Connecting the Dots with Pick's Theorem

Oxford Mathematician Kristian Kiradjiev has won the Graham Hoare Prize (awarded by the Institute of Mathematics and its Applications) for his article "Connecting the Dots with Pick's Theorem". The Graham Hoare Prize is awarded annually to Early Career Mathematicians for a brilliant Mathematics Today article. Kristian also won the award in 2017. Here he talks about his work.

"Pick's theorem is an example of a theorem that is not widely known but has surprising applications to various mathematical problems. At its essence, Pick's theorem is a geometrical result, but has quite a few algebraic implications.

Published in 1899 by Georg Alexander Pick, the theorem states that given any simple polygon whose vertices lie on an integer grid, its area, $A$, is calculated according to the following formula: \begin{equation} A=i+\frac{b}{2}-1, \label{eq:Pick} \end{equation} where $i$ is the number of grid points inside the polygon, $b$ is the number of grid points that lie on the boundary of the polygon, and by simple we mean a polygon without holes. For a polygon with $n$ holes, the formula generalises to \begin{equation} A=i+\frac{b}{2}-1+n. \label{eq:Pick1} \end{equation} A useful property of Pick's formula that we immediately note is that it is invariant under shearing of the lattice. Also, scaling the distance between grid points in one direction simply scales the area of the polygon. These two observations can be used to show that Pick's formula is valid for sheared (triangular) grids as well, provided we take account of any scaling in the direction perpendicular to the shearing one.

Pick's theorem can be used to tackle a number of problems in different fields of mathematics. To give a flavour of this, we present a geometric problem that is simple to state. Although square integer grids are ubiquitous in our life, it is a fact that one cannot draw one of the simplest figures, namely, an equilateral triangle with its vertices being grid points on such a lattice. Some standard proofs involve tedious algebra and trigonometry, whereas Pick's theorem proves the result immediately. Suppose we have drawn an equilateral triangle with sides of length $d$ on a square grid. Then, its area is given by the well-known formula $A=\sqrt{3} d^2/4$. Since the vertices of the triangle are integer points, then $d^2$ is an integer (by Pythagoras' theorem, for example), and, thus, the area is an irrational number. However, Pick's formula on square grids always gives a rational area. This contradiction proves the initial claim. The same argument shows that regular hexagons cannot be drawn on a square integer grid either. Another problem where Pick's theorem can be applied concerns some properties of the so-called Farey sequences.

One might think that Pick's theorem can be easily generalised to three (or more) dimensions for volume of solids, etc. However, in 1957, J. Reeve produced an example of what is now known as the Reeve tetrahedron, which shows that Pick's theorem does not have a direct analogue in three or more dimensions by devising a figure that can take many different values for its volume without changing the number of interior and boundary points. However, Pick's theorem has some close analogues in more than two dimensions such as the Ehrhart polynomials.

The figure above is a rather complicated polygon, called a Farey sunburst (because of its relation to Farey sequences). Its area can be readily calculated using Pick's theorem to be 48 square units."