# Geometric Folding Algorithms: Linkages, Origami, Polyhedra

### Course Description

This course focuses on the algorithms for analyzing and designing geometric foldings. Topics include reconfiguration of foldable structures, linkages made from one-dimensional rods connected by hinges, folding two-dimensional paper (origami), and unfolding and folding three-dimensional polyhedra. Applications to architecture, robotics, manufacturing, and biology are also covered in this course.

This is an advanced class on computational geometry focusing on folding and unfolding of geometric structures including linkages, proteins, paper, and polyhedra. Examples of problems considered in this field include:

- What forms of origami can be designed automatically by algorithms?
- What shapes can result by folding a piece of paper flat and making one complete straight cut?
- What polyhedra can be cut along their surface and unfolded into a flat piece of paper without overlap?
- When can a linkage of rigid bars be untangled or folded into a desired configuration?

Many folding problems have applications in areas including manufacturing, robotics, graphics, and protein folding. This class covers many of the results that have been proved in the past few years, as well as the several exciting open problems that remain open.

Erik Demaine. 6.849 Geometric Folding Algorithms: Linkages, Origami, Polyhedra, Fall 2012. (Massachusetts Institute of Technology: MIT OpenCourseWare), http://ocw.mit.edu (Accessed 25 Jan, 2015). License: Creative Commons BY-NC-SA
This curved-crease sculpture, created for the opening of the National Museum of Mathematics, demonstrates the intersection of origami, design, and mathematics that is at the heart of this course. (Erik Demaine and Martin Demaine.)

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### Video Lectures & Study Materials

Visit the official course website for more study materials: http://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-849-geometric-folding-algorithms-linkages-origami-polyhedra-fall-2012/

# Lecture Play Lecture
1 Lecture 1: Overview (1:19:50) Play Video
2 Class 1: Overview (50:52) Play Video
3 Lecture 2: Simple Folds (1:23:07) Play Video
4 Class 2: Universality & Simple Folds (53:05) Play Video
5 Lecture 3: Single-Vertex Crease Patterns (1:23:24) Play Video
6 Class 3: Single-Vertex Crease Patterns (46:25) Play Video
7 Lecture 4: Efficient Origami Design (1:26:28) Play Video
8 Class 4: Efficient Origami Design (49:20) Play Video
9 Lecture 5: Artistic Origami Design (1:26:22) Play Video
10 Class 5: Tessellations & Modulars (38:30) Play Video
11 Lecture 6: Architectural Origami (1:18:15) Play Video
12 Class 6: Architectural Origami (45:35) Play Video
13 Lecture 7: Origami is Hard (1:22:04) Play Video
14 Class 7: Origami is Hard (49:35) Play Video
15 Lecture 8: Fold & One Cut (1:20:57) Play Video
16 Class 8: Fold & One Cut (49:59) Play Video
17 Lecture 9: Pleat Folding (1:22:59) Play Video
18 Class 9: Pleat Folding (46:06) Play Video
19 Lecture 10: Kempe's Universality Theorem (1:18:19) Play Video
20 Class 10: Kempe's Universality Theorem (44:08) Play Video
21 Lecture 11: Rigidity Theory (1:24:39) Play Video
22 Class 11: Generic Rigidity (50:23) Play Video
23 Lecture 12: Tensegrities & Carpenter's Rules (1:24:23) Play Video
24 Class 12: Tensegrities (18:02) Play Video
25 Lecture 13: Locked Linkages (1:21:42) Play Video
26 Class 13: Locked Linkages (51:33) Play Video
27 Lecture 14: Hinged Dissections (1:21:26) Play Video
28 Class 14: Hinged Dissections (52:48) Play Video
29 Lecture 15: General & Edge Unfolding (1:20:50) Play Video
30 Class 15: General & Edge Unfolding (50:53) Play Video
31 Lecture 16: Vertex & Orthogonal Unfolding (1:22:08) Play Video
32 Class 16: Vertex & Orthogonal Unfolding (41:30) Play Video
33 Lecture 17: Alexandrov's Theorem (1:21:24) Play Video
34 Class 17: D-Forms (43:44) Play Video
35 Lecture 18: Gluing Algorithms (1:21:11) Play Video
36 Lecture 19: Refolding & Smooth Folding (1:25:19) Play Video
37 Class 19: Refolding & Kinetic Sculpture (31:21) Play Video
38 Lecture 20: Protein Chains (1:19:13) Play Video
39 Class 20: 3D Linkage Folding (38:47) Play Video
40 Lecture 21: HP Model & Interlocked Chains (1:09:46) Play Video