A sphere is an elliptic surface with constant positive curvature. A pseudosphere is a hyperbolic surface with constant negative curvature. This Pseudosphere is called a Breather. Click here to download some POV-Ray code for this image. You can also see this image described as an “Imploding Flower” on Chewxy’s Math Art
new version: POV-Ray 3.6.1, 6/21/06
old version: Mathematica 4.2, MathGL3d, POV-Ray 3.6.1, 9/30/04
(* runtime: 6 seconds *)
a = 0.498888; vmax = 47.1232; w = Sqrt[1 - a^2];
Breather[u_, v_] := Module[{d = a((w Cosh[a u])^2 + (a Sin[w v])^2)}, x = -u + 2w^2 Cosh[a u]Sinh[a u]/d; y = 2w Cosh[a u](-w Cos[v]Cos[w v] - Sin[v]Sin[w v])/d; z = 2w Cosh[a u](-w Sin[v]Cos[w v] +Cos[v]Sin[w v])/d; {x, y, z, {EdgeForm[], SurfaceColor[Hue[v/vmax]]}}];
ParametricPlot3D[Breather[u, v], {u, -10, 10}, {v, 0, vmax}, PlotPoints -> {49, 79}, Compiled -> False]
Links
- Virtual Math Museum – beautiful rendition of this surface by Luc Benard, winning entry on cover of Science Magazine’s 2006 Visualization Challenge
- POV-Ray Code – by Mike Williams, param.inc
- Cooling Air Towers – also have a hyperbolic shape website.
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Paul Nylander 
Hi Paul,
This is a really beautiful image. I recently learned of it through Clifford Pickover’s The Math Book. On my blog today, I’m posting about Pickover’s book, and about your image. I hope it’s okay with you that I’m putting a copy of your image there, with, of course, credit and a link back to you, see http://www.rudyrucker.com/blog/2009/08/28/pseudospheres/
Best,
Rudy