Dissection problem

In geometry, a dissection problem is the problem of partitioning a geometric figure (such as a polytope or ball) into smaller pieces that may be rearranged into a new figure of equal content. In this context, the partitioning is called simply a dissection (of one polytope into another). It is usually required that the dissection use only a finite number of pieces.

The Bolyai–Gerwien theorem states that any polygon may be dissected into any other polygon of the same area. It is not true, however, that any polyhedron has a dissection into any other polyhedron of the same volume. This process is possible, however, for any two honeycombs (such as cube) in three dimension and any two zonohedra of equal volume (in any dimension).

A dissection into triangles of equal area is called an equidissection. Most polygons cannot be equidissected, and those that can often have restrictions on the possible numbers of triangles. For example, Monsky's theorem states that there is no odd equidissection of a square.^[1]

See also

• Dissection puzzle
• Hilbert's third problem
• Hinged dissection

References

1. Stein, Sherman K. (March 2004), "Cutting a Polygon into Triangles of Equal Areas", The Mathematical Intelligencer 26 (1): 17–21, doi:10.1007/BF02985395, Zbl 1186.52015

External links

• David Eppstein, Dissection Tiling.