Multidimensional quadrangle condition and cuboctahedra in latin hypercubes
Anna A. Taranenko
TL;DR
This work generalizes the classical quadrangle condition from latin squares to $d$-dimensional latin hypercubes, using cuboctahedra as the higher-dimensional analogue of octahedra to quantify associativity. The authors prove a multidimensional quadrangle condition: the reconstruction of $2\times 2$ subcubes from bundles is possible precisely when every $2$-D plane is principally isotopic to a latin square isotopic to a group Cayley table, which characterizes maximal cuboctahedra. They show that Cayley tables of $d$-iterated groups are among the most associative and achieve the maximum cuboctahedra, and they provide explicit lower bounds for the number of cuboctahedra in latin squares and hypercubes, along with extensive enumeration data for small orders and dimensions. The results connect structural isotopy properties to combinatorial counts, yielding insights into the reducibility of highly structured hypercubes and offering computational benchmarks for small cases.
Abstract
The well-known quadrangle criterion states that a latin square is istopic to the Cayley table of a group if and only if all quadrangles spanned by the same triple of symbols coincide in the fourth symbol. Gowers and Long (2020) reformulated it 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 subcubes 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 must associative $d$-ary quasigroups have Cayley tables such that every $2$-dimensional plane is isotopic to a latin square 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 some computational results.