How small can the sum of the interior angles of a triangle in hyperbolic geometry get? Is there a lower bound?

I know that in Euclidean geometry, the sum of the interior angles of a triangle is exactly $$\pi$$.

In hyperbolic geometry, I know that the sum of the interior angles of a triangle is $$\leq \pi$$, and I know that there exist triangles in hyperbolic geometry with interior angles that sum to strictly less than $$\pi$$.

However, what I don't know is ...

How small can the sum of the interior angles of a triangle in hyperbolic geometry get? Is there a lower bound?

Is there a hyperbolic triangle with interior angle sum equal to, say, $$1/10000000$$? I suspect it is something nice like the angles have to add up to be at least $$\pi/2$$ but I don't know if that's true.

• I can't figure out a good way to represent it, but you can have a triangle with zero angle measure. In the Half plane representation of hyperbolic geometry, this consists of a half circle of Radius $2R$ with two half circles of radius $R$ inside it, all of them tangent to each other. See this wikipedia article for visuals. Dec 18 '19 at 3:22
• It is possible to create a triangle on the hyperbolic plane with three angles of measure 0. In fact this is an "ideal triangle." en.wikipedia.org/wiki/Ideal_triangle Dec 18 '19 at 3:24
• @CharlesHudgins Like this? i.imgur.com/scKQ8S6.png Dec 18 '19 at 3:25
• Yeah, although @DougM has already linked an article that has better pictures. It's a fair point that ideal triangles don't properly live in the hyperbolic plane. So the correct statement is probably that total angle measure $\theta$ satisfies $0 < \theta \leq \pi$. Dec 18 '19 at 3:27
• In the hyperbolic plane, one can have a triangle with angles $\alpha$, $\beta$ and $\gamma$ iff $\alpha,\beta,\gamma>0$ and $\alpha+\beta+\gamma<\pi$. Dec 18 '19 at 3:39

2 Answers

(Sorry if this should be a comment and not an answer: please advise). There is no lower bound. See, for example, Existence of triangles with three arbitrarily small angles in Archimedian Neutral Geometry, (e.g. in hyperbolic geometry). From baby Hartshorne for a rough sketch of a synthetic proof.

• Eh, probably should be a comment Dec 28 '19 at 22:54

Comment

Equilateral triangle on Poincare disc, each vertex zero (spike) angle.

Angle defect$$\pi.$$