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

TL;DR

This paper demonstrates that the Grünbaum--Rigby configuration is a special case of Kártészzi configurations and provides conditions for their geometric realizability with rotational symmetry.

Contribution

It establishes the isomorphism between the Grünbaum--Rigby configuration and a specific Kártészzi configuration and offers criteria for geometric realizability.

Findings

GR(21_4) is isomorphic to K(7;2,3)

Provides necessary and sufficient conditions for geometric realizability

Characterizes configurations with n-fold rotational symmetry

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.