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.[1]
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.
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?"[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.
| Multiplicity | Polyhedra 1 | Polyhedra 2 | Reference | |
|---|---|---|---|---|
| Tetrahedron | Cube | [3] | ||
| Tetrahedron | Cuboid (1x1x1.232) | [4] | ||
| 87 | Tetrahedron | Jonhson Solid J17 | [5] | |
| 37 | Tetrahedron | Jonhson Solid J84 | ||
| Cube | Tetramonohedron | [6] | ||
| Cube | 1x1x7 and 1x3x3 Cuboids | [7] | ||
| Cube | Octahedron (non-Regular) | |||
| Octahedron | Tetramonohedron | [8] | ||
| Octahedron | tetramonohedron | |||
| Octahedron | Tritetrahedron | [9] | ||
| Icosahedron | Tetramonohedron |
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.[10]
| Area | Multiplicity | Cuboid 1 | Cuboid 2 | Cuboid 3 | Reference | |
|---|---|---|---|---|---|---|
| 22 | 6495 | 1x1x5 | 1x2x3 | [11] | ||
| 22 | 3 | 1x1x5 | 1x2x3 | 0x1x11 | [12] | |
| 28 | 1x2x4 | √2x√2x3√2 | ||||
| 30 | 30 | 1x1x7 | 1x3x3 | √5x√5x√5 | [13] | |
| 30 | 1080 | 1x1x7 | 1x3x3 | |||
| 34 | 11291 | 1x1x8 | 1x2x5 | |||
| 38 | 2334 | 1x1x9 | 1x3x4 | |||
| 46 | 568 | 1x1x11 | 1x3x5 | |||
| 46 | 92 | 1x2x7 | 1x3x5 | |||
| 54 | 1735 | 1x1x13 | 3x3x3 | |||
| 54 | 1806 | 1x1x13 | 1x3x6 | |||
| 54 | 387 | 1x3x6 | 3x3x3 | |||
| 58 | 37 | 1x1x14 | 1x4x5 | |||
| 62 | 5 | 1x3x7 | 2x3x5 | |||
| 64 | 50 | 2x2x7 | 1x2x10 | |||
| 64 | 6 | 2x2x7 | 2x4x4 | |||
| 70 | 3 | 1x1x17 | 1x5x5 | |||
| 70 | 11 | 1x2x11 | 1x3x8 | |||
| 88 | 218 | 2x2x10 | 1x4x8 | |||
| 88 | 86 | 2x2x10 | 2x4x6 | |||
| 160 | 4x4x8 | √10x2√10x2√10 | ||||
| 532 | 7x8x14 | 2x4x43 | 2x13x16 | [14] | ||
| 1792 | 7x8x56 | 7x14x38 | 2x13x58 |
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.[15] 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.
| Area | Multiplicity | Polyhedra | Reference | |
|---|---|---|---|---|
| 14 | 29026 | All tricubes | [16] | |
| 14 | All tricubes | |||
| 18 | 68 | All tree-like tetracubes | [17] | |
| 22 | 23 pentacubes | [18] | ||
| 22 | 3 | 22 tree-like pentacubes | ||
| 22 | 1 | Non-planar pentacubes |
3D Simplicial polytope
| Area | Multiplicity | Polyhedra | Reference | |
|---|---|---|---|---|
| 8 | 1 | Both 8 face deltahedra | ||
| 10 | 4 | 7-vertex deltahedra | [19] |