Common net

Common net for both a octahedron and a Tritetrahedron.

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

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

Cuboids

Common net of a 1x1x5 and 1x2x3 cuboid

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

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

3D Simplicial polytope

Area	Multiplicity	Polyhedra	Reference
8	1	Both 8 face deltahedra	[8]
10	4	7-vertex deltahedra	[18]

References

1. 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.
2. Demaine, Erik D.; O'Rourke, Joseph (2007). Geometric folding algorithms: linkages, origami, polyhedra. Cambridge: Cambridge University Press. ISBN   978-0-521-85757-4.
3. 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.
4. 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. doi:10.1007/978-3-319-15612-5_26.
5. "Ryuuhei Uehara - Nonexistence of Common Edge Developments of Regular Tetrahedron and Other Platonic Solids - Papers - researchmap". researchmap.jp. Retrieved 2024-08-01.
6. 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 .
7. Demaine, Erik; O'Rourke (July 2007). Geometric Folding Algorithms: Linkages, Origami, Polyhedra. Cambridge University Press. ISBN   978-0-521-85757-4.
8. Weisstein, Eric. "Net".
9. 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. ISSN   0218-1959.
10. Mitani, Jun; Uehara, Ryuhei (2008). "Polygons Folding to Plural Incongruent Orthogonal Boxes" (PDF). Canadian Conference on Computational Geometry.
11. 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.
12. 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.
13. 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. ISSN   0218-1959.
14. Miller, George; Knuth, Donald. "Cubigami".
15. Mabry, Rick. "Ambiguous unfoldings of polycubes".
16. Miller, George. "Cubigami".
17. 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.
18. Mabry, Rick. "The four common nets of the five 7-vertex deltahedra".