Geometric Realization of Some Triangle-Free Combinatorial Configurations

Abstract

The main purpose of this paper is to illustrate the mutual benefit to combinatorics and geometry by considering a topic from both sides. Al-Azemi and Betten enumerate the distinct combinatorial (223) configurations that are triangle free. They find a very large number of such configurations, but when taking into account the automorphism group of each, they find two cases in which there is only a single configuration. On the heuristic assumption that an object that is unique in some sense may well have other interesting properties, the geometric counterparts of these configurations were studied. Several unexpected results and problems were encountered. One is that the combinatorially unique (223) configuration with automorphisms group of order 22 has three distinct geometric realizations by astral configurations.