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$\Delta ABC$ exists in hyperbolic geometry. What is the maximum value for $m\angle A+m\angle B+m\angle C$?

2 Answers2

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$\pi- \epsilon$.

The sum of the angles of a hyperbolic triangle comes out to $\pi - Area(\Delta ABC)$, so by making the area close to zero, your angles will be close to $\pi$ (think of $\epsilon$ as a small constant).

On the other hand, by making a very large hyperbolic triangle, the angles approach zero and the sides are nearly parallel. (Try this in a model of hyperbolic space, like the Poincare disc.)

Titus
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  • So in hyperbolic space, angles are dependent on area? Also, any resources you recommend me look at regarding hyperbolic triangles? –  Jul 16 '15 at 07:23
  • The angles of a triangle are related to its area. When you make the triangle very small, the local geometry is almost flat (just like very small areas of a sphere look flat). In flat space, triangles' angles sum to $\pi$, so you get almost the same thing.

    Going the other direction, as a vertex is pulled away from the others, its edges become nearly parallel. In euclidean space you can do this with only a single vertex, but in hyperbolic space you have enough 'room' to do it with all three.

     – Titus Jul 16 '15 at 07:29
  • Are you familiar with the Poincare disc or plane? Try forming a triangle in such a model and then drag its vertices towards the "boundary" to see the angles vanish as the area grows. – Titus Jul 16 '15 at 07:32
  • Also the exterior angle theorem ( https://en.wikipedia.org/wiki/Exterior_angle_theorem , the WEAK version) is valid in hyperbolic geometry – Willemien Jul 17 '15 at 08:42
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The area of a spherical triangle is given by $${\rm area}(\triangle)={\rm angle\ sum}-\pi\ ,$$ and for a hyperbolic triangle we have $${\rm area}(\triangle)=\pi-{\rm angle\ sum}(\triangle)\ .$$ Both formulas are a consequence of the famous Gauss-Bonnet theorem. It follows that in a hyperbolic triangle we have $${\rm angle\ sum}(\triangle)=\pi-{\rm area}(\triangle)\ .$$ As such triangles can have arbitrary small area $>0$ there is no hyperbolic triangle of maximal area, but we can say that the supremum of the angle sums is $\pi$. Note that small triangles of arbitrary shape can approximate this supremum.

Christian Blatter
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