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Let $L =\{y=0\}\cap\mathcal{H}^{2}$, and let $P=(3,2,2)$. Show that the parallel postulate fails in $\mathcal{H}^{2}$ by giving two lines $L',L'' \in \mathcal{H}^{2}$ with $P\in L',L''$ and $L\cap L'=L\cap L'' = \varnothing$.

I think a good idea would be to drop a perpendicular from $P$ onto $L$. So I have found this line will lie on $\Pi \cap \mathcal{H}^{2}$, where $\Pi$ is the plane given by $(t−1)−2x−3y=0$. I am really not sure how to proceed from here.

Jonas Meyer
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AlexBowring
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    What does $(3,2,2)$ mean? Have you tried this question yourself? Where did you attempts fall short? – Dan Rust Apr 20 '14 at 22:01
  • @DanielRust, I have no objection to waiting on the OP or closing the question, but it is fairly likely this is the hyperboloid model in the three dimensional Minkowski space, $z > 0$ and $z^2 - x^2 - y^2 = 1. $ http://en.wikipedia.org/wiki/Hyperboloid_model – Will Jagy Apr 20 '14 at 22:18
  • Meanwhile, in the online Warwick notes by C. Series, they give just one page on the hyperboloid model and then recommend: M. Reid and B. Szendroi Geometry and Topology, Cambridge Univ. Press, for beginners... – Will Jagy Apr 20 '14 at 22:37
  • I think a good idea would be to drop a perpendicular from $P$ onto $L$. So I have found this line will lie on $\Pi \cap \mathcal{H}^{2}$, where $\Pi$ is the plane given by $(t-1)-2x-3y=0$. I am really not sure how to proceed from here. Will Jagy I am not sure if you are mocking me but M.Reid lectured me on this subject and personally recommended his own book (unsurprisingly). Unfortunately his notes do not contain any numerical examples for any of his theories, which he leaves as exercises with no solutions. – AlexBowring Apr 20 '14 at 22:51
  • I've not worked with the hyperboloid model before but hopefully the following hint works as well in that model as it does in the disc/halfplane model. $L$ has two limit points on the boundary circle of hyperbolic space, can you find $L'$ with one limit point on the boundary equal to the first limit point of $L$ and going through the prescribed point, and a line $L''$ with a limit point on the boundary equal to the secon limit point of $L'$ also going through the prescribed point? Show these lines do not intersect $L$ on the interior of hyperbolic space. – Dan Rust Apr 20 '14 at 23:01
  • Thanks Dan, we have not really discussed this boundary circle. I am assuming it is the circle carved out by the hyperboloid in the $t=\infty$ plane (or along those lines). Your idea sounds reasonable but I am unsure how I would go about finding $L'$ and $L''$ this way. – AlexBowring Apr 20 '14 at 23:13

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