User:Noetico2/sandbox

= 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

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)

Polycubes
http://www.puzzlepalace.com/#/collections/9

https://cccg.ca/proceedings/2008/paper07full.pdf

https://www.youtube.com/watch?v=dLjCy6RmBN4

Deltahedra
3D Simplicial polytope