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arXiv · 2512.18872

The Gr\"unbaum--Rigby configuration as a special K\'arteszi configuration

Gábor Gévay·György Kiss·Tomaž Pisanski
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Abstract

In 1990, Branko Gr\"unbaum and John Rigby presented a 4-configuration, known today as the \emph{Gr\"unbaum--Rigby configuration}; it is denoted by $\mathrm{GR}(21_4)$. Independently and earlier, in 1986, Ferenc K\'arteszi published a paper in which he proved a theorem in real geometry that gives rise to a series of 4-configurations $\mathrm{K}(n;\ell,m)$. In an even earlier paper from 1964, he presented a figure which is essentially the same as that given by Gr\"unbaum and Rigby. In this paper, we explore some properties of the \emph{K\'arteszi configurations} and in particular show that $\mathrm{GR}(21_4)$ is isomorphic to $\mathrm{K}(7;2,3)$. We present a theorem that gives necessary and sufficient conditions on parameters $n,\ell,m$ such that the corresponding configuration $\mathrm{K}(n;\ell,m)$ is realisable as a geometric polycyclic configuration with $n$-fold rotational symmetry and no extra incidences.

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Gábor Gévay, György Kiss, Tomaž Pisanski. 2025-12-21. The Gr\"unbaum--Rigby configuration as a special K\'arteszi configuration. https://arxiv.org/abs/2512.18872

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