Edge-joined polygon with multiple principle shapes
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.
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.
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]
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.
^ abAraki, 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.
