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arXiv · 2511.17082

Multidimensional quadrangle condition and cuboctahedra in latin hypercubes

Anna A. Taranenko
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Abstract

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 maximal 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.

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Anna A. Taranenko. 2026-04-01. Multidimensional quadrangle condition and cuboctahedra in latin hypercubes. https://arxiv.org/abs/2511.17082

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