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.
Highlights
The main purpose of this paper is to illustrate the mutual benefit to combinatorics and geometry by considering a topic from both sides
By nk we mean an incidence structure of n distinct points and n distinct blocks, each block contains k points, each point belongs to k blocks, and each pair of distinct points belongs to at most one block
A triangle in a configuration is a set of three points P1, P2, P3, and three blocks B1, B2, B3, such j
Summary
One is that the combinatorially unique 223 configuration with automorphisms group of order 22 has three distinct geometric realizations by astral configurations. The determination of general conditions for a combinatorial configuration to be realizable by a geometric one is one of the deepest problems regarding configurations.
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