Constructing 5-Configurations with Chiral Symmetry

A 5-configuration is a collection of points and straight lines in the Euclidean plane so that each point lies on five lines and each line passes through five points. We describe how to construct the first known family of 5-configurations with chiral (that is, only rotational) symmetry, and prove that the construction works; in addition, the construction technique produces the smallest known geometric 5-configuration. In recent years, there has been a resurgence in the study of k-configurations with high degrees of geometric symmetry; that is, in the study of collections of points and straight lines in the Euclidean plane where each point lies on k lines and each line passes through k points, with a small number of symmetry classes of points and lines under Euclidean isometries that map the configuration to itself. 3-configurations have been studied since the late 1800s (see, e.g., [15, Ch. 3], and more recently [9, 12, 13]), and there has been a great deal of recent investigation into 4-configurations (e.g., see [1, 2, 5, 8, 14]). However, there has been little investigation into k-configurations for k > 4. Following [10], we say that a geometric k-configuration is polycyclic if a rotation by angle 2πi m for some integers i and m is a symmetry operation that partitions the points and lines of the configuration into equal-sized symmetry classes (orbits), where each orbit contains m points. If n = dm, then there are d orbits of points and lines under the rotational symmetry. The group of symmetries of such a configuration is at least cyclic. In many cases, the full symmetry group is dihedral; this is the case for most known polycyclic 4-configurations. A k-configuration is astral if it has b 2 c symmetry classes of points and of lines under rotations and reflections of the plane that map the configuration to itself. It has been conjectured that there are no astral 5-configurations, which would have 3 symmetry the electronic journal of combinatorics 17 (2010), #R2 1 classes of points and lines [6], [11, Conj. 4.1.1]; support for this conjecture was given in [3], where it was shown that there are no astral 5-configurations with dihedral symmetry. The existence of astral 5-configurations with only cyclic symmetry is still unsettled but is highly unlikely. Until recently, there were no known families of 5-configurations with a high degree of symmetry in the Euclidean plane. There were a few recently discovered examples in the extended Euclidean plane [12], [11, Section 4.1], but these are not polycyclic, since not all of the symmetry classes have the same number of points. The 5-configurations described in this paper have four symmetry classes of points and lines and chiral symmetry (that is, they have no mirrors of reflective symmetry); it is likely that they are as symmetric as possible. 1 2-astral configurations The construction of the 5-configurations begins with astral 4-configurations. Such a configuration, also known as a 2-astral configuration, may be described by the symbol