Monsky's theorem

In geometry, Monsky's theorem states that it is not possible to dissect a square into an odd number of triangles of equal area.[1] In other words, a square does not have an odd equidissection.

The problem was posed by Fred Richman in the American Mathematical Monthly in 1965, and was proved by Paul Monsky in 1970.[2][3][4]

Proof

Monsky's proof combines combinatorial and algebraic techniques, and in outline is as follows:

  1. Take the square to be the unit square with vertices at (0,0), (0,1), (1,0) and (1,1). If there is a dissection into n triangles of equal area then the area of each triangle is 1/n.
  2. Colour each point in the square with one of three colours, depending on the 2-adic valuation of its coordinates.
  3. Show that a straight line can contain points of only two colours.
  4. Use Sperner's lemma to show that every triangulation of the square into triangles meeting edge-to-edge must contain at least one triangle whose vertices have three different colours.
  5. Conclude from the straight-line property that a tricolored triangle must also exist in every dissection of the square into triangles, not necessarily meeting edge-to-edge.
  6. Use Cartesian geometry to show that the 2-adic valuation of the area of a triangle whose vertices have three different colours is greater than 1. So every dissection of the square into triangles must contain at least one triangle whose area has a 2-adic valuation greater than 1.
  7. If n is odd then the 2-adic valuation of 1/n is 1, so it is impossible to dissect the square into triangles all of which have area 1/n.[5]

Generalizations

Main article: Equidissection

The theorem can be generalized to higher dimensions: an n-dimensional hypercube can only be divided into simplices of equal volume, if the number of simplices is a multiple of n!.[2]

References

  1. ↑ Aigner, Martin; Ziegler, Günter M. (2010). "One square and an odd number of triangles". Proofs from The Book (4th ed.). Berlin: Springer-Verlag. pp. 131–138. doi:10.1007/978-3-642-00856-6_20.
  2. 1 2 Sperner's Lemma, Moor Xu
  3. ↑ Monsky, P. (1970). "On Dividing a Square into Triangles". The American Mathematical Monthly 77 (2): 161–164. doi:10.2307/2317329. MR 0252233.
  4. ↑ Kleber, M.; Vakil, R.; Stein, S. (2004). "Cutting a Polygon into Triangles of Equal Areas". The Mathematical Intelligencer 26: 17. doi:10.1007/BF02985395.
  5. ↑ Dissecting a square into triangles at the Wayback Machine (archived August 18, 2010)
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