Syllabus

Course Meeting Times

Lectures: 2 sessions / week, 1.5 hours / session

Open Problem Sessions (Optional): 1 session / week, 2 hours / session

Course Description

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

Automated design of new and complex origami, such as
- Freeform Origami, Origamizer, and Rigid Origami Simulator by Tomohiro Tachi
- TreeMaker by Robert J. Lang
Transforming robots by self-folding sheets or chains
How to fold robotic arms without collision
How to bend sheet metal into desired 3D shapes, such as
- Ununfoldable polyhedra with convex faces
- Orthostacks
Understanding how proteins fold

Major progress has 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 course 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.

Format

This year we will be experimenting with an inverted lecture format. Students will be expected to watch recorded, online lectures prior to attending class. In-class time will then be more interactive and dedicated to folding experiments, answering questions, collaborative projects, clarifying proofs, and exploring new results and applications.

We will also organize optional problem-solving sessions, during which we can jointly try to solve open problems in folding. In the past, these sessions have led to important new results and published papers, as well as class projects.

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 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.

Prerequisites

6.046J/18.410J Design and Analysis of Algorithms, or equivalent background in discrete mathematics and algorithms. Alternatively, permission from the instructor.

Textbooks

Required

Demaine, Erik, and Joseph O'Rourke. Geometric Folding Algorithms: Linkages, Origami, Polyhedra. Cambridge University Press, 2007. ISBN: 9780521857574.

Grading

Students will be given a small number of problem sets to complete during the first half of the course.

The other requirement for the course is a project, which can take the form of folding-inspired sculptures; formulations of clean, new open problems; implementations of existing algorithms; or well-written descriptions of one or more papers in the area. Projects can be purely mathematical (geometric) and/or theoretical computer science (algorithmic/complexity theoretic) and/or artistic. Students are required to complete a write-up of the project, and deliver a project presentation.