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Polycyclic Geometric Realizations of the Gray Configuration

Leah Wrenn Berman, Gábor Gévay, Tomaž Pisanski

TL;DR

The paper investigates geometric realizations of the Gray configuration, a $(27_3)$ configuration with Levi graph the Gray graph, under polycyclic symmetry. By analyzing the automorphism group $Aut\,G$ of the Gray graph, the authors classify semi-regular subgroups and derive all possible reduced Levi graphs, focusing on $\mathbb{Z}_3$, $\mathbb{Z}_3\times\mathbb{Z}_3$, $\mathbb{Z}_9$, and $\mathbb{Z}_9\rtimes\mathbb{Z}_3$. They construct explicit polycyclic realizations with $\mathbb{Z}_3$ symmetry corresponding to the Pappus RLG and the GG RLG, while proving that any straight-line $\mathbb{Z}_9$ realization would be a weak realization due to extra incidences; a topological/pseudoline realization exists for the $9$-fold case. The results illuminate how quotienting the Levi graph by semi-regular actions yields realizable geometric configurations and highlight limits of straight-line realizations under higher symmetry, pointing to broader questions about semisymmetric graphs and geometric realizability.

Abstract

The Gray configuration is a (27_3) configuration which typically is realized as the points and lines of the 3 x 3 x 3 integer lattice. It occurs as a member of an infinite family of configurations defined by Bouwer in 1972. Since their discovery, both the Gray configuration and its Levi graph (i.e., its point-line incidence graph) have been the subject of intensive study. Its automorphism group contains cyclic subgroups isomorphic to Z_3 and Z_9, so it is natural to ask whether the Gray configuration can be realized in the plane with any of the corresponding rotational symmetry. In this paper, we show that there are two distinct polycyclic realizations with Z_3 symmetry. In contrast, the only geometric polycyclic realization with straight lines and Z_9 symmetry is only a "weak" realization, with extra unwanted incidences (in particular, the realization is actually a (27_4) configuration).

Polycyclic Geometric Realizations of the Gray Configuration

TL;DR

The paper investigates geometric realizations of the Gray configuration, a configuration with Levi graph the Gray graph, under polycyclic symmetry. By analyzing the automorphism group of the Gray graph, the authors classify semi-regular subgroups and derive all possible reduced Levi graphs, focusing on , , , and . They construct explicit polycyclic realizations with symmetry corresponding to the Pappus RLG and the GG RLG, while proving that any straight-line realization would be a weak realization due to extra incidences; a topological/pseudoline realization exists for the -fold case. The results illuminate how quotienting the Levi graph by semi-regular actions yields realizable geometric configurations and highlight limits of straight-line realizations under higher symmetry, pointing to broader questions about semisymmetric graphs and geometric realizability.

Abstract

The Gray configuration is a (27_3) configuration which typically is realized as the points and lines of the 3 x 3 x 3 integer lattice. It occurs as a member of an infinite family of configurations defined by Bouwer in 1972. Since their discovery, both the Gray configuration and its Levi graph (i.e., its point-line incidence graph) have been the subject of intensive study. Its automorphism group contains cyclic subgroups isomorphic to Z_3 and Z_9, so it is natural to ask whether the Gray configuration can be realized in the plane with any of the corresponding rotational symmetry. In this paper, we show that there are two distinct polycyclic realizations with Z_3 symmetry. In contrast, the only geometric polycyclic realization with straight lines and Z_9 symmetry is only a "weak" realization, with extra unwanted incidences (in particular, the realization is actually a (27_4) configuration).

Paper Structure

This paper contains 10 sections, 3 theorems, 8 equations, 18 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. The automorphism group of the Gray configuration
  3. Finding possible quotient graphs and reduced Levi graphs from semi-regular subgroups
  4. Labeling elements of the Gray configuration and identifying the reduced Levi graphs
  5. A polycyclic realization of the Gray Configuration with the Pappus RLG
  6. A polycyclic realization of the Gray configuration with threefold rotational symmetry, using the $GG$ RLG.
  7. 9-fold symmetry of the Gray Graph and Gray Configuration
  8. 9-fold symmetry of the Gray graph
  9. Constructing a $\mathbb{Z}_{9}$ realization
  10. Conclusion and open questions

Key Result

Theorem 1

Assume that a geometric figure $\mathcal{F}$ changes continuously in such a way that Consider two points $P, P'\in \mathcal{F}$, both different from $O$. Then, if $P$ moves along a path $\ell$, then $P'$ moves along a path $\ell'$ such that $\ell'$ is an image of $\ell$ under a dilative rotation.

Figures (18)

  • Figure 1: The Gray graph. The square nodes correspond to lines and the circular nodes correspond to points; the coloring demonstrates rotational$\mathbb{Z}_{9}$ symmetry. This is further explored in Section \ref{['sect:9fold']}.
  • Figure 2: Spatial realization of the Gray configuration.
  • Figure 3: The five semi-regular quotient graphs of the Gray graph, and their IDs from Table \ref{['tab:subgroups']}. Since all quotients are bipartite, these graphs are all the possible RLGs of the Gray configuration.
  • Figure 4: The Gray Grid: the points and lines of the Gray configuration, viewed as points and lines on the $3 \times 3$ integer grid.
  • Figure 5: The Gray Grid labeled with the symmetry classes from Table \ref{['tab:SymmetryClassLabels']}.
  • ...and 13 more figures

Theorems & Definitions (10)

  • Definition 1
  • Remark 1
  • Remark 2
  • Definition 2
  • Theorem 1
  • Theorem 3
  • Lemma 4
  • proof
  • proof : Proof of Theorem \ref{['thm:nonrealizeZ9']}
  • Remark 3