geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

, a common net is a

net NET may refer to: Broadcast media United States * National Educational Television, the predecessor of the Public Broadcasting Service (PBS) in the United States * National Empowerment Television, a politically conservative cable TV network ...

that can be folded onto several

polyhedra In geometry, a polyhedron (: polyhedra or polyhedrons; ) is a three-dimensional figure with flat polygonal faces, straight edges and sharp corners or vertices. The term "polyhedron" may refer either to a solid figure or to its boundary su ...

. 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. proved that every convex polyhedron can be unfolded and refolded to a different convex polyhedron. 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?

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

{{Mathematics of paper folding Polyhedra