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I need to write up a quickie description of Hyperbolic Geometry for non-mathematicians. I am trying to say "Hyperbolic Geoemtry is the Geometry of the surface of a ____"

I remember that there is, in fact, a term for the surface I am thinking of. It is a surface of constant curvature -1, and the model of the Hyperbolic Plane on this surface lends itself very easily to seeing the "skinny triangles" so to speak (ones whose angle measures sum to less than 180). Does anyone remember what the hell this surface is called? I think it starts with a 't'? I honestly can't remember and Google isn't helping.

Thanks in advance, Stackexchange.

Ilan Weinschelbaum
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In general, a surface of constant negative curvature is called a pseudosphere, being the hyperbolic equivalent of a sphere. A quick search reveals that the particular shape you're likely thinking of, which is the unique solid of revolution of constant negative curvature, is a tractricoid. Though, notably, it is not globally isomorphic to the hyperbolic plane, as it contains a non-smooth "equator".

Milo Brandt
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  • YES! THE TRACTRICOID! thank you. I knew it started with a 't'.... – Ilan Weinschelbaum Sep 22 '14 at 22:25
  • Also isn't half the tractricoid, if you allow it to extend infinitely in all directions, isomorphic to the hyperbolic plane? We're getting into areas I have not explicitly studied, so I am not 100% confident here. But I was under the impression this was true... – Ilan Weinschelbaum Sep 22 '14 at 22:42
  • Hilbert's theorem precludes the existence of an immersion of the whole hyperbolic plane into Euclidean 3-space - so no such thing is possible; intuitively, if you cut along the singular equator and try to extend it, either you create an inflection point in the profile (0 curvature) or a "cup" (positive curvature). However, it might be possible to do something like identifying opposite points on the equator to get a model of the hyperbolic plane, but I'm not sure. – Milo Brandt Sep 22 '14 at 23:36
  • Thanks! I'll definitely read that theorem. Can't remember where I heard that before. – Ilan Weinschelbaum Sep 23 '14 at 03:12
  • A simple reasoning goes a bit like this: you cannot fit a surface of constant curvature into space of larger curvature. You can't fit a circle onto a sphere that has smaller radius than the circle. You can't fit an Euclidean line onto a sphere of any radius. Going up a dimension, a spherical space can only accomodate spheres with radius smaller or equal to the space itself, but not Euclidean planes, and Euclidean plane can't fit into any spherical space. And hyperbolic plane has negative curvature, it's LESS curved than Euclidean space, and so it can't fit in there. – Marek14 Aug 08 '17 at 08:42