Sparse line deletion constructions for symmetric 4-configurations

Leah Wrenn Berman; William H. Mitchell

Ars mathematica contemporanea, Volume 9 (2015) no. 2, pp. 165-186

Abstract

A 4-configuration is a collection of points and lines in the Euclidean plane such that each point lies on four lines and each line passes through four points. In this paper we introduce a new family of these objects. Our construction generalizes a 2010 result of Berman and Grünbaum in which suitable 4-configurations from the well-understood celestial family are altered to yield new configurations with reduced geometric symmetry groups. The construction introduced in 2010 removes every other line of a symmetry class from the celestial configuration; here we we give conditions under which every p-th line can be removed, for p ∈ {2, 3, 4, ⋯}. The geometric symmetry groups of the new configurations we obtain are of correspondingly smaller index as subgroups of the symmetry group of the underlying celestial configuration. These sparse constructions can also be repeated and combined to yield a rich variety of previously unknown 4-configurations. In particular, we can begin with a configuration with very high geometric symmetry—the dihedral symmetry of an m-gon for m quite large—and produce a configuration whose only geometric symmetry is 180 ∘  rotation.

Article information

DOI: 10.26493/1855-3974.478.bc1

Keywords: Configurations.