Finite Geometries and Latin Squares

The study of latin squares stretches far back into history and our fascination with them appears undiminished today, evidenced by its appearance in the popular Soduku puzzles. As we shall prove in this chapter, there are a very large number of \(n \times n\) latin squares. There are however, very few sets of mutually orthogonal latin squares. The problem of finding two mutually orthogonal latin squares can be rephrased in natural terms as the problem of lining up n regiments of n different ranking officers, on parade in an \(n \times n\) grid, such that in each row and column we find exactly one officer from each regiment and one officer of each of the n ranks. We will see that there is a solution to this problem for every \(n \neq 2,6\) . Finding larger sets of mutually orthogonal latin squares will lead us to consider finite geometries, incidence structures of points and lines in which the set of points and the set of lines are finite. We will consider properties of geometries defined from a finite vector space, focussing on affine and projective planes, as well as higher-dimensional projective spaces.