This lecture begins with a problem involving unfolding and refolding. Examples of smooth foldings and unfoldings are given, followed by a problem involving the wrapping of a sphere and a proof that the wrappings are contractive.

Lecture 19: Refolding & Smooth Folding

This lecture presents the hydrophobicity (HP) model for protein folding and optimization using parity cases. Interlocked 3D chains are presented through a table depicting different cases for combinations of open and closed chains.

Lecture 21: HP Model & Interlocked Chains

This lecture begins with how to construct a gluing tree. Combinatorial bounds and algorithms are proved for gluing results, which include the general case, edge-to-edge, and bounded sharpness. The different gluings for the cross are also shown.

Lecture 18: Gluing Algorithms

This class introduces recent research on flattening fixed-angle chains and addresses flipping of pockets in a polygon. Flaws and omissions in proofs on a bounding number of flips are presented along with a correct version of Bing and Kazarinoff's proof.

Class 20: 3D Linkage Folding

This class first covers research findings involving common unfoldings of boxes. Several examples of kinetic sculptures and machines are shown, including Theo Jansen's Strandbeests and Arthur Ganson's works at the MIT Museum.

Class 19: Refolding & Kinetic Sculpture

This class introduces the pita form and Alexandrov-Pogorelov Theorem. D-forms are discussed with a construction exercise, followed by a proof that D-form surfaces are smooth and are the convex hull of the seam. Rolling belts are addressed at the end.

Class 17: D-Forms

This lecture focuses on the folding of the backbone chain of proteins in relation to fixed-angle linkages. Four problems types (span, flattening, flat-state connectivity, locked) are presented, followed by the canonicalization of a producible chain.

Lecture 20: Protein Chains

This lecture addresses the mathematical approaches for solving the decision problem for folding polyhedra. A proof of Alexandrov's Theorem and later a constructive version of Alexandrov is presented. Gluing trees and rolling belts are introduced.

Lecture 17: Alexandrov's Theorem

This lecture begins with describing polyhedron unfolding for convex and nonconvex polygons. Algorithms for shortest path solutions and unfoldings are presented along with how to determine whether an edge unfolding exists.

Lecture 15: General & Edge Unfolding

This lecture explores algorithms for unfolding 2D chains including ODE, pointed pseudotriangulations, and the energy method. Locking rules are then extrapolated to address Spherical Carpenter's Rule, infinitesimally locked linkages, and locked 3D chains.

Lecture 13: Locked Linkages

Geometric Folding Algorithms: Linkages, Origami, Polyhedra

I was browsing classes MIT and I kind of accidentally pressed this one, I&#39;m really glad I did I watch the whole first lecture of Erik giving the overview for the class and it is fascinating. I did not know how interesting something like this is. I&#39;ll watch a video a day and look forward to seeing some of the class results from experiments. Definitely recommend checking this out.

Geo unfolding class

This class covers creases in context of smoothness and a proof from the lecture involving Taylor expansion. Algorithms for the numbers of folding operations necessary for an MV string are presented. The class ends with a hypar folding exercise.

Class 9: Pleat Folding

This lecture presents the fold and cut problem, and both the straight skeleton method and disk-packing method. Simple fold and cut is then generalized for folding surface of polyhedra.

Lecture 8: Fold & One Cut

This class begins with a folding exercise and demonstration involving Origamizer. A high-level overview of the mathematical constraints for Freeform and Rigid software are presented, followed by examples of origami robots and current open problems.

Class 6: Architectural Origami

This class begins with several examples of box-pleating and maze-folding. Clarifications on NP-hardness are provided with a walkthrough of a proof. Additional folding gadgets are introduced and non-simple folds are addresed.

Class 7: Origami is Hard

This lecture presents both the traditional and non-traditional style of origami with many visual examples. The lecture then moves onto applying tree method to design origami, and includes a design example for a crab using TreeMaker.

Lecture 5: Artistic Origami Design

This lecture introduces universal hinge patterns with the cube and maze gadget. NP-hardness problems involving partition and satisfiability are presented with examples of simple folds, global flat foldability, and disk packing.

Lecture 7: Origami is Hard

This class introduces more examples of origami models that use a variety of techniques and media. At the end of the session, the class participates in a folding exercise that uses business cards to make modular cubes.

Class 5: Tessellations & Modulars

This lecture presents Origamizer, freeform origami, and rigid origami applied to architectural and three-dimensional design contexts. Geometric constraints, demonstration videos, and physical models are shown for each portion of the lecture.

Lecture 6: Architectural Origami

This class begins with folded examples produced by TreeMaker and Origamizer. Explanation of the triangulation algorithm, checkerboard folding, the Lang Universal Molecule, and Origamizer tucking molecules are offered.

Class 4: Efficient Origami Design

This lecture continues to discuss the tree method and characterizing a uniaxial base. Another algorithm, Origamizer, is presented with introductory examples of folding a cube, checkerboard, and arbitrary polyhedra.

Lecture 4: Efficient Origami Design

This lecture begins with definitions of origami terminology and a demonstration of mountain-valley folding. Turn, hide, color reversal gadgets, proofs for folding any shape, Hamiltonian refinement, and foldability with 1D flat folding are presented.

Lecture 2: Simple Folds

This class reviews algorithms for testing flat-foldability for a 1D MV pattern and for single-vertex MV pattern. An exercise walks through determining local flat-foldability, and questions cover higher dimensions and motivations for flat-foldability.

Class 3: Single-Vertex Crease Patterns

This lecture explores the local behavior of a crease pattern and characterizing flat-foldability of single-vertex crease patterns. Kawasaki's theorem and Maekawa's theorem are presented as well as the tree method with Robert Lang's TreeMaker.

Lecture 3: Single-Vertex Crease Patterns

This class begins with a folding exercise of numerical digits. Questions discussed cover strip folding in the context of efficiency, defining pseudo-polynomial, seam placement, and clarifications about simple folds and flat-foldability.

Class 2: Universality & Simple Folds

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