Equidissection of a regular polygon is the problem of dissecting the polygon into triangles of equal area. Given a regular n-gon and a positive integer m, we ask if it is possible to equidissect the n-gon into m triangles. To answer this question, we make use of Sperner's lemma along with some elementary number theory. Sperner's lemma is a major combinatorial result that has found applications in topics such as fixed-point computation and fair division.
We shall see Sperner's lemma and its proof, and use it to solve the equidissection problem for regular polygons. We will also look at some polygons that cannot be equidissected into m triangles for any positive m. Finally, we will go over the equidissection problem for higher-dimensional hypercubes.
References:
- Monsky, P. (1970). On Dividing A Square Into Triangles. The American Mathematical Monthly, 77(2), 161–164. https://doi.org/10.1080/00029890.1970.11992441
- Stein, S. (2004). Cutting a Polygon into Triangles of Equal Areas. The Mathematical Intelligencer 26, 17–21. https://doi.org/10.1007/BF02985395
