Geometric Folding Algorithms: Linkages, Origami, Polyhedra

Sample Project Ideas

Below is a list of possible project ideas. The source of the project idea—extracted from lecture (L) or class (C) notes, or old (O) lecture notes from Fall 2010—is noted when applicable.

Design/Build/Art

SES #	PROJECT IDEAS
C06	Design and build a rigid origami structure.
C08	Design a fold-and-cut alphabet, preferably using a small number of simple folds. Fold-and-cut art à la Peter Callesen. Animate motion for 3D polyhedra flattening.
C09	How does real paper behave when folding a hypar?
L10	Create a Kempe-inspired linkage. Design linkages to draw letters of the alphabet.
L12	Design and build a tensegrity sculpture.
C14	Design elegant hinged dissections. Design / build reconfigurable furniture.

Coding

SES #	PROJECT IDEAS
—	Higher dimension folding visualizer.
L03	Implement local foldability tester which generates a M/V pattern.
C04	Implement algorithm to generate an arbitrary black/white pixel pattern using checkerboard results.
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 MacOS®/Linux®.
C08	Implement/improve on a fold-and-cut design tool, ideally including M/V state, degeneracy tool, and folded state. Possibly porting to Javascript™. Animate motion for 3D polyhedra flattening.
L10	Create a Kempe simulation.
C10	Implement Kempe with splines.
L11	JavaScript rigidity/over-bracing /pebble algorithm visualization tool.
C11	Improve Henneberg construction/puzzle applet, port to web/JavaScript.
L12	Make a virtual tensegrity simulator. Create a stress/lifting correspondence visualizer.
L13	Implement pointed pseudotriangulation algorithm. 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 and Alexandrov algorithm to automate case studies similar to square or Latin cross.

Open Problems

SES #	PROJECT IDEAS
—	Problem set 1, problem 2 many-layers version. What is the minimum number of creases needed to be removed to make a crease pattern flat-foldable?
L02	Pseudopolynomial upper/lower bounds for strip method of folding anything. Characterize possible seam placements.
C03	Characterize single-vertex flat-foldable 3D crease patterns.
L04	Optimal wrapping of other shapes by a square. Optimal wrapping of a cube with an x × y rectangle of paper. Do there exist other optimal wrappings of a cube by a square? Lower bounds for checkerboard folding.
C04	Optimal 2×2 checkerboard folding.
C06	For sufficiently small, rigid motion, is local foldability enough? Computational complexity of determining rigid foldability of crease patterns. 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). Is there a simpler proof of flat-foldability NP-hardness? 3×n map folding. [Hard]
L08	Prove a lower bound on number of creases in fold-and-cut related to local feature size. Higher dimensional fold-and-cut. Instantaneous flattening of polyhedral complexes. Connected configuration space of polyhedral piece of paper? Prove conjectures about linear and circular corridor density.
C08	Fold-and-cut with arcs of constant curvature. Can we continuously flatten nonconvex polyhedra? Prove conjectures about linear and circular corridor density.
L09	Do triangulated creases for hypars exist for all numbers of pleats and angles? Do circular pleats exist? [Hard] What is the maximum volume whose surface is a folding of a teabag.
C09	What creases work for regular k-gon pleats? Tight bounds for 1D pleat folding (allowing unfolding). Find an explicit example of a 1D M/V pattern which requires Ω(n/lg n) folds. 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 for paper with holes. 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] Characterize generic 3D rigidity. [Hard]
L13	Prove lower bound relating to feature size on number of steps to unfold polygon. Improve step bound for energy method to unfold polygon. Is there a unique minimum-energy configuration of a polygon? Are there nonlinear locked trees of less than 8 bars? Characterize locked linear trees. Is there a locked equilateral anything in 3D?
L14	Are there nonslender adornments that never lock?
C14	5D and higher dissections. Efficient algorithm to check for matching Dehn invariants. Any algorithm to find a dissection when one exists.
L15	Edge unfolding convex prismatoids. General unfolding polyhedra. [Hard] Can the star unfolding (or other edge/general unfoldings) be continuously bloomed? Edge unfolding a convex polyhedron into o(F) parts?
C15	Does inverted sun unfolding (source/star) avoid overlap? Does every Johnson solid have an edge zipper unfolding? Does every convex polyhedron have a general zipper unfolding? Which triangulated polyhedra are ununfoldable after attaching a witch’s hat to each face? Are 12-face polyhedra unununfoldable? Can prismatoids or even prismoids be fully band unfolded? Continuous blooming of star unfolding, sun unfolding, all edge unfoldings, all unfoldings, or orthogonal polyhedra.
L16	Vertex unfolding convex polyhedra. [Hard] Grid unfolding orthogonal polyhedra. [Hard]
C16	Convex-faced vertex-ununfoldable polyhedron. Unfolding hexagonal polyhedra.
L17	Prove dependence of algorithms for Alexandrov’s Theorem on feature size.
C17	Algorithm for Burago-Zalgaller Theorem guaranteeing nonconvex polyhedron for any gluing.
L18	Complexity of whether a polygon of paper can be glued into a convex polyhedron.
L19	Which polyhedra have common unfoldings? Are there two polycubes with no common grid unfolding? Close the genus gap for nonorthogonal polyhedra with orthogonal faces. Minimum perimeter (and area) folding of a sphere.
L20	Complexity of 3D min/max span. Flat-state connectivity of open chain, orthogonal tree, etc. Locked equilateral equiangular fixed-angle chain?
L21	PTAS or APX-hardness for optimal folding in HP model? Unique foldings in nonsquare HP model? Minimum number of cuts to unlock an n-bar open chain? Smallest k-chain that interlocks with a 2-chain?
O21	Complexity of shortest flip sequence. Maximum number of flipturns. Characterize infinitely deflatable polygons.