Noob question about hyperbolic geometry

[Dale Dough]

OK, I have no formal math education besides some basic HS calculus, but the subject interests me. However, I have never been a big fan of constructing proofs or working out complicated equations - I just want to have some understanding of the juicy stuff that boggles my mind. But lately I found myself drawn to the subject more than usual.

I understand that the (only) way to define 'area' in hyperbolic geometry is to use triangles as units, because perfect squares cannot exist. The area of a triangle is some constant (dependent upon the curvature of the space?) multiplied by the amount that the sum of the angles is short of 180 degrees.

This implies that no body can have an area greater than or equal to 180 degrees times that constant - it is always possible to imagine a triangle big enough to enclose that area.

So, how about the entire plane of hyperbolic two-dimensional (or any-dimensional, I guess) space? A spherical plane curves back onto itself and thus has a limited area. As the sphere becomes bigger, the plane approaches an Euclidean plane, which has an unlimited area. So how can a hyperbolic plane have a limited area, and be smaller than en Euclidean plane?

[pzhon]

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I understand that the (only) way to define 'area' in hyperbolic geometry is to use triangles as units, because perfect squares cannot exist.

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No. You can use other definintions, but the area of a triangle is particularly simple to compute.

There are quadrilaterals with equal angles and equal sides. The angles are just smaller than 90 degrees. Whether you call these squares or not doesn't matter.

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The area of a triangle is some constant (dependent upon the curvature of the space?) multiplied by the amount that the sum of the angles is short of 180 degrees.

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Yes. It is standard to assume the curvature is -1, not just an unspecified negative constant, and to measure the angles in radians. The constant you mention is -1/Gaussian curvature, which simplifies to 1.

Similarly, if you are looking at spherical geometry, you can assume the points are on a sphere of radius 1, and the area of a triangle is the excess of the sum of the angles over pi. The areas you compute will correspond to the normal notion of surface area. For example, the triangle consisting of the intersection of the unit sphere with the positive orthant (vertices (1,0,0), (0,1,0), (0,0,1)) has area pi/2+pi/2+pi/2 - pi = pi/2, and that's really 1/8 of the surface area of a unit sphere.

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This implies that no body can have an area greater than or equal to 180 degrees times that constant - it is always possible to imagine a triangle big enough to enclose that area.

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No. You can decompose any polygon into triangles, but in hyperbolic geometry, some polygons are too large to fit inside any triangle. For example, any polygon with an angle defect greater than pi can't fit inside any triangle, since its area is too large.

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So how can a hyperbolic plane have a limited area, and be smaller than en Euclidean plane?

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A hyperbolic plane has infinite area.

[Dale Dough]

I just had it stuck in my mind that any area could be enclosed within a triangle. I was confused because I decided to set this straight using some dramatic illogical approach.

I guess I better start looking into some elementary boring proofs...

Thanks.

[wooly_chicken]

math is all about proofs. without them you'd never make it anywhere.

[srjunkacct]

Since a Simpsons quote is always appropriate:

It should be obvious to even the most dim-witted individual who holds an advanced degree in hyperbolic topology, n'gee, that Homer Simpson has stumbled into...the third dimension. (http://www.snpp.com/episodes/3F04.html)

People who study hyperbolic 3-manifolds must get a kick out of this episode.