Folding Polyiamonds into Octahedra

@article{Bolle2021FoldingPI,
  title={Folding Polyiamonds into Octahedra},
  author={Eva Bolle and Linda Kleist},
  journal={Comput. Geom.},
  year={2021},
  volume={108},
  pages={101917}
}

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TLDR
A linear-time dynamic programming algorithm to fold a tree-shaped polyomino into a constant-size polycube, in some models is given and the triangular version of the problem is considered, characterizing which polyiamonds fold into a regular tetrahedron.

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TLDR
It is demonstrated that a 3×3 square can fold into a unit cube using horizontal, vertical, and diagonal creases on the 6× 6 half-grid, implying that all tree-shaped polyominoes with at least nine squares fold intoA unit cube.

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We present two universal hinge patterns that enable a strip of material to fold into any connected surface made up of unit squares on the 3D cube grid—for example, the surface of any polycube. The

Polygons Folding to Plural Incongruent Orthogonal Boxes

TLDR
There are more than two thousands such polygons of several sizes found by a randomized algorithm that repeatedly produces many nets of orthogonal boxes at random, and matches them in a huge hash table.

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TLDR
This paper gives self-overlapping general unfoldings of Platonic solids other than the tetrahedron (i.e., a cube, an octahedron, a dodecahedrons, and an icosahedron), and edge unfoldsings of some Archimedean solids.

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TLDR
It is shown here that all 23 tree-like pentacubes have no such common unfolding, although 22 of them have a common unfolding.

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TLDR
This paper first shows that there is an orthogonal polygon that fold to three boxes of size 1×1×5, 1 × 2 × 3, and 0 × 1 × 11, which solves the open problem mentioned in literature and shows some polygons that can fold to two incongruent Orthogonal boxes in more general directions.

Universal Hinge Patterns for Folding Orthogonal Shapes

TLDR
It is an open question which polycubes are rigidly foldable from a particular cube gadget, though it seems through simple empirical testing that the authors' cube gadgets fold rigidly in isolation (when making one-cubepolycubes).

Common developments of three incongruent boxes of area 30

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TLDR
An affirmative answer to the open problem of whether there exists an orthogonal polygon that folds into three boxes of positive volume is given and it is shown how to construct an infinite number of orthogsonal polygons that fold into three incongruent Orthogonal boxes.