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A standard fact that underlies the Fenchel-Nielsen coordinates on Teichmuller space is the fact that for all triples $(a,b,c)$ of positive real numbers, there exists a unique hyperbolic hexagon whose angles are all right angles such that in their natural circular order the sides are $(a,x,b,y,c,z)$ for some positive real numbers $x,y,z$.

I know two sources that claim to prove this. In Hubbard's book "Teichmuller theory, volume 1" he has a proof, but it rests on Exercise 3.5.5 (which he describes as "surprisingly tricky", and which I can only solve with a really terrible and unenlightening calculation). In Farb-Margalit's "Primer on mapping class groups", there is also a proof, but they essentially assert something equivalent to that exercise without proof.

Question: Does anyone know a source with a complete proof? Even better, a proof that minimizes terrible calculations.

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  • $\begingroup$ "Ratcliffe, Foundations of Hyperbolic Manifolds is decent. (See the section on hyperbolic trigonometry.) " quote from comment at math.stackexchange.com/questions/1076278/… $\endgroup$
    – Will Jagy
    Commented Nov 1, 2021 at 1:33
  • $\begingroup$ @WillJagy: I had forgotten about that one. Looking at it, it does give all the details, but via a gigantic wall of formulas (sadly, par for the course in that book). Hopefully someone will know a more enlightening way of doing it... $\endgroup$
    – Lisa
    Commented Nov 1, 2021 at 1:38
  • $\begingroup$ Apparently Thurston gave a very simple "Law of Sines" for right angled hexagons, see mathoverflow.net/questions/278365/… I will check, my guess is Thurston gave a proof that I would, well, like. $\endgroup$
    – Will Jagy
    Commented Nov 1, 2021 at 2:45
  • $\begingroup$ library.msri.org/books/gt3m/PDF/2.pdf $\endgroup$
    – Will Jagy
    Commented Nov 1, 2021 at 2:49
  • $\begingroup$ Alright, Thurston concludes with a couple of diagrams after presenting the hyperbolic Laws of sines, and two cosines. Those are in my little article zakuski.utsa.edu/~jagy/papers/Intelligencer_1995.pdf $\endgroup$
    – Will Jagy
    Commented Nov 1, 2021 at 2:58

3 Answers 3

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My personal favorite proof is described well in this blog post (which attributes it to Hermann Karcher, though I first heard a version of it back in graduate school, so it should probably just be called folklore). It is entirely synthetic and calculation-free.

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Section 7.19 (Hexagons) of Beardon's book The geometry of discrete groups gives a proof.

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  • $\begingroup$ Sam, does it show Thurston's claim about right angled pentagons, if $A,B$ are lengths of two consecutive sides, and $D$ the side that has no common vertex with either of those two, then $\cosh D = \sinh A \sinh B \; ? \; $ This is 2.6.17 on the final page of library.msri.org/books/gt3m/PDF/2.pdf , where 2.6.18 is his Law of Sin(h)es for right angled hexagons. It ought to come from clever drawings and a little trig, I don't see it. $\endgroup$
    – Will Jagy
    Commented Nov 1, 2021 at 17:53
  • $\begingroup$ Dear Will - Yes. It is in Section 7.18 (Pentagons). $\endgroup$
    – Sam Nead
    Commented Nov 2, 2021 at 16:40
  • $\begingroup$ Thank you. I have books about non-Euclidean plane geometry owing to contact with Hartshorne and Greenberg. Neither book appears to do any pentagons and hexagons, but both Lambert and Saccheri quadrilaterals. Oh, I found Beardon and printed out 7.17-7.19 $\endgroup$
    – Will Jagy
    Commented Nov 2, 2021 at 17:25
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A proof of this that I enjoyed reading appears as lemma 3.6 in Thurston's Work on Surfaces, by Fathi, Laudenbach, and Poénaru.

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