Highly Incident Configurations with Chiral Symmetry

Abstract

A geometric k-configuration is a collection of points and straight lines in the plane so that k points lie on each line and k lines pass through this point. We introduce a new construction method for constructing k-configurations with non-trivial dihedral or chiral (i.e., purely rotational) symmetry, for any k ≥ 3; the configurations produced have 2 k−2 symmetry classes of points and lines. The construction method produces the only known infinite class of symmetric geometric 7-configurations, the second known infinite class of symmetric geometric 6-configurations, and the only known 6-configurations with chiral symmetry.

An infinite class of movable 5-configurations

“…This technique was extended in [7] to the Circumcircle Construction Lemma. Although the lemma can be stated as a more general incidence theorem [8], we state it as follows in order to facilitate the main construction in Section 4.…”

“…Suppose the elements of each symmetry class of elements are labelled cyclically counterclockwise, beginning from some chosen 0th ele- [3,5,8,9]. The Crossing Spans Lemma and its associated reduced Levi graph "gadget" are shown in Figure 2.…”

“…In this paper, we present a new construction that produces infinitely many new geometric 5-configurations which are movable: that is, there is at least one continuous degree of freedom in the construction while fixing 4 points in general position. This construction significantly generalizes the construction presented in [9] and removes the need to complete the construction via a continuity argument, instead providing an entirely ruler-and-compass construction for those configurations, given an initial m-gon. The new construction technique uses two elementary geometric lemmas, the Circumcircle Construction Lemma and the Crossing Spans Lemma, which previously have been used separately in other configuration construction techniques.…”

“…Two constructions that produce infinite families of 5-configurations with a reasonably small number of points and lines are known [7,9]. The (48 5 ) configuration shown in Figure 1a is the smallest known geometric 5-configuration and is the smallest example of the construction in [9]; a reasonably small example of the construction discussed in [7] is shown in Figure 1b (the smallest example is not intelligible at small scale).…”

“…The (48 5 ) configuration shown in Figure 1a is the smallest known geometric 5-configuration and is the smallest example of the construction in [9]; a reasonably small example of the construction discussed in [7] is shown in Figure 1b (the smallest example is not intelligible at small scale). In his monograph on configurations [14, Section 4.1], Grünbaum spends only 5 pages (mostly pictures) discussing the little that is known about 5-configurations.…”