Geometric Folding Algorithms: Linkages, Origami, Polyhedra

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

This lecture continues with open problems involving general unfoldings of polyhedra and proof of vertex unfolding using construction of facet-paths. Approaches for unfolding orthogonal polyhedra, grid unfolding, and folding convex polyhedra are presented.

This class begins with defining handles and holes, and the Gauss-Bonnet Theorem applied to convex polyhedra. Algorithms for zipper unfolding, edge ununfoldable polyhedra, square-packing, band unfolding, and blooming of convex polyhedra are presented.

This class reviews covers topologically convex vertex-ununfoldable cases and unfolding for orthogonal polyhedra, including the approach of heavy-light decomposition. The class also reviews Cauchy's Rigidity Theorem.

This lecture introduces adorned chains and slender chains. Proofs involving these definitions, as well as locked polygons and hinged dissections, are presented.

This class focuses on hinged dissections. Examples of hinged dissections and several built, reconfigurable applications are offered Pseudopolynomials, triangulation, and 3D dissections are then discussed.

This class reviews Carpenter's Rule and properties of pseudotriangulation. Various proofs are presented, which cover topics including non-zero stresses, linear and equilateral locked trees, and unfolding of 4D chains.

This lecture covers infinitesimal rigidity and motion, and tensegrity systems as an extension of rigidity theory. The rigidity matrix, equilibrium stress, and duality are introduced, and a proof to Carpenter's Rule Theorem is presented.

This lecture begins with a review of linkages and classifying graphs as generically rigid or flexible. Conditions for minimally generic rigid graphs are presented with degree-of-freedom analysis. Proofs of Henneberg and Laman characterizations are given.

This class covers how the pebble algorithm works with first a proof of the 2k property, and then 2k-3. Generic rigidity and the running time of the algorithm is discussed, and software simulations running the algorithm are shown.

This class covers several examples of tensegrity structures and in Freeform software. A question on linear programming's application to the motions and stresses is addressed.

This class presents open problems involving holes, sliding linkages, and generalizations of Kempe. A proof for the semi-algebraic sets for Kempe is presented and various origami axioms are given. The class ends with a continuation of hypar folding.

This lecture begins by defining folding motion by a series of folded state, linkages, graphs, and configuration space. A proof of Kempe's Universality Theorem is presented along with Kempe's gadgets, and also the Weierstrass Approximation Theorem.

This lecture introduces the hyperboloic paraboloid, hyparhedra, and the circular pleat. Topics include triangulated folding of the hypar, how paper folds between creases, and Gaussian curvature. Various proofs involving straight creases are given.

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.

This class begins with a demonstration of software for fold and cut. Odd-degree vertices, and a comparison of skeleton method and tree method are discussed. Clarifications on the disk-packing method with a definition for the number of disks are given.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

This class introduces the inverted structure of the class, the topics covered in the course and its motivation. Examples of applications are provided, along with types and characterizations of geometric objects, and foldability and design questions.

This lecture introduces the topics covered in the course and its motivation. Examples of applications are provided, types and characterizations of geometric objects, foldability and design questions, and results. Select open problems are also introduced.