6.849: Geometric Folding Algorithms: Linkages, Origami, Polyhedra (Fall 2020)
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Project
The project is the most important requirement of the course.
It can take several forms:

Design/create artwork, furniture, architecture, sculpture, tool,
illustration, or other object based on the ideas in the class.
Your work should be both aesthetically compelling and technically grounded
(though the latter need not be explicitly visible).
You may use any medium you wish, including virtual;
the challenge of working with your format will be taken into consideration.

Theoretical contribution to the field: tackle/solve a problem, formulate interesting open problems or conjectures, etc.
You should not feel pressure in terms of grades to produce results, but you
should spend substantial time thinking and trying to solve the problem.
On the other hand, from past experience, we expect many such results to be
produced during class time, and you are encouraged to turn those results into
your project(s).

Survey a few papers on a related topic (not already wellcovered by the class or textbook).

Design a possible new lecture for a future edition of this class.

Write/record an accessible tutorial for teaching folding to a broad audience.

Substantially improve the Wikipedia articles for several topics related to the class.
Recommended only if you have existing experience editing Wikipedia and with
its guidelines.
In this case, your project writeup and presentation should summarize the
changes you made, why you made the decisions you did, and any challenges you
ran into, in addition to linking to new articles and change diffs.

Implement/visualize one or more algorithms or results,
or make a tool to help explore an open problem,
from this class or on related topics.
We encourage such implementations to be designed as web pages,
with code written in
CoffeeScript
(see our guide
for Python programmers) or
JavaScript.
You are encouraged to relate the final project to your research interests, and
you will not be limited to the topics discussed in class.
Deadlines and Format
 Project proposals are due Thursday, October 15, 2020 at 7pm Eastern,
in place of a regular problem set,
and should be submitted
via Coauthor.
See the forthcoming Problem Set for details.
 The course staff will then give feedback on your proposal, including any
tips and direction we can think of,
and decide whether to approve your project's theme.
 If you decide to change your project direction substantially, it is
required that you check in with the course staff to get feedback/approval
on the new direction.
You should not change the topic of the project, but you may change the form
of the project (such as writing a survey if you fail to bring a
theoretical contribution).
 Subsequent problem sets will mainly consist of
project progress reports to encourage regular progress.
 The project presentation is due on Tuesday, December 1, 2020.
 Because this is earlier than the project paper deadline, you're expected
to have made less progress.
You should at least give a clear description of the problem
you are tackling and what you plan to do.
 You must submit your slides in PDF format,
and you may also demo any software you wrote or objects you built.
 You must also record your presentation and submit it as a link
(Zoom recording or YouTube unlisted video).
See details below.
 By MIT policy, the project paper is due on day of the last
regularly scheduled lecture of this class,
Monday, December 7, 2020.
No extensions are possible.
 This should be a wellwritten document describing the problem
you tackled (be it an artistic, implementation, mathematical, or writing
challenge), what approaches you took, what difficulties you encountered,
and what results you obtained,
in addition to citing the relevant literature.
 Aim for on the order of 10 pages, say in the range 5–20 pages.
 If your project involves writing software, then you should submit the
source code as well; see details below.
Collaboration
Collaboration is strongly encouraged, especially for research
projects—this
is often the key to successful research in theoretical computer science.
You can work in a group of students in the class if you find common interests.
(Students listening to the class will likely have less time to
devote, but they are welcome to participate in a project too.)
In particular, we expect that many projects will naturally grow out of the
collaborations during the interactive portions of class.
We have higher expectations of projects by larger groups.
You are also welcome to collaborate with anyone outside the class,
including your research supervisor (if you have one)
and including the course staff.
The only constraint for the class is that your own contribution should be
substantial enough, both in terms of solving problems and writing it up.
(To evaluate “substantial enough”, talk to the course staff.)
In any case, collaborators should be clearly marked in the project proposal,
paper, and presentation.
If you work on a team, you will be required to
post a short private Coauthor message
(in the appropriate thread)
summarizing your own contributions to the
project(s) you were involved in (when submitting the project).
This will let us ensure that everyone makes substantialenough contributions.
Multiple Projects
You are allowed to work on multiple projects, possibly with different teams.
Just make sure that the total amount of work you do sums to at least one
“person project unit”.
We suggest having one as your main, or one or a small number
to which you will make major contributions, along with any number of
projects to which you will make minor contributions.
We also require that you do some of the writing on at least one project paper,
and participate in at least one of the project presentations.
In particular, we'd like projects tackling/solving open problems
to match papers as much as possible.
For example, instead of working on one project about two small papers, write
two smaller projects. Conversely, instead of doing a project about your
piece of a larger paper, be a part of a large project to write the entire
paper. It's also fine to write up other people's results from open problem
solving if they're OK with it (in particular, if they're not planning to do so
for their class project) — help with writing is an important
contribution to any paper!
Presentation
A project with k presenters will have 8 + 2k minutes
for their presentation.
So:
 1person projects can use 10 minutes
 2person projects can use 12 minutes
 3person projects can use 14 minutes
Presentations are short, so be efficient in what you cover (and be sure to
split the time roughly evenly between speakers). Focus on the problem you
tackled, why it's interesting, and what results you came up with, and don't get
bogged down in details. It's OK to go a little under time,
but do not go over time. Practice
your talk, both to measure and to optimize time spent.
To submit your presentation, attach a PDF document
to the relevant thread on Coauthor,
and then edit that post to have your project's title,
to atmention everyone involved in the project (including yourself),
and to add a link to your presentation recording.
(If you wish, you can also attach a PowerPoint file, or link to Google
Slides or other cloud service, but PDF makes it easy to view the slides
within Coauthor.)
Your presentation recording can be done in one of two ways:
 Easy option: Zoom recording
 Create a Zoom meeting for your project group,
and select "Record to Cloud".
(MIT's Zoom license supports this feature; if you're using a different
Zoom license, make sure it also supports this feature.)
 After you end the meeting, the meeting creator will
get an email about the recording.
 Open the link, and do the following:
 Click the "Share" button, and select "Publicly"
(under "Share this recording").
This will let students from other organizations see your video.
 Click "Copy shareable link" and paste that link into Coauthor.
 Advanced option: YouTube unlisted video
 Record the video how you want (e.g., using
OBS Studio),
 Optionally edit the video how you want
(e.g., using Adobe Premiere).
 Upload the video to your YouTube channel as an Unlisted video.
 Include the direct link to that video on Coauthor.
 Do not attach a video file to Coauthor.
Your uploaded slides and video link will be visible to your fellow students,
so that they can watch your talk as well.
Project Submission
To submit your final project, attach a PDF document to
the relevant thread on Coauthor,
and then edit that post to atmention everyone involved in the project
(including yourself), and to add a link to the source files
(LaTeX, figures, code, etc.).
We require putting all source files into a repository on
our MIT Github organization,
or an Overleaf document with link sharing turned on,
and including a link in your post.
If you're unfamiliar with Github, here are some basic instructions:
 Create a new repository with title similar to your project (if you don't have one already), marked Private.
 If you're already using or know how to use Git,
push your repository to the resulting remote.
 To just upload your files, go to
https://github.com/68492020/YOURREPONAME/upload/master
and drag your files in.
Ideas
Many of the inclass discussions/results are suitable for extension into
projects.
In addition, here is a list of possible project ideas, or seeds of project
ideas, extracted from
Fall 2012 lecture (L) and class (C) notes
and Fall 2010 lecture notes (O).
You can also see some
sample past projects from Fall 2012.
Your project proposal would need to flesh out these ideas into a project scale
(in particular, significantly larger than a problem set, and
in proportion to your group size).
Design/Build/Art
 C06: Design and build a rigid origami structure
 C08: Foldandcut art à la Peter Callesen
 C08: Animate motion for 3D polyhedra flattening
 C09: How does real paper behave when folding a hypar?
 L12: Design and build a tensegrity sculpture
 C14: Design elegant hinged dissections
 C14: Design/build reconfigurable furniture
Coding
 L03: Implement local foldability tester which generates a M/V pattern
 C05: Improve/extend the interface or capabilities of Tess. Possibly 3D animation through interfacing with Rigid Origami Simulator
 C06: Port Tomohiro Tachi's software to platforms other than Windows
 C08: Implement/improve on a foldandcut design tool, ideally including M/V state, degeneracy tool, and folded state.
 C08: Animate motion for 3D polyhedra flattening
 Higher dimension folding visualizer
 C10: Implement Kempe with splines
 L11: JavaScript rigidity/overbracing/pebble algorithm visualization tool
 C11: Improve Henneberg construction/puzzle applet, port to web/JavaScript
 L12: Make a virtual tensegrity simulator
 L12: Create a stress/lifting correspondence visualizer
 L13: Implement pointed pseudotriangulation algorithm
 L13: Implement (infinitesimal) locked linkage tester/designer tool
 C14: Implement hinged dissection animator: slender adornments,
general algorithm, and/or polyform algorithm
 C15: Implement continuous blooming algorithms
 L16: Implement orthogonal polyhedra unfolding
 L18: Combine gluing algorithm + Alexandrov algorithm to automate
case studies similar to square or Latin cross
Open Problems
 C03: Characterize singlevertex flatfoldable 3D crease patterns
 L04: Optimal wrapping of other shapes by a square
 L04: Optimal wrapping of a cube with an x × y rectangle of paper
 L04: Lower bounds for checkerboard folding
 C06: For sufficiently small, rigid motion, is local foldability enough?
 C06: Can a paper shopping bag be unfolded from the flat state by adding extra creases?
 C07: Universal folding of polyhedra other than boxes (e.g., polyoctahedra)
 C07: 3×n map folding [HARD]
 L08/C08: Prove conjectures about linear and circular corridor density
 L08: Prove a lower bound on number of creases in foldandcut related to local feature size
 L08: Higher dimensional foldandcut
 L08: Connected configuration space of polyhedral piece of paper?
 C08: Can we continuously flatten nonconvex polyhedra?
 L09: Do triangulated creases for hypars exist for all numbers of pleats and angles?
 L09: Do circular pleats exist? [HARD]
 L09: What is the maximum volume whose surface is a folding of a teabag (doubly covered square)?
 C09: What creases work for regular kgon pleats?
 C09: Tight bounds for 1D pleat folding (allowing unfolding)
 C09: Find an explicit example of a 1D M/V pattern which requires Ω(n/lg n) folds
 C09: Computational complexity of finding the shortest fold sequence to produce a given 1D M/V pattern (allowing unfolding)
 L10: Characterize when there are folding motions to folded states when the paper has holes [HARD]
 L10: Does adding a finite number of creases suffice to allow a folding motion between two folded states if the target folded state does not touch itself?
 L11: Develop a faster 2D rigidity testing algorithm, or prove a lower bound [HARD]
 L11: Characterize generic 3D rigidity [HARD]
 L13: Prove lower bound relating to feature size on number of steps to unfold polygon
 L13: Improve step bound for energy method to unfold polygon
 L13: Is there a unique minimumenergy configuration of a polygon?
 L13: Are there nonlinear locked trees of less than 8 bars?
 L13: Characterize locked linear trees
 L13: Is there a locked equilateral chain/tree in 3D?
 L14: Are there nonslender adornments that never lock?
 C14: Dissection in 5D and higher dimensions
 C14: Efficient algorithm to check for matching Dehn invariants
 C14: Any algorithm to find a 3D dissection when one exists
 L15: Edge unfolding convex prismatoids
 L15: General unfolding polyhedra [HARD]
 L15: Edge unfolding a convex polyhedron into o(F) parts?
 C15: Does inverted sun unfolding (source/star) avoid overlap?
 C15: Does every Johnson solid have an edge zipper unfolding?
 C15: Does every convex polyhedron have a general zipper unfolding?
 C15: Which triangulated polyhedra are ununfoldable after attaching
a witch's hat to each face?
 C15: Are 12face polyhedra unununfoldable?
 C15: Continuous blooming of star unfolding, sun unfolding,
all edge unfoldings, all unfoldings, or orthogonal polyhedra
 L16: Vertex unfolding convex polyhedra [HARD]
 L16: Grid unfolding orthogonal polyhedra [HARD]
 C16: Convexfaced vertexununfoldable polyhedron
 C16: Unfolding hexagonal polyhedra
 L17: Prove dependence of algorithms for Alexandrov's Theorem on feature size
 C17: Algorithm for BuragoZalgaller Theorem guaranteeing nonconvex polyhedron for any gluing
 L18: Complexity of whether a polygon of paper can be glued into a convex polyhedron
 L19: Which pairs of polyhedra have common unfoldings?
 L19: Are there two polycubes with no common grid unfolding?
 L19: Close the genus gap for nonorthogonal polyhedra with orthogonal faces
 L19: Minimum perimeter (and area) folding of a sphere
 L20: Flatstate connectivity of open chain, orthogonal tree, etc.
 L20: Locked equilateral equiangular fixedangle chain?
 L21: PTAS or APXhardness for optimal folding in HP model?
 L21: Unique foldings in nonsquare HP model?
 L21: Minimum number of cuts to unlock an nbar open chain?
 L21: Smallest kchain that interlocks with a 2chain?
 O21: Complexity of shortest flip sequence
 O21: Maximum number of flipturns
 O21: Characterize infinitely deflatable polygons