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Does $\exists$ on the hyperbolic plane, a convex quadrilateral $Q$ and a convex pentagon $P$ with the same angle sum? I found this question to be rather interesting.

Richard Carpenter
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2 Answers2

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If you know about ideal polygon then this is easy because every ideal polygon has angle sum zero.

tessellation
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  • I thought of that but is an ideal polygon convex? – Richard Carpenter Apr 27 '13 at 06:05
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    If boundaries of ideal polygon are geodesics then t is convex. This is true in general i.e. if you have a polygon with geodesic boundary then it is convex.

    P.S.: Here convexity means in hyperbolic sense not in euclidean sense i.e. any two point in the polygon can be joined by an hyperbolic geodesic which lies in that polygon.

     – tessellation Apr 27 '13 at 06:09
  • I see. I have one more question. Does the quadrilateral or the pentagon have all right interior angles? I would assume it cant be the quadrilateral since that would imply a rectangle which does not exist. – Richard Carpenter Apr 27 '13 at 06:14
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    Theorem:Gauss-Bonnet: An n-gon with angles $\alpha_1,\alpha_2,...,\alpha_n$ has area $(n-2)\pi-(\alpha_1,\alpha_2,...,\alpha_n).$ Now if a quadrilateral has all interior angle $\pi/2$ then its area is zero which in not possible. But for a pentagon this case is possible because an ideal pentagon ha all angles 0 and if we start to draw smaller pentagon the interior angles will became small up-to angle $3\pi/5$ which is greater than $\pi/2.$ Hence there will be a polygon yu desired. This logic applies for all polygon with more than 4 sides. – tessellation Apr 27 '13 at 06:40
  • How interesting. So what you are saying is that in hyperbolic geometry, a pentagon can have all right interior angles? – Richard Carpenter Apr 27 '13 at 06:44
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    Yes I am. Actually if you take a regular pentagon of area $\pi/2$ which will exists by the above argument its all angles will be $\pi/2$. – tessellation Apr 27 '13 at 06:52
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Hint: a regular quadrilateral can have any value of interior angles between 0 and π/2, and a regular pentagon can have any value of interior angles between 0 and 3π/5.

Incnis Mrsi
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