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I know some good books in elementary hyperbolic geometry but they have not 'good' exercises.by good exercises I mean nontrivial beautiful problems.we have many resources in of this problems in Euclidean geometry do you know any resource for hyperbolic geometry?

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  • $\begingroup$ by elementary I mean an undergraduate course $\endgroup$ – ali Jan 25 '18 at 15:48
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Judging by the lack of answers, I guess that no such resource exists...

If you want something to force you to think about hyperbolic geometry, and gain informal intuitions about it, my HyperRogue could be appropriate. The closest to traditional geometric problems (though still rather far) is probably the land of Camelot, where your task is to find the center of a large hyperbolic circle.

Some problems I had to solve while writing HyperRogue:

  • From what point do we have to look at the hyperboloid model of a hyperbolic plane, to see it in the Poincaré disk model? In the Klein model?

  • A hyperbolic plane $A$ is isometrically embedded into $\mathbb{H}^3$ as an equidistant surface of radius $r$. We are looking at it from a point in distance $d$ above the base plane of $A$. For what values od $r$ and $d$ does $A$ look as in the Poincaré disk model? In the Klein model?

  • Consider the truncated order-7 triangular tiling. Compute the number of tiles in distance of $d$ steps from a given heptagon or hexagon. (hard)

  • Consider the truncated order-7 triangular tiling. We can create a shape by combining $n$ tiles together. Suppose that we can tile the hyperbolic plane with our shape. Show that $n$ is divisible by 10. (easy)

  • What are the important lengths (edge, distance from hexagon vertex to center, distance from heptagon vertex to center, etc.) in the truncated order-7 triangular tiling? (I have not actually solved this problem -- HyperRogue finds these with binary search, but I think it can be done)

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