----- Original Message ---- From: Fred lunnon <fred.lunnon@gmail.com> To: math-fun <math-fun@mailman.xmission.com> Sent: Wednesday, February 13, 2008 4:03:45 PM Subject: Re: [math-fun] Homotheties of the hyperbolic plane ... Which reminds me of another similarly off-the-wall query. In the Poincar\'e model ("circle" including line and point in Euclidean plane) circles within the disc correspond to Circles; circles orthogonal to the disc correspond to Lines; circles tangent to the disc correspond to Horocycles; circles meeting to the disc correspond to Equidistant Curves; but circles outside the disc correspond to what in Hyperbolic space? WFL _______________________________________________ There is one more circle, the boundary of the disk. It is infinity, the set of directions in the hyperbolic plane. As for the "outside", it seems to be a parallel universe, another hyperbolic plane. On the Poincare sphere, pick a circle, call it "infinity". It separates the sphere into two caps, which can be made into two hyperbolic planes. Conformal maps on the sphere which preserve infinity become congruences in the hyperbolic planes. Reflection through infinity exchanges the two universes and establishes a correspondence between their objects. A circle orthogonal to infinity reflects onto itself, so its two hyperbolic lines correspond. On the other hand, a circle intersecting infinity nonorthogonally reflects onto a different circle, and so its two equidistant curves are noncorresponding. No doubt much more can be said. In particular, many parallel universes can coexist on the Poincare sphere, and be related via groups of conformal mappings. Gene ____________________________________________________________________________________ Looking for last minute shopping deals? Find them fast with Yahoo! Search. http://tools.search.yahoo.com/newsearch/category.php?category=shopping