topic for presenting in hyperbolic geometry For my course work, i have to give a presentation of 20-30 min presentation in hyperbolic geometry. Can any one suggest some topic(or any interesting theorem) in this area.I want to present some thing which will be interesting to the audience.It will be helpful if references are also suggested.
I have taken a standard course in hyperbolic geometry. My exposure in maths includes algebra, real and functional analysis, complex analysis, number theory.
 A: Well, in the hyperbolic plane of constant curvature $-1,$ you can square the circle with compass and straightedge. To be precise:
(A) there is a countably infinite set of circle/square pairs of the same area, where both the radius of the circle and the edge of the square are constructible lengths 
However, there is no one-way procedure in either direction:
(B) there are  circle/square pairs of the same area, where the circle is constructible but the square not,
(C) there are  circle/square pairs of the same area, where the square is constructible but the circle not. 
Oh, a square means a four sided convex figure with the same four edge lengths and four equal vertex angles.
A: My post may be not as @Will's very interesting, but since I had a presentation like you would have, so I am willing to tell you my experience:


*

*Collect some exciting facts in Hyperbolic Geometry (H.G). For example those ones which are obvious for us like the Parallel postulate and then try to show its inconsistency in H.G theoretically. I think, there are some good theorems in this book, chapter 6 in especial.

*After doing above suggestion, use a nice software like this one or this one to examining the theorems graphically. I think, these jobs would make your presentation exciting.
