Quadratical quasigroups and Mendelsohn designs

Let the product of points [Formula: see text] and [Formula: see text] be the vertex [Formula: see text] of the right isosceles triangle for which [Formula: see text] is the base, and [Formula: see text] is oriented anticlockwise. This yields a quasigroup that satisfies laws [Formula: see text], [Formula: see text] and [Formula: see text]. Such quasigroups are called quadratical. Quasigroups that satisfy only the latter two laws are equivalent to perfect Mendelsohn designs of length four ([Formula: see text]). This paper examines various algebraic identities induced by [Formula: see text], classifies finite quadratical quasigroups, and shows how the square structure of quadratical quasigroups is associated with toroidal grids.