Polycyclic Geometric Realizations of the Gray Configuration

TL;DR

This paper investigates geometric realizations of the Gray configuration, demonstrating the existence of two distinct Z_3 symmetric realizations and showing that Z_9 symmetric realizations are only weak, with additional unintended incidences.

Contribution

It provides the first explicit constructions of polycyclic realizations of the Gray configuration with Z_3 symmetry and clarifies limitations for Z_9 symmetric realizations.

Findings

Two distinct Z_3 symmetric realizations are constructed.

Z_9 symmetric realization is only weak, with extra incidences.

The Gray configuration's symmetries influence its geometric realizations.

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).