Can anyone suggest a good way to decorate a pseudosphere? Just putting a grid on it, as in the image at the top of Paolo Lazzarini's page http://www.paololazzarini.it/geometria_sulla_sfera/modelli_noneu_start.htm, isn't bad, but I'm looking for something snazzier to serve as the visual "brand" for my blog. Tim Hoffman's image http://www.math.kobe-u.ac.jp/DIVERSIONS/TIMH1.JPG appeals to me more, but I don't know what curves are being approximated. Can any of you figure out what's going on in this image? I like Figure 6' in I. Tadao's "Hyperbolic Non-Euclidean World and Figure-8 Knot" site http://web1.kcn.jp/hp28ah77/us20_pseu.htm, but I'm suspicious. Tadao writes "Fig. 6' shows that the red concentric circles are certainly at regular intervals and that all the white circles are surely the same in size and placed at normal, regular intervals." But is this really true? It's unclear to me why the even spacing assertion holds when one crosses the cut-line. In any case, what I'm hoping for ultimately is an animated gif showing a decorated half-pseudosphere rotating about a vertical symmetry axis (looking something like a sorcerer's cap). I imagined at first that one of Escher's circle-limit pictures could be transported from the Poincare disk to the pseudosphere, but I don't think that that can work. The pseudosphere is obtained from the Poincare disk by cutting out a part of it and then gluing parts of the boundary together; this gluing doesn't play nicely with the symmetry groups of reflection tilings of the hyperbolic plane. (Or maybe it does, and I'm not thinking clearly.) Ditto for Apollonian gaskets and the like. Jim Propp