Geometric Features of Latin Squares by  dr. ir. Y.M. de Haan

We live in a universe of patterns

Ian Stewart

 

Summary

The geometric structures of Latin squares are studied in this document. It is shown that the full collection of Latin squares of a certain order can be partitioned according to (equal) geometric structure. The full sets of 576 4x4 Latin squares and 161280 of order 5 can be stepwise reduced to no more than 5 and 18 really different Latin squares.
Approaches to perform this reduction process include: extending the notion of a Latin square to a cyclic or periodic structure, establishing symmetry relations between different Latin squares, and utilizing that the the distributions of same-symbol sites in Latin squares follow a relatively small number of geometric patterns (2, 4 and 10 for orders 4, 5 and 6). The numbers of ways in which these "Latin patterns" can be superposed to form Latin squares are shown to be rather limited as well.