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- 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?
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| L02 |
- Pseudopolynomial upper/lower bounds for strip method of folding anything.
- Characterize possible seam placements.
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| C03 |
- Characterize single-vertex flat-foldable 3D crease patterns.
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| 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.
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| C04 |
- Optimal 2×2 checkerboard folding.
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| 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?
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| 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]
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| 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.
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| C08 |
- Fold-and-cut with arcs of constant curvature.
- Can we continuously flatten nonconvex polyhedra?
- Prove conjectures about linear and circular corridor density.
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| 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.
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| 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).
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| 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?
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| L11 |
- Develop a faster 2D rigidity testing algorithm, or prove a lower bound. [Hard]
- Characterize generic 3D rigidity. [Hard]
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| 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?
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| L14 |
- Are there nonslender adornments that never lock?
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| C14 |
- 5D and higher dissections.
- Efficient algorithm to check for matching Dehn invariants.
- Any algorithm to find a dissection when one exists.
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| 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?
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| 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.
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| L16 |
- Vertex unfolding convex polyhedra. [Hard]
- Grid unfolding orthogonal polyhedra. [Hard]
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| C16 |
- Convex-faced vertex-ununfoldable polyhedron.
- Unfolding hexagonal polyhedra.
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| L17 |
- Prove dependence of algorithms for Alexandrov's Theorem on feature size.
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| C17 |
- Algorithm for Burago-Zalgaller Theorem guaranteeing nonconvex polyhedron for any gluing.
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| L18 |
- Complexity of whether a polygon of paper can be glued into a convex polyhedron.
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| 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.
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| L20 |
- Complexity of 3D min/max span.
- Flat-state connectivity of open chain, orthogonal tree, etc.
- Locked equilateral equiangular fixed-angle chain?
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| 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?
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| O21 |
- Complexity of shortest flip sequence.
- Maximum number of flipturns.
- Characterize infinitely deflatable polygons.
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