The Grünbaum--Rigby configuration as a special Kárteszi configuration

Abstract

In 1990, Branko Grünbaum and John Rigby presented a 4-configuration, known today as the \emph{Grünbaum--Rigby configuration}; it is denoted by $\mathrm{GR}(21_4)$. Independently and earlier, in 1986, Ferenc Kárteszi 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 Grünbaum and Rigby. In this paper, we explore some properties of the \emph{Kárteszi 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.