The Apollonian tiling of the plane into circles is analyzed with respect to its group properties. The relevant group, which is non-compact and discrete, is found to be identical to the symmetry group of a particular geometric tree-graph in hyperbolic three-space. A linear recursive method to compute the radii is obtained. Certain modiications of the problem are investigated, and relations to ot...

In this paper we show that the equivalence of the Apollonian metric and its inner metric remains unchanged by the removal of a point from the domain. For this we need to assume that the complement of the domain is not contained in a hyperplane. This improves a result of the authors wherein the same conclusion was reached under the stronger assumption that the domain contains an exterior point.

he present study sets the psychological type profile and temperament of 190 Catholic priests serving in England, Wales, Ireland alongside 1,298 reported by Ruppart 1985. In current 68% preferred introversion, 61% sensing, 53% feeling, 84% judging, confirming findings other recent studies Australia Italy a trend towards away from feeling. 55% presented as Epimethean Temperament (SJ) 21% Apolloni...