Common net
In geometry, a common net is a net that can be folded onto several polyhedra. To be a valid common net, there should not exist any non-overlapping sides, and the resulting polyhedra must be connected through faces. Examples of these particular nets in the research date back to the end of the 20th century; despite that, not many examples have been found. Two classes, however, have been deeply explored, regular polyhedra and cuboids. The search for common nets is usually made by either an extensive search or the overlapping of nets that tile the plane. Demaine et al. (2013) proved that every convex polyhedron can be unfolded and refolded to a different convex polyhedron. There are types of common nets: strict edge unfoldings and free unfoldings. Strict edge unfoldings refer to common nets where the different polyhedra that can be folded use the same folds: to fold one polyhedra from the net of another, there is no need to make new folds. Free unfoldings refer to the opposite case when we can create as many folds as needed to enable the folding of different polyhedra. Multiplicity of common nets refers to the number of common nets for the same set of polyhedra.
In geometry, a common net is a net that can be folded onto several polyhedra. To be a valid common net, there should not exist any non-overlapping sides, and the resulting polyhedra must be connected through faces. Examples of these particular nets in the research date back to the end of the 20th century; despite that, not many examples have been found. Two classes, however, have been deeply explored, regular polyhedra and cuboids. The search for common nets is usually made by either an extensive search or the overlapping of nets that tile the plane.
Demaine et al. (2013) proved that every convex polyhedron can be unfolded and refolded to a different convex polyhedron.[1]
There are types of common nets: strict edge unfoldings and free unfoldings. Strict edge unfoldings refer to common nets where the different polyhedra that can be folded use the same folds: to fold one polyhedra from the net of another, there is no need to make new folds. Free unfoldings refer to the opposite case when we can create as many folds as needed to enable the folding of different polyhedra.
Multiplicity of common nets refers to the number of common nets for the same set of polyhedra.
Regular polyhedra
[edit]Open problem 25.31 in Geometric Folding Algorithm by Rourke and Demaine 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?[2]
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.
| Multiplicity | Polyhedra 1 | Polyhedra 2 | Reference |
|---|---|---|---|
| Tetrahedron | Cube | [3] | |
| 87 | tetrahedron | gyroelongated square bipyramid (J17) | [4] |
| 37 | tetrahedron | snub disphenoid (J84) | [4] |
| Cube | tetramonohedron | [5] | |
| Cube | 1x1x7 and 1x3x3 cuboids | [6] | |
| cube | non-regular octahedron | [3] | |
| octahedron | tetramonohedron | [7] | |
| octahedron | tetramonohedron | [5] | |
| octahedron | tritetrahedron | [8] | |
| icosahedron | tetramonohedron | [5] |
Non-regular polyhedra
[edit]Cuboids
[edit]
Common nets of cuboids have been deeply researched, mainly by Uehara and coworkers. To the moment, common nets of up to three cuboids have been found, It has, however, been proven that there exist infinitely many examples of nets that can be folded into more than one polyhedra.[9]
| Area | Multiplicity | Cuboid 1 | Cuboid 2 | Cuboid 3 | Reference |
|---|---|---|---|---|---|
| 22 | 6495 | 1x1x5 | 1x2x3 | [10] | |
| 22 | 3 | 1x1x5 | 1x2x3 | 0x1x11 | [11] |
| 28 | 1x2x4 | √2x√2x3√2 | [11] | ||
| 30 | 30 | 1x1x7 | 1x3x3 | √5x√5x√5 | [12] |
| 30 | 1080 | 1x1x7 | 1x3x3 | [12] | |
| 34 | 11291 | 1x1x8 | 1x2x5 | [10] | |
| 38 | 2334 | 1x1x9 | 1x3x4 | [10] | |
| 46 | 568 | 1x1x11 | 1x3x5 | [10] | |
| 46 | 92 | 1x2x7 | 1x3x5 | [10] | |
| 54 | 1735 | 1x1x13 | 3x3x3 | [10] | |
| 54 | 1806 | 1x1x13 | 1x3x6 | [10] | |
| 54 | 387 | 1x3x6 | 3x3x3 | [10] | |
| 58 | 37 | 1x1x14 | 1x4x5 | [10] | |
| 62 | 5 | 1x3x7 | 2x3x5 | [10] | |
| 64 | 50 | 2x2x7 | 1x2x10 | [10] | |
| 64 | 6 | 2x2x7 | 2x4x4 | [10] | |
| 70 | 3 | 1x1x17 | 1x5x5 | [10] | |
| 70 | 11 | 1x2x11 | 1x3x8 | [10] | |
| 88 | 218 | 2x2x10 | 1x4x8 | [10] | |
| 88 | 86 | 2x2x10 | 2x4x6 | [10] | |
| 160 | 4x4x8 | √10x2√10x2√10 | [11] | ||
| 532 | 7x8x14 | 2x4x43 | 2x13x16 | [13] | |
| 1792 | 7x8x56 | 7x14x38 | 2x13x58 | [13] |
*Non-orthogonal foldings
Polycubes
[edit]The first cases of common nets of polycubes found was the work by George Miller, with a later contribution of Donald Knuth, that culminated in the Cubigami puzzle.[14] It’s composed of a net that can fold to all 7 tree-like tetracubes. All possible common nets up to pentacubes were found. All the nets follow strict orthogonal folding despite still being considered free unfoldings.
| Area | Multiplicity | Polyhedra | Reference |
|---|---|---|---|
| 14 | 29026 | All tricubes | [15] |
| 14 | All tricubes | [10] | |
| 18 | 68 | All tree-like tetracubes[14] | [16] |
| 22 | 23 pentacubes | [17] | |
| 22 | 3 | 22 tree-like pentacubes | [17] |
| 22 | 1 | Non-planar pentacubes | [17] |
Deltahedra
[edit]3D Simplicial polytope
| Area | Multiplicity | Polyhedra | Reference |
|---|---|---|---|
| 8 | 1 | Both 8 face deltahedra | [8] |
| 10 | 4 | 7-vertex deltahedra | [18] |
References
[edit]- ^ Demaine, Erik D.; Demaine, Martin L.; Itoh, Jin-ichi; Lubiw, Anna; Nara, Chie; OʼRourke, Joseph (2013-10-01). "Refold rigidity of convex polyhedra". Computational Geometry. 46 (8): 979–989. doi:10.1016/j.comgeo.2013.05.002. hdl:1721.1/99989. ISSN 0925-7721.
- ^ Demaine, Erik D.; O'Rourke, Joseph (2007). Geometric folding algorithms: linkages, origami, polyhedra. Cambridge: Cambridge University Press. ISBN 978-0-521-85757-4.
- ^ a b Shirakawa, Toshihiro; Horiyama, Takashi; Uehara, Ryuhei (March 28–30, 2011). "Construction of Common Unfolding of a Regular Tetrahedron and a Cube" (PDF). 27th European Workshop on Computational Geometry (EuroCG 2011). Morschach, Switzerland. pp. 47–50.
- ^ 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. Vol. 8973. Springer. pp. 294–305. doi:10.1007/978-3-319-15612-5_26. ISBN 978-3-319-15611-8.
- ^ a b c "Ryuuhei Uehara - Nonexistence of Common Edge Developments of Regular Tetrahedron and Other Platonic Solids - Papers - researchmap". researchmap.jp. Retrieved 2024-08-01.
- ^ Xu, D.; Horiyama, T.; Shirakawa, T.; Uehara, R. (2017). "Common developments of three incongruent boxes of area 30". Computational Geometry. 64 (7): 1–12. doi:10.1016/j.comgeo.2017.03.001.
- ^ Demaine, Erik; O'Rourke (July 2007). Geometric Folding Algorithms: Linkages, Origami, Polyhedra. Cambridge University Press. ISBN 978-0-521-85757-4.
- ^ a b Weisstein, Eric. "Net".
- ^ Shirakawa, Toshihiro; Uehara, Ryuhei (February 2013). "Common Developments of Three Incongruent Orthogonal Boxes". International Journal of Computational Geometry & Applications. 23 (1): 65–71. doi:10.1142/S0218195913500040. hdl:10119/11480. ISSN 0218-1959.
- ^ a b c d e f g h i j k l m n o p q Mitani, Jun; Uehara, Ryuhei (2008). "Polygons Folding to Plural Incongruent Orthogonal Boxes" (PDF). Canadian Conference on Computational Geometry.
- ^ a b c Abel, Zachary; Demaine, Erik; Demaine, Martin; Matsui, Hiroaki; Rote, Günter; Uehara, Ryuhei. "Common Developments of Several Different Orthogonal Boxes". The 23rd Canadian Conference on Computational Geometr: 77–82. hdl:10119/10308.
- ^ a b Xu, Dawei; Horiyama, Takashi; Shirakawa, Toshihiro; Uehara, Ryuhei (August 2017). "Common developments of three incongruent boxes of area 30". Computational Geometry. 64: 1–12. doi:10.1016/j.comgeo.2017.03.001. ISSN 0925-7721.
- ^ a b Shirakawa, Toshihiro; Uehara, Ryuhei (February 2013). "Common Developments of Three Incongruent Orthogonal Boxes". International Journal of Computational Geometry & Applications. 23 (1): 65–71. doi:10.1142/S0218195913500040. hdl:10119/11480. ISSN 0218-1959.
- ^ a b Miller, George; Knuth, Donald. "Cubigami".
- ^ Mabry, Rick. "Ambiguous unfoldings of polycubes".
- ^ Miller, George. "Cubigami".
- ^ a b c Aloupis, Greg; Bose, Prosenjit K.; Collette, Sébastien; Demaine, Erik D.; Demaine, Martin L.; Douïeb, Karim; Dujmović, Vida; Iacono, John; Langerman, Stefan; Morin, Pat (2011). "Common Unfoldings of Polyominoes and Polycubes". In Akiyama, Jin; Bo, Jiang; Kano, Mikio; Tan, Xuehou (eds.). Computational Geometry, Graphs and Applications. Lecture Notes in Computer Science. Vol. 7033. Berlin, Heidelberg: Springer. pp. 44–54. doi:10.1007/978-3-642-24983-9_5. ISBN 978-3-642-24983-9.
- ^ Mabry, Rick. "The four common nets of the five 7-vertex deltahedra".