What are the interesting applications of hyperbolic geometry?

I am aware that, historically, hyperbolic geometry was useful in showing that there can be consistent geometries that satisfy the first 4 axioms of Euclid's elements but not the fifth, the infamous parallel lines postulate, putting an end to centuries of unsuccesfull attempts to deduce the last axiom from the first ones.

It seems to be, apart from this fact, of genuine interest since it was part of the usual curriculum of all mathematicians at the begining of the century and also because there are so many books on the subject.

However, I have not found mention of applications of hyperbolic geometry to other branches of mathematics in the few books I have sampled. Do you know any or where I could find them?

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There are others on this site who are far more capable of giving a pertinent answer to this than me. While you're waiting for one to appear, you could have a glance at the Wikipedia page on the modular group and its relationship to hyperbolic geometry which is certainly one of the principal sources of interest. – t.b. Dec 23 '11 at 20:54
Uniformization (en.wikipedia.org/wiki/Uniformization_theorem) is among the most important results in the theory of Riemann surfaces – user8268 Dec 23 '11 at 23:21
Again, I'm not the right person to give a full-fledged answer here, but I would like to make the case for perhaps looking at the question in a slightly different way. From my point of view, hyperbolic geometry isn't an "idea that has applications" as much as a phenomenon that pops up throughout mathematics. I would classify both of the results already mentioned (geometrization conjecture and the uniformization theorem) as examples of hyperbolic geometry as a phenomenon (i.e. the idea that lots of manifolds are naturally hyperbolic) rather than examples of applications. – NKS Dec 23 '11 at 23:31
One thing I'll throw out there while waiting for someone who knows more about this than I do is that the field of geometric group theory is all about studying the ways in which groups can be assigned a geometry, which is often hyperbolic. The geometry of the group has algebraic consequences; for instance hyperbolic groups have solvable word problem. – NKS Dec 23 '11 at 23:44
I wrote a comment that disappeared so I'll write it again and hope it doesn't show up twice. Marilyn vos Savant wrote a book claiming Wiles' proof of Fermat's Last Theorem was wrong because it relied on hyperbolic geometry. She was led to this (absurd) conclusion by Wiles' use of modular forms, as referenced by t.b. a few comments up. – Gerry Myerson Dec 24 '11 at 7:00

Maybe this isn't the sort of answer you were looking for, but I find it striking how often hyperbolic geometry shows up in nature. For instance, you can see some characteristically hyperbolic "crinkling" on lettuce leaves and jellyfish tentacles:

My guess as to why this shows up again and again (and I am certainly not a biologist here, so this is only speculation) is that hyperbolic space manages to pack in more surface area within a given radius than flat or positively curved geometries; perhaps this allows lettuce leaves or jellyfish tentacles to absorb nutrients more effectively or something.

EDIT: In response to the OP's comment, I'll say a little bit more about how these relate to hyperbolic geometry.

One way to detect the curvature of your surface is to look at what the surface area of a circle of a given radius is. In flat (Euclidean) space, we all know that the formula is given by $A(r) = \pi r^2$, so that there is a quadratic relationship between the radius of your circle and the area enclosed. Off the top of my head, I don't know what the formula is for a circle inscribed on the sphere (a positively-curved surface) is, but we can get an indication that circles in positive curvature enclose less area than in flat space as follows: the upper hemisphere on a sphere of radius 1 is a spherical circle of radius $\pi/2$, since the distance from the north pole to the equator, walking along the surface of the sphere, is $\pi/2$. In flat space, this circle would enclose an area of $\pi^3/4 \approx 7.75$. But the upper hemisphere has a surface area of $2 \pi \approx 6.28$.

By contrast, in hyperbolic space, a circle of a fixed radius packs in more surface area than its flat or positively-curved counterpart; you can see this explicitly, for example, by putting a hyperbolic metric on the unit disk or the upper half-plane, where you will compute that a hyperbolic circle has area that grows exponentially with the radius.

So what happens when you have a hyperbolic surface sitting inside three-dimensional space? Well, all that extra surface area has to go somewhere, and things naturally "crinkle up". If you are at all interested, you can crochet hyperbolic planes (see, for instance, this article of David Henderson and Daina Taimina), and you'll see how this happens in practice.

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Vi Hart did some (inconclusive) experiments on why dried fruit slices curl up into hyperbolic shapes. – Rahul Dec 24 '11 at 5:12
Can you explain in what sense these exemples are related to hyperbolic geometry? – Vincent L. Dec 28 '11 at 0:12

My personal pick is the way hyperbolic geometry is used in network science to reason about a whole lot of strange properties of complex networks:

Krioukov et al.: Hyperbolic Geometry of Complex Networks

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There are several applications of hyperbolic surfaces in crystallography, in particular, to periodic minimal surfaces.

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Hyperbolic geometry has been used to construct models of the human vision system and of "color space." Here is one reference: http://www.perceptionweb.com/abstract.cgi?id=p060221

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Zooming a camera out from one portion of a handout, and zooming in on another, as efficiently and smoothly as possible. See Dror Bar-Natan's talk: The hardest math I've ever really used.

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One application that I know of: Hyperbolic polyhedra can be used to obtain a formula that allows you to compute a discretized version of a conformal map. See Discrete conformal maps and ideal hyperbolic polyhedra by Bobenko, Pinkall and Springborn.

These conformal maps in turn can be used e.g. to flatten 3D surfaces in a conformal way for 2D parametrization. Nice images included in this paper. I myself use these conformal maps to convert ornaments between euclidean and hyperbolic geometry, which I guess hardly counts as “other branches of mathematics”.

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On cosmological scales, it's not unlikely that the universe, or large regions of it, has a spatial geometry that is hyperbolic. The average spatial curvature of the universe is within error bars of zero, so it could actually be negative.

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In this paper by Benjamin Bakker and Jacob Tsimerman on the Frey-Mazur conjecture, they use the computation of the volume of certain hyperbolic manifolds in a crucial way.

http://arxiv.org/pdf/1309.6568v1.pdf

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I think a lot of this is missing the point. Hyperbolic geometry isn't just a cool trick that has a couple of applications, it's something that automatically falls out of the mathematics when you're studying geometry, and as such it has direct applications to all sorts of fields.

A relevant result here is the uniformization theorem for Riemann surfaces, which says that the universal cover of a Riemann surface has to be one of the following:

• the Riemann sphere,
• the Euclidean plane, or
• the hyperbolic plane.

In particular, the hyperbolic plane is the universal cover of every Riemann surface of genus two or higher.

This fact is centrally important all over mathematics. This is why you have to learn about hyperbolic geometry to study modular forms in number theory, for instance.

And note that the Poincare conjecture was resolved by establishing Thurston's geometrization conjecture, which is just the three-dimensional analogue of uniformization.

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You can look at the following arcticle on "what is hyperbolic geometry" by Mahan mj...

http://www.asiapacific-mathnews.com/03/0301/0001_0005.pdf

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I am not a mathematician, but my "Barron's dictionary of Mathematics Terms' Second Edition 1995 copyright edited by Douglas Dowing says one example of hyperbolic function use is "catenary"; catenary: study of the curve formed by a flexible rope hanging between two points.

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Hyperbolic geometry and hyperbolic trig functions are two different things. – smithkm Jul 12 '15 at 19:52