Common Net

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

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

Cuboids

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

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	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

3D Simplicial polytope

Area	Multiplicity	Example	Polyhedra 1	Polyhedra 2	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

[:0-1] Toshihiro Shirakawa, Takashi Horiyama, and Ryuhei Uehara, 27th European Workshop on Computational Geometry (EuroCG 2011), 2011, 47-50.

[2] Koichi Hirata, Personal communication, December 2000

[:1-3] 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)

[:2-4] 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, Cham. https://doi.org/10.1007/978-3-319-15612-5_26

[:3-5] 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

[6] Xu D., Horiyama T., Shirakawa T., Uehara R., Common developments of three incongruent boxes of area 30, Computational Geometry, 64, 8 2017

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