--WR Somsky: One thing you need to be very careful of when working w/ gears in non-euclidean spaces is that the circumferences -- which must be integral multiples of the gear tooth-spacing -- are NOT proportional to the radii. In euclidean space, the circumfrences and radii are proportional, so we can get by just working w/ the radii as integral multiples of some unit radius. --WDS: yes, already knew that. In fact, in S2, circumf = 2*pi*sin(angular radius) = 2*pi*(euclidean radius) where note, conveniently, the euclidean radius is what (in my earlier email) I noted got transformed rationally by inversive maps. In H2, circum = 2*pi*sinh(angular radius). -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)