6.5310: Geometric Folding Algorithms: Linkages, Origami, Polyhedra (Spring 2025)

Prof. Erik Demaine TAs Josh Brunner, Jenny Diomidova

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Overview

the algorithms behind building TRANSFORMERS and designing ORIGAMI

Whenever you have a physical object to be reconfigured, geometric folding often comes into play. This class is about algorithms for analyzing and designing such folds. Motivating applications include

automated design of new and complex origami
using 2D fabrication technology to manufacture complex 3D objects
transforming robots by self-folding sheets or chains
how to fold robotic arms without collision
how to bend sheet metal into desired 3D shapes
understanding how proteins fold

Major progress have been made in recent years in many of these directions, thanks to a growing understanding of the mathematics and algorithms underlying folding. Nonetheless, many fundamental questions remain tantalizingly unsolved. This class covers the state-of-the-art in folding research, including a variety of open problems, enabling the student to do research and advance the field.

Inverted Lectures

This class features inverted lectures: most material is covered in video lectures recorded in 2010 and 2012 (already watched by over 30,000 people), which you can conveniently play at faster speed than real time. In-class time will be focused on collaborative problem solving with your fellow students and the course staff. Particularly unusual is that the problems we'll solve in groups will include a choose-your-own-mix of problem-set style problems with known solutions, design problems for those who love to create, coding problems for those who love programming, and open research problems that no one knows the answer to, with the goal of publishing papers about whatever we discover. (Past offerings of this class have led to over a dozen published papers.) You can work on whatever type of problem most interests you. To facilitate collaboration, we'll be using our open-source software platform called Coauthor, along with GitHub for (optional) coding.

Topics

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:

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.

Textbook and Readings

The textbook for the class is Geometric Folding Algorithms: Linkages, Origami, Polyhedra by Erik Demaine and Joseph O'Rourke, published by Cambridge University Press (2007). It is available for free from MIT libraries.

Additional recommended reading:

Origametry: Mathematical Methods in Paper Folding by Tom Hull (MIT access, via institutional login to MIT)
Origami Design Secrets: Mathematical Methods for an Ancient Art (2nd edition) by Robert Lang (MIT access)
Twists, Tilings, and Tessellations: Mathematical Methods for Geometric Origami by Robert Lang (MIT access)
Introduction to Computational Origami by Ryuhei Uehara (MIT access)

Specifics

Class Time:	Tuesdays and Thursdays at 11:00am–12:30pm (or until 1pm for those who want to stick around)
Class Room:	MIT room 32-082
First Class:	Tuesday, February 4, 2025
Office Hours:	Tuesdays and Thursday at 12:30pm–1pm
Professor:	Erik Demaine, edemaine at mit.edu
TAs:	Josh Brunner, brunnerj at mit.edu Jenny Diomidova, diomidova at mit.edu
Design Advisor:	Alfonso Parra Rubio, aprubio at mit.edu
Staff Email:	65310-staff at csail.mit.edu
Units:	3-0-9
Prerequisites:	6.1220 or equivalent background in discrete mathematics and algorithms. Alternatively, permission from the instructor.
Credit:	EECS, AAGS (Theoretical CS Concentration)

Participating

We welcome both undergraduate and graduate students from all universities. Participation must be in person; there is no online component (beyond the freely available video lectures).

If you are interested in attending the class, for credit or as a listener, please do the following:

Preregister if you're from MIT/Harvard.
Join the 65310-students mailing list.
Sign up for an account on Coauthor.
Sign up for an account on GitHub.com if you don't have one (this is different from github.mit.edu).
Fill out this signup form.

Requirements & Grading

There are six components to the class, some of which are required:

required Watch all video lectures, measured by thumbs-upping completion questions on Coauthor (see your stats).
required Participate during all (or most) in-person classes (email 65310-staff about exceptions)
required Participate in class discussion, measured as coauthoring at least one nontrivial Coauthor post each week. See your statistics view to check for zero weeks.
(Example of a trivial post for a solved problem: "Apply methods from lecture." Example of a nontrivial post for a solved problem: Describing the solution techniques you tried but failed. Deleted and unpublished posts do not count, but private posts do.)
choice Problem sets. Problem sets will be posted weekly.
choice Project write-up and presentation based on in-class work.

We're experimenting with a flexible grading scheme this semester:

20% for watching lectures [required]
40% for in-class participation [required]
50% for problem sets [optional]
50% for a project (or 25% for a small project) [optional]

The percentage weights sum to more than 100%. To compute your overall grade, we will reduce the percentage weight on your lowest-scoring problem set or project (whose weight isn't already zero) until the total weight reaches 100%. This lets you make up for lost points in one component by working more on the other components. In particular, three reasonable approaches to taking this class are to do problem sets and no project (20% + 40% + 50% = 110%, so still plenty of room to make up mistakes); one normal project and no problem sets (also 110%); or one small project and half the problem sets (also 110%). You can also do any combination, including multiple projects.

One additional rule is that you cannot pass the class without both watching most of the lectures and in-class participation most weeks (according to the fairly minimal requirements above), which is the meaning of "required".

Each component will be graded on a 100-point scale, then weighted as described above, then summed to produce your overall score. Your letter grade (A, B, etc.) will be assigned according to this score (where 90+ = A, 80+ = B, 70+ = C, 60+ = D, <60 = F). However, we reserve flexibility in assigning the annotation (+, −, or straight letter) according to other factors such as going above and beyond in participation or just scraping by.

Past and Future

This class was offered in Fall 2020 (as 6.849), Spring 2017 (as 6.849), Fall 2012 (as 6.849), Fall 2010 (as 6.849), Fall 2007 (as 6.885), and Fall 2004 (as 6.885). You might be interested in the lecture notes, problem sets, etc. from those offerings.

required	Watch all video lectures, measured by thumbs-upping completion questions on Coauthor (see your stats).
required	Participate during all (or most) in-person classes (email 65310-staff about exceptions)
required	Participate in class discussion, measured as coauthoring at least one nontrivial Coauthor post each week. See your statistics view to check for zero weeks. (Example of a trivial post for a solved problem: "Apply methods from lecture." Example of a nontrivial post for a solved problem: Describing the solution techniques you tried but failed. Deleted and unpublished posts do not count, but private posts do.)
choice	Problem sets. Problem sets will be posted weekly.
choice	Project write-up and presentation based on in-class work.