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In geometry, a common net is a net that can be folded onto several polyhedra. To be a valid common net, there shouldn't exist any non-overlapping sides and the resulting polyhedra must be connected through faces. The research of examples of this particular nets dates 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 of common nets is usually 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.

There can be types of common nets, strict edge unfoldings and free unfoldings. Strict edge unfoldings refers to common nets where the different polyhedra that can be folded use the same folds, that is, 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?"

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.

Non-regular polyhedra

Cuboids

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.

*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. 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.

Deltahedra

3D Simplicial polytope

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. ^ Toshihiro Shirakawa, Takashi Horiyama, and Ryuhei Uehara, 27th European Workshop on Computational Geometry (EuroCG 2011), 2011, 47-50.
4. Koichi Hirata, Personal communication, December 2000
5. ^ 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
6. ^ "Ryuuhei Uehara - Nonexistence of Common Edge Developments of Regular Tetrahedron and Other Platonic Solids - Papers - researchmap". researchmap.jp. Retrieved 2024-08-01.
7. Xu D., Horiyama T., Shirakawa T., Uehara R., Common developments of three incongruent boxes of area 30, Computational Geometry, 64, 8 2017
8. 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)
9. ^ Weisstein, Eric. "Net".
10. 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.
11. ^ Mitani, Jun; Uehara, Ryuhei (2008). "Polygons Folding to Plural Incongruent Orthogonal Boxes" (PDF). Canadian Conference on Computational Geometry.
12. ^ 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.
13. ^ 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.
14. ^ 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.
15. ^ Miller, George; Knuth, Donald. "Cubigami".
16. Mabry, Rick. "Ambiguous unfoldings of polycubes".
17. Miller, George. "Cubigami".
18. ^ 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.
19. Mabry, Rick. "The four common nets of the five 7-vertex deltahedra".

• Polyhedra