Geometric constructions for symmetric 3-configurations

Leah Wrenn Berman¹

¹University of Alaska Fairbanks

The electronic journal of combinatorics, Tome 20 (2013) no. 3

Résumé

A geometric 3-configuration is a collection of points and straight lines, typically in the Euclidean plane, in which every point has 3 lines passing through it and every line has 3 points lying on it, and we say that such configuration is symmetric if there are non-trivial isometries of the plane that map the configuration to itself. Many symmetric 3-configurations may be easily constructed with computer algebra systems using algebraic techniques: e.g., constructing a number of symmetry classes of points and lines, by various means, and then determining the position of a final class of points or lines by solving some polynomial equation. In contrast, this paper presents a number of ruler-and-compass-type constructions for exactly constructing various types of symmetric 3-configurations, as long as the vertices of an initial regular m-gon are explicitly provided. In addition, it provides methods for constructing chirally symmetric 3-configurations given an underlying unlabelled reduced Levi graph, for extending these constructions to produce dihedrally symmetric 3-configurations, and for constructing 3-configurations corresponding to all 3-orbit and 4-orbit reduced Levi graphs that contain a pair of parallel arcs. Notably, most of the configurations described are movable: that is, they have at least one continuous parameter.

Détail

DOI : 10.37236/2341

Classification : 51E30, 05B30
Mots-clés : configurations, discrete geometry, incidence geometry