On some packings of the hyperbolic plane by incongruent hypercyclic domains (Q1315119)

In an earlier paper [Period. Math. Hung. 10, 217-229 (1979; Zbl 0413.52008)] the author studied packings of the hyperbolic plane by congruent equidistant-curves. Now he studies packings by incongruent equidistant-curves whose altitudes assume three different values. Three equidistant-curves with altitudes \(0<\ell_ 1<\ell_ 2<\ell_ 3\) determine an arc-triangle obtained by pushing the equidistant-curves together; the main lemmas (which are of independent interest) involve inequalities on the areas of these arc-triangles. There are fine diagrams which make the analysis ``visible''. The paper concludes with a table giving the densities of packings when \(\ell_ 1\), \(\ell_ 2\), \(\ell_ 3\) are proportional to \(c\), \(2c\), \(3c\) for \(c=0.1,0.5,1,2\). These densities can be made close to \({3\over\pi}\), which is the maximal density of the horocyclic packings.