Of course! The hyperboloid has positive curvature (relative to its embedding in R^3), and the Poincaré disk has negative curvature, so I should've known something was, er, fishy. (This is, gallingly, a mistake I believe I have made before; I wonder how many more times I'll have to make it before I stop doing it every decade or so?) Anyway, thanks Adam! Jim On Saturday, March 21, 2015, Adam P. Goucher <apgoucher@gmx.com> wrote:
The hyperbolic plane isn't isometric to the hyperboloid (except when you impose an `inner product' with a (+, +, -) signature to the ambient space), so the fish would appear really distorted.
Sincerely,
Adam P. Goucher
Sent: Saturday, March 21, 2015 at 7:26 PM From: "James Propp" <jamespropp@gmail.com <javascript:;>> To: math-fun <math-fun@mailman.xmission.com <javascript:;>> Subject: [math-fun] Fish on a hyperboloid
Is it really possible that no one's rendered Escher's hyperbolic fish on the surface of a hyperboloid, where they would have more room to swim and feel altogether happier?
I went to images.google.com but didn't find an image of such an object.
I realize that a picture of such a surface would not do its symmetries justice. Maybe someone has created an animation that permits 3D rotation? That'd be almost as good as a tangible object. And if it permitted zooming in and zooming out, it'd be even better than the "real thing".
Jim Propp
P.S. Insert your own "snakes on a plane" joke here. _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com <javascript:;> https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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