Multidimensional Quadrangle Condition and Cuboctahedra in Latin Hypercubes

The well-known quadrangle criterion states that a latin square is isotopic to the Cayley table of a group if and only if all quadrangles spanned by the same triple of symbols coincide on the fourth symbol. Gowers and Long (2020) reformulated this result in the following way: the Cayley tables of the most associative quasigroups have the maximum number of octahedra. In the present paper, we state the multidimensional quadrangle condition for $d$-dimensional latin hypercubes in terms of the reconstruction of submatrices of order 2 from a bundle of $d+1$ entries and in terms of the maximum number of cuboctahedra. In particular, we show that the most associative $d$-ary quasigroups have Cayley tables such that every 2-dimensional plane is isotopic to a latin square that is principally isotopic to the Cayley table of a group. We also estimate the number of cuboctahedra in latin squares and hypercubes from below and provide computational results.