Chiral Astral Realizations of Cyclic 3-Configurations

Abstract

A cyclic $(n_{3})$ configuration is a combinatorial configuration whose automorphism group contains a cyclic permutation of the points of the configuration; that is, the points of the configuration may be considered to be elements of ${\mathbb {Z}}_{n}$, and the lines of the configuration as cyclic shifts of a single fixed starting block [0, a, b], where $a, b \in {\mathbb {Z}}_{n}$. We denote such configurations as $\mathrm{Cyc}_{n}(0,a,b)$. One of the fundamental questions in the study of configurations is that of geometric realizability. In the case where $n = 2m$, it is combinatorially possible to divide the points and lines of the configuration into two classes according to parity, so it is natural to ask whether the configuration can be realized using those classes. We provide methods for producing geometric realizations of configurations $\mathrm{Cyc}_{2m}(0,a,b)$ that have two symmetry classes under the maximal rotational subgroup of the geometric symmetry group (that is, chiral astral realizations), and we provide a number of constraints on a and b that guarantee such a realization exists. Experiments on up to 500 points suggest that, with the exception of some small sporadic examples and a single infinite family $\mathrm{Cyc}_{2(k+1)}(0,1,k)$, $k \ge 3$ and k odd, all cyclic $(2m_{3})$ configurations are realizable as geometric chiral astral configurations using the methods described in this paper.