Table of Contents
ISRN Geometry
Volume 2012 (2012), Article ID 560760, 10 pages
http://dx.doi.org/10.5402/2012/560760
Research Article

Geometric Realization of Some Triangle-Free Combinatorial Configurations 𝟐 𝟐 πŸ‘

Department of Mathematics, University of Washington, Seattle, WA 98195-4350, USA

Received 17 April 2012; Accepted 15 May 2012

Academic Editors: E. Gutkin, B. Mohar, and J. S. Snoeyink

Copyright © 2012 Branko Grünbaum. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.