Latin Polytopes

Latin squares are well studied combinatorial objects. In this paper we generalize the concept and propose new objects like Latin triangles, free Latin squares, Latin tetrahedra, free Latin cubes, etc. We start with a classic definition of Latin squares followed by one based on the concept of latinized board. A Latin square appears then as a combinatorial design whose points are geometric and whose lines (the rows and columns) are invariant under the symmetries of the square. The generalization that follows proceeds by 1. broadening this geometric symmetry 2. considering more general configurations of points and 3. admitting lines that intersect more freely. The resulting concept is the Latin board. Finally, we particularize Latin boards to define Latin polytopes, Latin polygons and Latin polyhedra.