Danzer's configuration revisited

TL;DR

This paper explores the geometric and combinatorial properties of Danzer's DCD(4) configuration, part of an infinite series, and investigates its symmetries and realizations.

Contribution

It revisits Danzer's configuration within an infinite series and examines its geometric realizations and symmetry properties.

Findings

No geometric realizations with 5- or 7-fold rotational symmetry.

Found a point-circle realization with dihedral symmetry D_7.

Conjecture on the absence of certain symmetric realizations.

Abstract

We revisit the configuration of Danzer DCD(4), a great inspiration for our work. This configuration of type (35_4) falls into an infinite series of geometric point-line configurations DCD(n). Each DCD(n) is characterized combinatorially by having the Kronecker cover over the Odd graph $O_n$ as its Levi graph. Danzer's configuration is deeply rooted in Pascal's Hexagrammum Mysticum. Although the combinatorial configuration is highly symmetric, we conjecture that there are no geometric point-line realizations with 7- or 5-fold rotational symmetry; on the other hand, we found a point-circle realization having the symmetry group $D_7$, the dihedral group of order 14.