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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.

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    $\begingroup$ Tilings of the hyperbolic plane are well-studied but also have a bunch of things that people don't know well. If you can find a copy of The Symmetries Of Things, it gives criteria for a particular set of symmetry generators (to be vague) to actually define a hyperbolic tiling, and it's chock full of presentation-friendly images of tilings. $\endgroup$ Commented Mar 22, 2013 at 20:41

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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.

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  • $\begingroup$ what do you mean by hyperbolic plane of constant curvature -1? Also can you give some reference about this.Thanks $\endgroup$
    – Phani Raj
    Commented Mar 21, 2013 at 18:50
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    $\begingroup$ See Marvin's article downloadable from mathdl.maa.org/mathDL/22/… as well as zakuski.utsa.edu/~jagy/papers/Intelligencer_1995.pdf The part about curvature $-1$ just means that we take Gauss's constant $k,$ sometimes called the "distance scale," as $k=1.$ The main book reference is the 4th edition of Marvin Jay Greenberg, Euclidean and Non-Euclidean Geometries, this material is not in the 3rd edition. $\endgroup$
    – Will Jagy
    Commented Mar 21, 2013 at 19:08
  • $\begingroup$ @PhaniRaj, commented with some references, forgot at sign. $\endgroup$
    – Will Jagy
    Commented Mar 21, 2013 at 19:11
  • $\begingroup$ @WillJagy: May I answer this old question below, Will? Thanks $\endgroup$
    – Mikasa
    Commented Dec 22, 2013 at 2:37
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    $\begingroup$ @WillJagy: Thanks Will. I hope the OP can return and see this post, however, that presentation's done and the story was ended month ago. $\endgroup$
    – Mikasa
    Commented Dec 22, 2013 at 3:05
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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.

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