Überdeckung der euklidischen Ebene durch zwei Einheitskreispackungen. (Covering of the Euclidean plane by two unit circles packings) (Q1086846)

The author discusses the following problem for packings of unit circles in the euclidean plane \(E^ 2\). Given such a packing P, when can we find another packing Q of unit circles such that \(P\cup Q\) cover \(E^ 2\). It is proved that Q exists if and only if P is a lattice packing, for which the underlying lattice has a basis \(\{\) a,b\(\}\) with \(| a| =| b| =2\) and 60\(\circ \leq |\sphericalangle (a,b)| \leq 90\circ\); then Q can be chosen as the translate of P by \((a+b)\). As a consequence, a covering C of \(E^ 2\) by unit circles splits into two packings of unit circles if and only if C is a lattice covering, for which the underlying lattice G has a basis \(\{a,(a+b)\}\), with a,b\(\in G\), \(| a| =| b| =2\) and 60\(\circ \leq |\sphericalangle (a,b)| \leq 90\circ.\) The paper presents parts of the author's Dissertation (Erfurt, 1984).