Gene writes [of the hyperbolic plane H^2]: << That no nontrivial homothety exists is clear from the observation that a space of constant nonzero curvature possesses an intrinsic length scale, namely the inverse of its curvature.

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Perhaps this is the best short description of why this is impossible. In any case, the impossibility of a bijection H^2 -> H^2 that increases all distances by a factor of (say) 2 brings up an interesting question: Can one quantify just how close one can get to such a map? --Dan _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele