Latin Polytopes

Latin squares are well studied combinatorial objects. In this paper the concept is generalized and new designs are proposed (Latin triangles, Latin tetrahedra, etc.) that feature similar properties.

The paper starts with a classic definition of Latin squares followed by one based on concepts of modern design theory. A Latin square appears then as a combinatorial design whose points are geometric. Its rows and columns are now symmetric lines that intersect in specific ways, while its “labelled lines” intersect the former also in a particular manner.

The generalization that follows proceeds by 1. broadening the inherent symmetry of the Latin square 2. considering more general configurations of points and 3. admitting symmetric and labelled lines that intersect more freely. The resulting concept is the Latin board. Finally, this object is particularized to define Latin polytopes, Latin polygons and Latin polyhedra.

A Latin tetrahedron. Fold the picture along the black lines to create a tetrahedron.
Now each band between lines of the same color contains all numbers from 1 to 16.