geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ...

, Monsky's theorem states that it is not possible to dissect a

square In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90- degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length a ...

into an odd number of

triangle A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, any three points, when non- colline ...

s of equal area. In other words, a square does not have an odd

equidissection In geometry, an equidissection is a partition of a polygon into triangles of equal area. The study of equidissections began in the late 1960s with Monsky's theorem, which states that a square cannot be equidissected into an odd number of triang ...

. The problem was posed by Fred Richman in the '' American Mathematical Monthly'' in 1965, and was proved by Paul Monsky in 1970.

Proof

Monsky's proof combines

combinatorial Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many ap ...

and

algebra Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary ...

ic techniques, and in outline is as follows: Square_triangulation

#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''. #Colour each point in the square with one of three colours, depending on the 2-adic valuation of its coordinates. #Show that a straight line can contain points of only two colours. #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. #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. #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. #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''.

Optimal dissections

By Monsky's theorem it is necessary to have triangles with different areas to dissect a square into an odd number of triangles. Lower bounds for the area differences that must occur to dissect a square into an odd numbers of triangles and the optimal dissections have been studied.

Generalizations

The theorem can be generalized to higher dimensions: an ''n''-dimensional hypercube can only be divided into

simplices In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given dimension. ...

of equal volume, if the number of simplices is a multiple of ''n''!.

References

{{reflist Euclidean plane geometry Theorems in discrete geometry Geometric dissection