User:Noetico2/sandbox
Common Net
[edit]![](http://upload.wikimedia.org/wikipedia/commons/thumb/d/d7/TritetrahedronAmbiguousNet_1000.svg/400px-TritetrahedronAmbiguousNet_1000.svg.png)
In geometry, a common net refers to nets that can be folded onto several polyhedra. To be a valid common net there can't exist any no overlapping sides and the resulting polyhedra must be connected trough faces. The research of examples of this particular nets dates back to the end of the XX century, despite that, not many examples have been found. Two classes, however, have been deeply explored, regular polyhedra and cuboids. The search of common nets is usuallt made by either extensive search or the overlapping of nets that tile the plane.
Demaine et al. proved that every convex polyhedron can be unfolded and refolded to a different convex polyhedron (10.1016/j.comgeo.2013.05.002).
Regular Polyhedra
[edit]Open problem 25.31 in Geometric Folding Algorithm by Rourke and Demained reads:
"Can any Platonic solid be cut open and unfolded to a polygon that may be refolded to a different Platonic solid? For example, may a cube be so dissected to a tetrahedron?"
This problem has been partially solved by Shirakawa et al. with a fractal net that is conjectured to fold to a tetrahedron and a cube.
https://courses.csail.mit.edu/6.849/fall10/lectures/L17_images.pdf
Multiplicity | Example | Polyhedra 1 | Polyhedra 2 | Reference |
---|---|---|---|---|
Tetrahedron | Cube | Construct of Common Development of Regular Tetrahedron and Cube [1] | ||
Figure 25.51 Geometric Folding Algorithme | Tetrahedron | Cuboid (1x1x1.232) | Hirata, 2000[2] [3] | |
87 | https://jgaa.info/getPaper?id=386 | Tetrahedron | Jonhson Solid J17 | Common Unfolding of Regular Tetrahedron and Johnson-Zalgaller Solid[4] |
37 | https://jgaa.info/getPaper?id=386 | Tetrahedron | Jonhson Solid J84 | Common Unfolding of Regular Tetrahedron and Johnson-Zalgaller Solid[4] |
Cube | Tetramonohedron | Nonexistence of Common Edge Developments of Regular Tetrahedron and Other Platonic Solids (Horiyama & Uehara 2010)[5] | ||
Cube | 1x1x7 and 1x3x3 Cuboids | Common developments of three incongruent boxes of area 30[6] | ||
https://link.springer.com/chapter/10.1007/978-981-15-4470-5_4 | Cube | Octahedron (non-Regular) | Construct of Common Development of Regular Tetrahedron and Cube [1] | |
Octahedron | Tetramonohedron | Figure 25.50 Geometric Folding Algorithm [3] | ||
Octahedron | tetramonohedron | Nonexistence of Common Edge Developments of Regular Tetrahedron and Other Platonic Solids (Horiyama & Uehara 2010) [5] | ||
Image on top | Octahedron | Tritetrahedron | ||
Icosahedron | Tetramonohedron | Nonexistence of Common Edge Developments of Regular Tetrahedron and Other Platonic Solids (Horiyama & Uehara 2010) [5] |
Non regular Polyhedra
[edit]Cuboids
[edit]![](http://upload.wikimedia.org/wikipedia/commons/thumb/f/f1/Cuboid_common_net.png/395px-Cuboid_common_net.png)
For cuboids the question wether a polygon that can fold into four or more orthogonal boxes exist is still a open problem. It has, however, been proven that there exist infinetly many examples of nets that can be folded into more than one polyhedra (https://www.worldscientific.com/doi/abs/10.1142/S0218195913500040)
Area | Multiplicity | Example | Cuboid 1 | Cuboid 2 | Cuboid 3 | Reference |
---|---|---|---|---|---|---|
22 | 6495 | 1x1x5 | 1x2x3 | Polygons Folding to Plural Incongruent Orthogonal Boxes | ||
22 | 3 | 1x1x5 | 1x2x3 | 0x1x11 | Common Developments of Several Different Orthogonal Boxes | |
30 | 30 | 1x1x7 | 1x3x3 | √5x√5x√5 | Common developments of three incongruent boxes of area 30 | |
30 | 1080 | 1x1x7 | 1x3x3 | Common developments of three incongruent boxes of area 30 | ||
34 | 11291 | 1x1x8 | 1x2x5 | Polygons Folding to Plural Incongruent Orthogonal Boxes | ||
38 | 2334 | 1x1x9 | 1x3x4 | Polygons Folding to Plural Incongruent Orthogonal Boxes | ||
46 | 568 | 1x1x11 | 1x3x5 | Polygons Folding to Plural Incongruent Orthogonal Boxes | ||
46 | 92 | 1x2x7 | 1x3x5 | Polygons Folding to Plural Incongruent Orthogonal Boxes | ||
54 | 1735 | 1x1x13 | 3x3x3 | Polygons Folding to Plural Incongruent Orthogonal Boxes | ||
54 | 1806 | 1x1x13 | 1x3x6 | Polygons Folding to Plural Incongruent Orthogonal Boxes | ||
54 | 387 | 1x3x6 | 3x3x3 | Polygons Folding to Plural Incongruent Orthogonal Boxes | ||
58 | 37 | 1x1x14 | 1x4x5 | Polygons Folding to Plural Incongruent Orthogonal Boxes | ||
62 | 5 | 1x3x7 | 2x3x5 | Polygons Folding to Plural Incongruent Orthogonal Boxes | ||
64 | 50 | 2x2x7 | 1x2x10 | Polygons Folding to Plural Incongruent Orthogonal Boxes | ||
64 | 6 | 2x2x7 | 2x4x4 | Polygons Folding to Plural Incongruent Orthogonal Boxes | ||
70 | 3 | 1x1x17 | 1x5x5 | Polygons Folding to Plural Incongruent Orthogonal Boxes | ||
70 | 11 | 1x2x11 | 1x3x8 | Polygons Folding to Plural Incongruent Orthogonal Boxes | ||
88 | 218 | 2x2x10 | 1x4x8 | Polygons Folding to Plural Incongruent Orthogonal Boxes | ||
88 | 86 | 2x2x10 | 2x4x6 | Polygons Folding to Plural Incongruent Orthogonal Boxes | ||
532 | 7x8x14 | 2x4x43 | 2x13x16 | Common developments of three incongruent orthogonal boxes | ||
1792 | 7x8x56 | 7x14x38 | 2x13x58 | Common developments of three incongruent orthogonal boxes |
Polycubes
[edit]http://www.puzzlepalace.com/#/collections/9
https://cccg.ca/proceedings/2008/paper07full.pdf
https://www.youtube.com/watch?v=dLjCy6RmBN4
Area | Multiplicity | Example | Polyhedra 1 | Polyhedra 2 | Polyhedra 3 | Reference |
---|---|---|---|---|---|---|
14 | 29026 | https://lsusmath.rickmabry.org/rmabry/dodec/ambiguous/ambiguous-polycubes1-1-3--L1.html | Tricubes | http://www.puzzlepalace.com/#/collections/9 | ||
18 | Cubigami 7 | All tree-like tetracubes | http://www.puzzlepalace.com/#/collections/9 | |||
22 | https://courses.csail.mit.edu/6.849/fall10/lectures/L17_images.pdf | 22 tree-like pentacubes | https://core.ac.uk/download/pdf/9590509.pdf | |||
22 | https://courses.csail.mit.edu/6.849/fall10/lectures/L17_images.pdf | Non-planar pentacubes | https://core.ac.uk/download/pdf/9590509.pdf |
Deltahedra
[edit]3D Simplicial polytope
Area | Multiplicity | Example | Polyhedra 1 | Polyhedra 2 | Polyhedra 3 | Reference | |
---|---|---|---|---|---|---|---|
8 | On top | octahedron | Tritetrahedron | ||||
10 | 4 | https://lsusmath.rickmabry.org/rmabry/dodec/delta/common.html | 5/7 7-vertex deltahedra | Rick Mabry, https://lsusmath.rickmabry.org/rmabry/dodec/delta/common.html | |||
https://www.facebook.com/groups/puzzlefun/posts/10158369093625152/ | https://www.tessellation.jp/hiyoku | ||||||
You may enjoy this file, the 261 edge unfoldings of the truncated tetrahedron, computed by my student Emily Flynn: cs.smith.edu/~orourke/Unf/TruncTetra261Unfs.pdf . I just discovered a week ago that one of them (only one!) can be zipped to another convex polyhedron. –
Joseph O'Rourke |
https://www.science.smith.edu/~jorourke/Unf/TruncTetra261Unfs.pdf |
- ^ a b Toshihiro Shirakawa, Takashi Horiyama, and Ryuhei Uehara, 27th European Workshop on Computational Geometry (EuroCG 2011), 2011, 47-50.
- ^ Koichi Hirata, Personal communication, December 2000
- ^ a b Demaine, Erik; O'Rourke (July 2007). Geometric Folding Algorithms: Linkages, Origami, Polyhedra. Cambridge University Press. ISBN 978-0-521-85757-4.
{{cite book}}
: CS1 maint: date and year (link) - ^ a b Araki, Y., Horiyama, T., Uehara, R. (2015). Common Unfolding of Regular Tetrahedron and Johnson-Zalgaller Solid. In: Rahman, M.S., Tomita, E. (eds) WALCOM: Algorithms and Computation. WALCOM 2015. Lecture Notes in Computer Science, vol 8973. Springer, Cham. https://doi.org/10.1007/978-3-319-15612-5_26
- ^ a b c Horiyama, T., Uehara, R. (2010). Nonexistence of Common Edge Developments of Regular Tetrahedron and Other Platonic Solids, Information Processing Society of Japan, vol 2010. https://researchmap.jp/read0121089/published_papers/22955748?lang=en
- ^ Xu D., Horiyama T., Shirakawa T., Uehara R., Common developments of three incongruent boxes of area 30, Computational Geometry, 64, 8 2017