Fred wrote: << On 2/13/08, [I] wrote: << I don't think I ever considered this before, but it seems I just figured out all possible ways such that, given a real number c > 0, one can construct a bijection of the hyperbolic plane H^2 to itself that multiplies all distances by c: f: H^2 -> H^2 such that dist(f(x),f(y)) = c dist(x,y) for all x, y in H^2. (If the word isn't "homothety", then I withdraw it.) I wonder if anyone else would care to tackle the same question.

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... [Fred's argument goes here] ... The conclusion is that there are no nontrivial hyperbolic homotheties --- which is not too surprising, since (a) we would otherwise surely have heard of them already; and (b) there are none in spherical space.

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That's the conclusion I reached. My reasoning wasn't entirely from scratch: It's well known that the isometries of the hyperbolic plane H^2 are precisely its angle-preserving bijections. (I.e., bijections that are conformal or anti-conformal.) But any bijection of a manifold that multiplies distance by a fixed constant c must be angle-preserving, and so in this case is an isometry of H^2. Thus c = 1. --Dan P.S. I agree with Fred's sociological reasoning that if non-isometric homotheties of H^2 exzisted, we would have heard of them. But I was somewhat surprised that they don't, since if the Euclidean plane R^2 is infinite, H^2 seems "more" infinite, in that there is so much extra area. With my naive intuition, it felt like there ought to be more flexibility to arrange the points as desired. But the nonexsitence is entirely consistent with and also follows from the fact that a regular hyperbolic polygon that tiles H^2 with a fixed number per vertex -- e.g., say a hyperbolic square that occurs 5 per vertex -- can occur only in one exact size and shape: its side length is uniquely determined. If the fictional homotheties existed, this would not be true. _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele