculpture: Theo Jansen’s Strandbeests, Arthur Ganson.

Detailed Descriptions of Lecture 19 and Class 19 (PDF)

Notes for Lecture 19 (PDF - 7.5MB)

Slides for Lecture 19 (PDF - 1.1MB)

Video of Lecture 19

Notes for Class 19 (PDF - 1.6MB)

Slides for Class 19 (PDF - 1.7MB)

Video of Class 19

20

3D linkage folding: ribosomes, HP protein folding NP-hardness, flattening is strongly NP-hard, flips, flipturns, deflations, pops, popturns.
This class covers both L20, and L21 (optional).

Detailed Descriptions of Lectures 20 and 21, and Class 20 (PDF)

Notes for Lecture 20 (PDF - 9.4MB)

Slides for Lecture 20 (PDF - 1.2MB)

Video of Lecture 20

Notes for Lecture 21 (PDF - 8.4MB)

Slides for Lecture 21 (PDF)

Video of Lecture 21

Notes for Class 20 (PDF - 5.7MB)

Slides for Class 20 (PDF - 1.3MB)

Video of Class 20

21–24 Student presentations No lecture resources for these sessions No class resources for these sessions

Course Info

Fall 2012
Learning Resource Types
Lecture Videos
Problem Sets with Solutions
Projects
Instructor Insights