From Orthogonal Latin Squares to Finite Projective Planes

Finite projective planes and mutually orthogonal latin squares (also known as Graeco-Latin squares) are among the most interesting and beautiful mathematical visualizations, and they are also related to each other in this way:

A complete set of orthogonal latin squares of order n exists if and only if a finite projective plane of order n exists.

The above is true when n ≥3. Therefore:

The relationship can be difficult to visualize at first. On the left, a complete set of (n-1, so in this case 2) mutually orthogonal latin squares of order 3 is represented by the inner and outer “dimensions” of each square. Each color, in each dimension, appears once in each row and once in each column, and the pair of inner and outer colors that appears in each of the 9 squares is unique. On the right, a finite projective plane of order 3 has 13 points and 13 lines. Each line intersects with 4 points, and each point intersects with 4 lines. The diagrams seem quite different, but one can be constructed from the other. Let’s start with the latin squares.

The squares in this form use only 3 colors, no more than necessary. The difference between the two “dimensions” is represented by the inner and outer colors of each square. Nevertheless, we can choose to use another set of 3 colors for the inner dimension.

This set of latin squares actually has four variables: the inner dimension, the outer dimension, the rows, and the columns. So we can also add colors to represent the row and columns.

We can now switch to a different type of diagram, while keeping the same color coding so we can easily see the connection. The 9 squares, with 4 variables each, become 9 points, with 4 lines intersecting each point, and 3 points intersecting each line. The rows are parallel with each other and the columns are parallel with each other. Some of the diagonal lines have to be curvy, but the three lines in each of the two sets of diagonals are still considered to be parallel with each other. This is the affine plane of order 3.

In affine geometry, as in Euclidean geometry, parallel lines do not intersect. To move from the affine plane to the projective plane, we add a “horizon line.” Just as parallel train tracks appear to meet at a point on the horizon, so each set of parallel lines will meet at a point. So we will add a point for the 3 rows to meet, a point for the 3 columns to meet, and a point for each of the two sets of curvy diagonals to meet. These 4 points will be connected with each other by the “horizon line” or line at infinity.

Now we have progressed from a complete set of mutually orthogonal latin squares of order 3 to a finite projective plane of order 3. All that remains, if we wish, is to rearrange the points and lines to create a more symmetrical and attractive diagram. I am indebted to Petr Vojtěchovský for the idea of arranging the diagram in this way.