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The Grünbaum--Rigby configuration as a special Kárteszi configuration

Gábor Gévay, György Kiss, Tomaž Pisanski

TL;DR

Addresses the relationship between the Grünbaum–Rigby $(GR(21_4))$ configuration and Kárteszi's $(n;\ell,m)$ configurations. The authors present a self-contained Kárteszi theorem using regular $n$-gons, define the three-parameter family $K(n;\ell,m)$, and derive a criterion for when these configurations realize without extra incidences, relying on Poonen–Rubinstein results on astral configurations. A key result is the identification of $(GR(21_4))$ with $K(7;2,3)$ and the explicit description of when $K(n;\ell,m)$ remains free of extra incidences (with the smallest nontrivial extra-incidence example $K(12;4,5)$). The work bridges historical constructions, clarifies realizability conditions for celestial 4-configurations, and suggests extensions to broader polycyclic families.

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.

The Grünbaum--Rigby configuration as a special Kárteszi configuration

TL;DR

Addresses the relationship between the Grünbaum–Rigby configuration and Kárteszi's configurations. The authors present a self-contained Kárteszi theorem using regular -gons, define the three-parameter family , and derive a criterion for when these configurations realize without extra incidences, relying on Poonen–Rubinstein results on astral configurations. A key result is the identification of with and the explicit description of when remains free of extra incidences (with the smallest nontrivial extra-incidence example ). The work bridges historical constructions, clarifies realizability conditions for celestial 4-configurations, and suggests extensions to broader polycyclic families.

Abstract

In 1990, Branko Grünbaum and John Rigby presented a 4-configuration, known today as the \emph{Grünbaum--Rigby configuration}; it is denoted by . 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 . 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 is isomorphic to . We present a theorem that gives necessary and sufficient conditions on parameters such that the corresponding configuration is realisable as a geometric polycyclic configuration with -fold rotational symmetry and no extra incidences.

Paper Structure

This paper contains 4 sections, 6 theorems, 16 equations, 6 figures.

Table of Contents

  1. Introduction
  2. From Kárteszi Theorem to Kárteszi configurations
  3. When $\mathrm{K}(n;\ell,m)$ contains extra incidences
  4. Conclusion

Key Result

Proposition 1

The Kárteszi configuration $\mathrm{K}(n;\ell,m)$ has no extra incidence if and only if there is no integer $x$ such that the 2-celestial 4-configuration (= astral configuration) $n \#(\ell,m,x)$ exists.

Figures (6)

  • Figure 1: The Grünbaum--Rigby configuration $\mathrm{GR}(21_4)$.
  • Figure 2: Drawing of $\mathrm \mathrm{K}(7;2,3)$ taken from the paper Kar65 of Kárteszi.
  • Figure 3: Example of an application of Kárteszi Theorem for $n = 13,\, \ell = 3$ and $m = 5$ producing $\mathrm K(13;3,5)$.
  • Figure 4: Illustration for Lemma \ref{['lem:rotate']}.
  • Figure 5: The smallest combinatorial Kárteszi configuration $\mathrm{K}(12;4,5)$ has a geometric representation with extra incidences.
  • ...and 1 more figures

Theorems & Definitions (9)

  • Proposition 1
  • Theorem 2: Kárteszi, 1964 Kar65
  • Lemma 3
  • proof
  • Theorem 4: Poonen and Rubinstein, 1998
  • Lemma 5
  • proof
  • Theorem 6
  • proof