Geometric constructions for 3-configurations with non-trivial geometric symmetry

作者：Berman Leah Wrenn^*

来源：Electronic Journal of Combinatorics, 2013, 20(3): P9.

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; that is, it is an (n(3)) configuration for some number n of points and lines. We will 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.

• 出版日期2013-8-2