Coordinatizing n_3 configurations

Abstract

Given an n 3 configuration, a one-point extension is a technique that constructs (n + 1) 3 configurations from it. A configuration is geometric if it can be realized by a collection of points and straight lines in the plane. Given a geometric n 3 configuration with a planar coordinatization of its points and lines, a method is presented that uses a one-point extension to produce (n+1) 3 configurations from it, and then constructs geometric realizations of the (n + 1) 3 configurations. It is shown that this can be done using only a homogeneous cubic polynomial in just three variables, independent of n. This transforms a computationally intractable problem into a computationally practical one.

The configurations (13_3)

“…One of these is the Pappus configuration. Rational coordinatizations of them can be used as starting points for the one-point extension algorithm of [8]. Incidence tables of these three configurations are given in Table 1.…”

“…Given a list of (n 3 ) configurations with rational coordinatizations, the software that generates the configurations ((n + 1) 3 ) by one-point extensions also finds rational coordinatizations. The algorithm is described in [8]. This software was used to find rational coordinatizations of most of the (10 3 ) configurations, and all of the (11 3 ) and (12 3 ) configurations.…”

“…Two recent reference books on configurations are Grünbaum [4] and Pisanski-Servatius [10]. The current paper builds upon the author's previous papers [7,8], where one-point extensions and a coordinatization algorithm are presented.…”

“…Table 2: Integer coordinatizations of the configurations (9 3 ) extension only extends the incidence structure of an (n 3 ) configuration, not the coordinatization. Using the coordinatization algorithm of [8], when a geometric configuration is derived by a one-point extension from a smaller geometric configuration, the planar drawing of the smaller configuration can usually be extended to a drawing of the configuration in question.…”

One-point extensions in n_3 configurations

“…The significance of this construction is that if the n 3 configuration is geometric, with a given coordinatization, then there is usually a simple method to extend the coordinatization to the (n + 1) 3 configuration, that is, the (n + 1) 3 configuration will also be geometeric. This is too long to include here, it will be the subject of another paper, currently in preparation [8].…”