[Erik holding a box-pleated MIT]
sample frame from lecture video (L05)

6.849: Geometric Folding Algorithms: Linkages, Origami, Polyhedra (Spring 2017)

Prof. Erik Demaine; Martin Demaine; Dr. Jason Ku; TAs Adam Hesterberg & Jayson Lynch

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Video Lectures

Cxx indicate class sessions / contact hours, where we solve problems related to the listed video lectures.
Lxx indicate video lectures from Fall 2010 (with a different numbering).
The Lxx videos are required viewing before attending the Cxx class listed above them.
Extra Videos are optional extra videos from Fall 2012 (with a different numbering), if you want to know more

C01 Feb. 8 [+] Class format: Inverted lectures, problem solving, requirements.
Content overview: Applications to robotics, graphics, mechanics, manufacturing, medical, aerospace, biology, sculpture, architecture. Linkages, paper, polyhedra. Foldability/analysis vs. design/synthesis. Universality, decision, and hardness results. Sample results.
[Notes] [Slides] [Extra Video]

In our first class, I'll give a whirlwind tour of the entire class in about an hour, talk about the format of the class, and then we'll solve some problems! In general, the format of the class will focus on problem solving — both on solving solved problems (like you would on a pset) but collaboratively in class, and on solving unsolved problems that no one knows the answer to. In addition to mathematical problems, we'll have design and programming problems/challenges.

To facilitate problem solving, we'll be using (open-source) software I wrote called Coauthor. To that end:

  1. Please sign up for an account now here: https://coauthor.csail.mit.edu/6.849-2017/
  2. Please fill out this survey so that we can give you permission to use Coauthor, and get a sense of who is taking the class: https://goo.gl/forms/QzEyhK2lhcHcUZy33
  3. Please bring a laptop (or tablet) to class.

In this first class, we'll work on problems related to strip folding.

C02 Feb. 15 [+] Review: Simple origami: Simple folds, strip folding, single-vertex crease patterns.
NP-hardness: Reductions, weak vs. strong, ruler folding, simple folding.
[Notes] [Slides]

I'll give an introduction to NP-hardness — roughly what it means (likely no efficient algorithm to solve a problem) and how to prove problems NP-hard (by reductions). In particular, I'll show NP-hardness proofs for

  • Ruler Folding (fold a 1D hinge pattern down to a given length), and
  • Simple Folding (fold a given crease pattern by a sequence of simple folds).

We'll also distinguish between weak and strong NP-hardness, which depends on how large the numbers in the problem are allowed to get.

Then we'll work on problems related to simple folds.

L01 [+] Origami intro: Piece of paper, crease pattern, mountain-valley assignment.
Universality: Folding any shape (silhouette or gift wrapping).
Simple folds: 1D flat-foldability characterization, 2D map-folding algorithm.
[Notes] [Slides] [Video]
[Extra Video]
This lecture kicks off a series of lectures about origami. It focuses on a relatively simple kind of origami, called “simple folds”, which involve folding along one straight line by ±180 degrees. These are some of the most “practical” types of folds from an automated manufacturing standpoint, are well-studied mathematically, and a good warmup before we get to complex origami folds which fold many creases at once.

On the design side, we'll see how simple folds are enough to fold any 2D shape, and with slightly more general folds, we can fold any 3D shape even with a two-color pattern on the surface.

On the foldability side, we'll see how to efficiently determine whether a crease pattern with mountains and valleys indicated can be folded flat in two interesting cases: 1D pieces of paper (in other words, parallel creases in a strip of paper) and 2D rectangular maps.

L02 [+] Single-vertex crease patterns: Characterizations of flat-foldable crease patterns and mountain-valley patterns, combinatorics of the latter, local flat foldability is easy.
Tree method of origami design: Introduction, uniaxial base, demo.
[Notes] [Slides] [Video]
[Extra Video]
This lecture is about the local behavior of flat folding around each vertex of a crease pattern. In other words, we study each vertex individually, by characterizing all single-vertex crease patterns and mountain-valley patterns that are flat foldable. Then we look at how to combine multiple vertices into a "locally foldable" crease pattern.

We also get started on the tree method of origami design, developed by many Japanese origami designers over the years, and turned into an algorithm and computer program TreeMaker by Robert Lang. This method has been the most successful in transforming complex origami design, and we'll cover more of it next lecture.

C03 Feb. 22 [+] NP-hardness: Flat foldability, disk packing.
Origami design: TreeMaker, Origamizer, cube/sphere wrapping.
[Notes] [Slides]

I'll show (at a pretty high level) two more proofs of NP-hardness:

  1. flat foldability of deciding whether a given crease pattern, or a given mountain-valley pattern
  2. disk packing, which corresponds to the tree method for star trees

I'll also mention some new results in cube/sphere wrapping.

Then we'll work on problems related to seam placement.

L03 [+] Efficient origami design: Tree method, TreeMaker, uniaxial base, active path, rabbit-ear molecule, universal molecule, Margulis Napkin Problem; cube folding, checkerboard folding; Origamizer, watertight, tuck proxy. [Notes] [Slides] [Video]
[Extra Video]
This lecture is all about efficient origami design. We saw in Lecture 1 how to fold anything impractically. Now we'll see how to fold many shapes practically.

First up is the tree method, whose software implementation TreeMaker I demoed at the end of Lecture 2. I'll describe how it lets us fold an optimum stick-figure (tree) origami base, although computing that optimum is actually NP-complete (as we'll see in Lecture 6). This algorithm is used throughout modern complex origami design; I'll show some examples by Robert Lang and our own Jason Ku.

Second we'll look at a simple, fully understood case: the smallest square to fold a cube.

Third we'll look at a classic problem that we made progress on recently: folding an n × n checkerboard from the smallest bicolor square.

Finally we'll look at the latest and most general method, Origamizer, for folding any polyhedron reasonably efficiently. Here we don't have a nice theoretical guarantee on optimality, but the method works well in practice, provably always works, and has other nice features such as watertightness.

L04 [+] Architectural Origami: Origamizer, Freeform Origami, Rigid Origami Simulator, cylindrical origami, thick origami
(guest lecture by Tomohiro Tachi)
[Slides] [Video]
[Extra Video]
Origami, the art of folding a sheet of paper into various forms without stretching or cutting, can be applied to structural engineering and design purposes. The applications include the forming of a 3D surface without assembling multiple parts and the construction of kinetic structures such as retractable roofs, openings, temporary shelters, and space structures. In the design process of such applied origami, it is very difficult for the designer to control the form to fit design contexts while preserving the necessary functionalities of the original patterns. Therefore, without sufficient knowledge or intelligent design systems, the resulting designs would end up in either just a mere copy and paste of an existing origami pattern or an "origami inspired" design which is not using the properties of origami in functional ways.

This lecture will present my recent studies on computational origami algorithms and interactive systems to enable architectural designs. The topics include the algorithm for origamizing arbitrary polyhedral surfaces, freeform variation method of different types of origami patterns, and rigid origami theory, design, and physical implementation. The proposed systems are freely available from my website (http://www.tsg.ne.jp/TT/software/): Rigid Origami Simulator, Origamizer, and Freeform Origami. These systems enable origami designs for architecture, i.e., architectural origami, as well as an origami bunny.

C04 Mar. 1 [+] Specialized origami design: Origami art, universal hinge patterns, fold and one cut.
Guest Lecture: Chuck Hoberman
[Notes] [Slides]

We'll have our first (of several) guest lectures: Chuck Hoberman. Currently teaching at Harvard GSD, Chuck is a brilliant mechanism designer, famous for the expanding Hoberman sphere and many architectural installations. See some of his work.

Then we'll work on problems related to fold & cut and extrusion folding.

L05 [+] Artistic origami design: sampling of representational origami art; tree method and its use.
(guest lecture by Jason Ku)
[Slides] [Video]
[Extra Video]
This is a guest lecture by Jason Ku, president of OrigaMIT (the MIT origami club), PhD student in mechanical engineering, and prominent origami designer.

His lecture will be about his perspectives on artistic origami design. Lecture 3 outlined the tree method algorithm of origami design. Now we'll see how this method applies in real-world examples.

The first half of the lecture will introduce the artistic side of representational origami. In origami, as in many disciplines, familiarity with the canon of work that already exists can be quite useful in understanding avenues for future creative development. Thus we'll cover the works and styles of a sampling of the world's most renowned paper folders.

The second half of the lecture will focus on the actual design of representational origami art. We will briefly review tree theory, weigh the pros and cons of this design method, and emphasize the relationships between a tree, a circle/river packing, and the locus of possible hinge crease in a uniaxial base. Finally, we'll go through the process of designing an origami model with, then without, the help of TreeMaker.

L06 [+] Universal hinge patterns: box pleating, polycubes; orthogonal maze folding. (first 25 minutes of video)
OPTIONAL — NP-hardness: introduction, reductions; simple foldability; crease pattern flat foldability; disk packing (for tree method).
[Notes] [Slides] [Video]
[Extra Video]
This lecture covers two main topics:

First, continuing our theme from Lecture 3 on efficient origami design, we'll see how subsets of a single hinge pattern are enough to fold any orthogonal shape made up of cubes, whereas other approaches use a completely different set of creases for each origami model you want. In general, we can fold n cubes from an O(n) × O(n) square of paper. In the special case of “orthogonal mazes”, we can waste almost no paper, with the folding only a small constant factor smaller than the original piece of paper. You can try out this yourself.

Second, we'll see a few ways in which origami is hard. Specifically, I'll give a brief, practical introduction to NP-hardness, and prove three origami problems NP-hard:

  • folding a given crease pattern via a sequence of simple folds;
  • flat folding a given crease pattern (using any folded state);
  • optimal design of a uniaxial base, even when the tree is just a star.
L07 [+] Fold and one cut: history, straight-skeleton method, disk-packing method, simple folds, higher dimensions, flattening polyhedra. [Notes] [Slides] [Video]
[Extra Video]
This lecture is about my first work in computational origami: folding a piece of paper flat so that one complete straight cut makes a desired pattern of cuts (and resulting polygonal shapes). The problem has a long history (back to the 1700s) and possible applications to airbag folding through a problem called flattening. We'll see two different methods for this problem, each with connections to the tree method of origami design: the first generalizes the universal molecule to nonconvex polygons, but loses the ability to control the shadow tree; the second uses disk packing (but no rivers) and universal molecules for triangles and quadrangles. I'll also talk about a brand new result that started from this class three years ago: what shapes can you make only with simple folds?
C05 Mar. 8 [+] Advanced origami: Pleat folding, curved creases.
Linkage design: Kempe.
Guest Lecture: Duks Koschitz
[Notes] [Slides]

We'll have Duks Koschitz (from Pratt) giving a guest lecture about David Huffman's curved crease origami, which was the subject of his PhD thesis at MIT in Dept. of Architecture. Huffman was a pioneer of curved-crease folding, with beautiful work; he was also an MIT student and professor. His work was recently acquired by MIT Museum and is being cataloged.

Then we'll work on problems related to fold & cut.

L08 [+] Pleat folding: “Hyperbolic paraboloid”, circular pleat, self-folding origami, plastic deformation and elastic memory; triangulation, interval arithmetic; how paper folds between creases, ruled surfaces, torsal, straight creases stay straight, polygons stay flat; nonexistence.
Inflation: Teabag problem, inflating polyhedra.
Curved creases: Recreating David Huffman's curved-crease folding.
[Notes] [Slides] [Video]
[Extra Video]
This lecture is primarily about pleat folding, specifically the so-called “hyperbolic paraboloid” model common in origami and originating at the Bauhaus in the late 1920s. Surprisingly, this origami model does not exist: it's impossible to fold anything nonflat with a crease pattern consisting of concentric squares and diagonals. We'll prove this, using a new theory for how paper folds in between creases, in particular showing that straight creases stay straight when folded, and interior polygons stay flat when folded. By contrast, we'll show how the “hyperbolic paraboloid” folds fine after adding specific diagonal creases to the trapezoidal faces.

Afterward, we'll briefly look at a couple of related topics: how to maximally inflate a teabag or other polyhedral surface by folding, and what kind of foldings are possible with curved creases. On the latter topic, I'll overview a big ongoing project in which we're trying to reconstruct the mathematics underlying a beautiful series of sculptures folded by David Huffman from the 1970s–1990s.

L09 [+] Folding motions: folded states vs. motions; universal foldability of polygonal paper.
Linkages to sign your name: graphs vs. linkages vs. configurations, configuration space; Kempe Universality Theorem, original proof, bug, corrections, generalizations and strengthenings.
[Notes] [Slides] [Video]
[Extra Video]
This lecture kicks off the beginning of linkage folding. We'll segue from origami by thinking about folding motions of pieces of paper, and prove that it's always possible to reach any folded state of a polygonal piece of paper. How many extra creases do we need? This unsolved question leads to problems in the linkage-folding domain.

Then we'll focus on Kempe's Universality Theorem: there's a linkage to sign your name, or more mathematically, trace any polynomial curve. This proof introduces a bunch of cool gadgets, in particular to add angles and multiply (or divide) angles by constant factors.

C06 Mar. 15 [+] Rigidity: Generic, infinitesimal, tensegrity.
Linkage universality: Carpenter's Rule Theorem.
Guest Lecture: Tom Hull
[Notes] [Slides]

Prof. Tom Hull is visiting from Western New England University, speaking about the mathematics of self-folding and rigid origami. Tom is one of the original origami mathematicians — he and I were PhD students at the same time (though in different countries), and he introduced me to all the people and papers in the field.

Then we'll work on problems related to rigidity.

L10 [+] Rigidity theory: Rigidity, generic rigidity, minimal generic rigidity, Henneberg characterization, Laman characterization, polynomial-time algorithm, convex polyhedra. [Notes] [Slides] [Video]
[Extra Video]
This lecture is about rigidity theory, which is about telling when a linkage can fold at all. This field goes back to mechanical engineering of the 18th and 19th centuries, with applications to structural engineering and architecture (getting buildings and bridges to stand up), biology (understanding which parts of a protein still move after folding up), and linkage folding itself (beyond just “does it move at all?”, as we'll see in the next lecture).

We'll cover two main theorems characterizing “generically” rigid graphs in 2D. Henneberg's Theorem, from 1911, gives a nice and direct characterization, but it's hard to turn into an algorithm. Laman's Theorem, from 1970, is intuitively harder to work with, but turns into a fast (quadratic-time) algorithm.

Unfortunately, this is all just for 2D, and we don't know any good characterizations for generic rigidity in 3D. I'll briefly describe some nice theorems about the rigidity of convex polyhedra in 3D, though, which in particular explain why Buckminster Fuller's geodesic domes stand up.

L11 [+] Rigidity theory: Infinitesimal rigidity, rigidity matrix.
Tensegrity theory: tensegrities, equilibrium stress, duality, polyhedral lifting, Maxwell-Cremona Theorem.
Locked linkages: Carpenter's Rule Theorem.
[Notes] [Slides] [Video]
[Extra Video]
This lecture continues our tour through rigidity theory, introducing two more big ideas: infinitesimal rigidity and tensegrities. Infinitesimal rigidity is a nice way to capture the generic case using linear algebra, and it nicely generalizes (and is efficiently computable) in any dimension. Tensegrities (a term coined by Buckminster Fuller) allow the addition of struts (which prevent compression) and cables (which prevent expansion) in addition to bars (which prevent both—which is what we've mostly been thinking about). A nice special case of tensegrities are “spider webs”, which turn out to relate to the algorithmic design of origami tessellations.

Although not obviously related, the rigidity tools we build up allow us to prove the existence of actual folding motions between any two configurations of “chain linkages” (whose graphs are paths or cycles) whose edges aren't allowed to cross. This Carpenter's Rule Theorem (which was essentially my PhD thesis) kicks off our coverage of understanding when linkages can “lock”.

C07 Mar. 22 [+] Linkage universality: Locked linkages and unfolding algorithms.
Hinged dissections: Universality.
Guest Lecture: Samuel Felton
[Notes] [Slides]

This class we have Prof. Samuel Felton from Northeastern University guest lecturing about folding robots. Sam is an expert in folding robots; in particular, he was the lead author on the self-folding robot from Science (which was his PhD work at Harvard). It should be fun!

Then we'll work on problems related to locked linkages.

L12 [+] Locked linkages: Algorithms for unfolding 2D chains, pseudotriangulation, energy; rigid folding of single-vertex origami; locked trees, infinitesimally locked linkages, Rules 1 and 2; locked 3D chains, knitting needles. [Notes] [Slides] [Video]
[Extra Video]
This lecture is about locked linkages. Continuing on from the Carpenter's Rule Theorem from last lecture, which says that 2D chains can't lock, we'll see three different algorithms for folding 2D chains. Each algorithm has varying levels of expansiveness, symmetry, and efficiency, with applications to 2D robot-arm motion planning. We'll also see an application of a spherical version of the Carpenter's Rule Problem to rigid folding of single-vertex origami. Then we'll tour the world of locked 2D trees, which has had significant progress recently. To this end, I'll describe the extensive technology for proving 2D linkages to be locked. Finally we'll briefly look at locked 3D chains, which relates to protein folding.
L13 [+] Hinged dissections: Locked and unlocked chains of planar shapes, adorned chains, slender adornments, slender implies not locked, Kirszbaun's Theorem, locked triangles with apex angle > 90°; existence of hinged dissections, refinement. [Notes] [Slides] [Video]
[Extra Video]
This lecture is about chains of polygons, or other 2D shapes, connected together by hinges in 2D. These structures are classically called “hinged dissections”. On the foldability side, we'll see some surprisingly general situations, called “slender adornments”, where these chains cannot lock, building on the Carpenter's Rule Theorem and expansiveness (Lecture 11). We'll also see some examples that do lock, building on our theory of infinitesimally locked linkages and Rules 1 and 2 (Lecture 12). On the design side, we'll show that we can actually design hinged dissections that fold into any finite collection of desired polygonal shapes, using slender adornments to guarantee foldable motions.
C08 Apr. 5 [+] Polyhedron unfolding.
Guest Lecture: Benjamin Parker
[Notes] [Slides]

We will have a guest lecture by Benjamin Parker, one of my favorite artists in the world of origami tessellations. You can see some of his work on his website, http://www.brdparker.com/.

Then we'll work on problems related to polyhedron unfolding.

L14 [+] Polyhedron unfolding: Edge unfoldings, general unfoldings, big questions, curvature, general unfolding of convex polyhedra, source unfolding, star unfolding, edge-unfolding special convex polyhedra, fewest nets, edge-ununfoldable nonconvex polyhedra. [Notes] [Slides] [Video]
[Extra Video]
This lecture begins our coverage of polyhedron (un)folding. Specifically, we'll talk about how to cut open a polyhedral surface into one piece without overlap so that that piece can be cut out of sheet material and assembled into the 3D surface. What distinguishes this field from origami is that we're not allowed to cover any part of the surface more than once. Two of the biggest open problems in geometric folding are here:
  1. Does every convex polyhedron have an edge unfolding? This is also the oldest problem, implicit back to 1525.
  2. Does every polyhedron (without boundary) have a general unfolding? This is one of my favorite open problems.
I'll describe the various attacks and partial results toward solving the first problem, as well as the reverse situations where we know much more: general unfolding of convex polyhedra and edge unfolding of general polyhedra.
L15 [+] Polyhedron unfolding: Vertex unfolding, facet paths, generally unfolding orthogonal polyhedra, grid unfolding, refinement, Manhattan towers, orthostacks, orthotubes, orthotrees.
Polyhedron folding: Cauchy's Rigidity Theorem, Alexandrov's uniqueness of folding.
[Notes] [Slides] [Video]
[Extra Video]
This lecture continues the theme of unfolding polyhedra, and kicks off our coverage of folding polygons into polyhedra.

On the unfolding side, we'll cover “vertex unfolding”, which is a variation on edge unfolding kind of like hinged dissections. We'll prove that this type of unfolding exists, even for nonconvex polyhedra, provided every face is a triangle. Then we'll cover recent breakthroughs in general unfolding, for orthogonal polyhedra.

On the folding side, we'll prove Cauchy's Rigidity Theorem: convex polyhedra have exactly one convex realization (viewing faces as rigid and edges as hinges). Then we'll show how to extend this to Alexandrov's Uniqueness Theorem: if you glue up the boundary of a polygon, there's at most one convex polyhedron you can make. (Next lecture we'll see how to actually get one.)

C09 Apr. 12 [+] Polyhedron folding.
Guest Lecture: Yongquan Lu
[Notes] [Slides]

YQ Lu will give a guest lecture about his work in paper sliceforms, as embodied by his software Sliceform Studio [https://www.sliceformstudio.com/]. See his resulting artwork at https://www.sliceformstudio.com/gallery.html.

Then we'll work on problems related to gluings and sphere wrapping.

L16 [+] Polyhedron folding: Decision problem, enumeration problem, combinatorial problem, nonconvex solution, convex polyhedral metrics, Alexandrov gluings, Alexandrov's Theorem, Bobenko-Izmestiev constructive proof, pseudopolynomial algorithm, ungluable polygons, perimeter halving, gluing tree, rolling belts. [Notes] [Slides] [Video]
[Extra Video]
This lecture dives into the problem of folding polygons into polyhedra. The focus here is on folding convex polyhedra, though there is one nice result about the nonconvex case by Burago & Zalgaller.

The main tool in this area is called Alexandrov's Theorem, from 1941, which characterizes when a gluing of the boundary of a polygon will result in a convex polyhedron; plus, as we saw last lecture, that convex result is always unique. We'll sketch a proof of this theorem as well as recent algorithms for finding the convex polyhedron.

With this tool in hand, we'll explore some different properties of gluings. Some polygons, in fact "most" in a certain sense, have no Alexandrov gluings. Convex polygons, on the other hand, always do. Some polygons have infinitely many gluings, but this always happens in a controlled way with a few "rolling belts". Along the way we'll see gluing trees, a useful tool for analyzing gluings that we'll use in the next lecture for algorithms to find gluings.

L17 [+] Polyhedron folding: Combinatorial type of gluing, exponential upper and lower bounds for combinatorially distinct gluings, polynomial upper bound for polygons of bounded sharpness, dynamic program for edge-to-edge gluing, including polynomial-time decision, exponential-time dynamic program for general gluing; case studies of Latin cross, equilateral triangle, and square. [Notes] [Slides] [Video]
This lecture continues our discussion of gluing polygons up and folding them into convex polyhedra, namely, via Alexandrov gluings. Now we'll see algorithms to actually find Alexandrov gluings, as well as give good bounds on how many there can be. Then I'll describe a few case studies: the Latin cross, the equilateral triangle, and the square.
C10 Apr. 19 [+] Specialized polyhedron folding. Protein folding.
Jason Ku: Fabricating folded structures, designing crease patterns to satisfy boundary conditions
[Notes] [Slides]

Jason Ku will give a lecture on how to fabricate folded structures using thick materials, and how to design crease patterns that satisfy boundary conditions. You can find both his academic and artistic folding work at http://jasonku.mit.edu/.

Then we'll work problems about boundary conditions, common unfolding, and protein folding.

L18 [+] Polyhedron refolding: Dissection-like open problem, regular tetrahedron to box, Platonic solids to tetrahedra, box to box, polycubes, orthogonal unfoldings with nonorthogonal foldings.
Smooth polyhedron folding: Smooth Alexandrov, D-forms, ribbon curves.
Smooth polyhedron unfolding: Smooth prismatoids.
Smooth origami: wrapping smooth surfaces with flat paper, Mozartkugel, contractive mapping, Burago-Zalgaller Theorem (crinkling/crumpling), stretched path, stretched wrapping, source wrapping, strip wrapping, petal wrapping, comb wrapping, Pareto curve.
[Notes] [Slides] [Video]
[Extra Video]
This lecture is a big collection of fun results related to unfolding, refolding, and smooth folding:
  • common unfolding of a regular tetrahedron and near-cube
  • common unfoldings of Platonic solids and near-regular tetrahedra
  • common unfoldings of boxes of different sizes
  • common unfoldings of many polycubes
  • orthogonal unfoldings with nonorthgonal foldings
  • smooth Alexandrov's theorem, applied to smooth convex shapes (D-forms)
  • unfolding smooth convex polyhedra (prismatoids)
  • wrapping (paper folding) smooth surfaces like spheres, using a new definition of origami, where distances can shrink instead of necessarily staying the same
The last section has practical applications to computational confectionery, reducing the material usage in wrappings of spherical chocolates such as Mozartkugel. Yum!
L19 [+] Protein folding: Fixed-angle linkages, tree, and chains; span; flattening; flat-state connectivity, disconnectivity of orthogonal partially rigid trees, connectivity of orthogonal open chains; locked fixed-angle chains; producible protein (fixed-angle) chains, ribosome, β-producible chains, helix-like canonical configuration, flat states are producible, producible states are connected. [Notes] [Slides] [Video]
[Extra Video]
This lecture is about fixed-angle linkages in 3D, which have the constraint that the angles (in addition to the lengths) must remain fixed at all times. Fixed-angle linkages model the mechanics of chemical bonds between atoms in a molecule. In particular, the backbone of a biological protein is a fixed-angle tree in this model, and can be approximated by a fixed-angle chain. We'll cover several results about fixed-angle linkages, all motivated by questions about protein folding:
  • Span: What's the farthest or nearest you can fold the endpoints of a fixed-angle chain?
  • Flattening: When does a fixed-angle chain have a non-self-intersecting flat state?
  • Flat-state connectivity: When can a fixed-angle chain be folded without self-intersection between every pair of flat states?
  • Locked: When is a fixed-angle linkage locked? In particular, we'll show that all states of a fixed-angle chain producible by a simple model of the ribosome (which includes all flat states) can reach each other.
A lot is known about each question, though many problems remain open.
C11 Apr. 26 [+] Protein folding.
Wrapup Session.
[Notes] [Slides]

This class is the last one in which we solve problems. Specifically, we have a couple of problems related to protein folding in the HP model, and a few about interlocked linkages. In addition, I'll summarize the past and future of the class:

PAST: I'll review what we accomplished in all of our problem solving during the semester.

FUTURE: I'll present the 1-slide summaries of the presentations to come in the final two class sessions (C12 and C13).

L20 [+] Protein folding: HP model of protein folding, NP-hardness and approximation, unique optimal foldings, protein design.
Interlocked 3D chains: Lubiw's problem, unlocking 2-chains, unlocking two 3-chains and 2-chains, interlocking 3-chains, interlocking 3-chain with 4-chain, interlocking 4-chain with triangle, interlocking 3-chain with quadrangle, topological vs. geometric arguments, rigid and fixed-angle variations.
[Notes] [Slides] [Video]
This lecture continues our discussion on protein folding, this time focusing on simple theoretical models of the forces, rather than the mechanics, behind protein folding. In particular, we'll see the HP model, a lattice model capturing the hydrophobia of certain amino, which try to hide from the surrounding water. Finding the optimal folding a given protein is NP-complete, but there are some decent constant-factor approximations, and it's also not known whether protein design is similarly hard. We'll see one step in the direction of design: guaranteeing a unique optimal folding.

Then we'll turn to a fun problem of interlocked linkages. It's known that you need five bars to lock an open chain, but can you interlock multiple chains each with less than five bars? The answer is yes, with a nearly complete characterization of the possibilities. One consequence we'll see is how to cut a chain of n bars into around n/2 pieces that are not interlocked, while n/4 pieces are necessary.

C12 May 3 Student Presentations I
C13 May 10 Student Presentations II


Videography: Martin Demaine Equipment: Video shot with a Canon VIXIA HG21 camera using Manfrotto 701HDV Pro Fluid Video Mini Head on Manfrotto 055XPROB Aluminum Tripod Legs. Audio recorded using Countryman IsoMax E6 EarSet Microphone on camera using Sennheiser EW 112-p G3 wireless microphone system. See our guide to recording video lectures.
Editor: Erik Demaine
Subtitles: OCW