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I've been studying Ring Theory, and there is a marvellous chapter about Constructible Numbers and Euclidean Geometry (LEQUAIN, Y et al. Elementos de Álgebra) and I began wondering: What about the Hyperbolic Geometry?

I studied the Hyperbolic Plane in three ways: Without models, Half-Plane Poincaré Model and Disk Poincaré Model. But I'm not sure about this question: What are the analogue in the hyperbolic plane for euclidean non-graduated ruler and compass that can make the Constructible Numbers?

Any help would be appreciated! Thanks!

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A length $t$ in the hyperbolic plane can be constructed if and only if $\sinh t$ is a length that can be constructed in the Euclidean plane. The constructible angles in the hyperbolic plane are exactly the same as those in the Euclidean plane.

My Article

Place to download Marvin's article click on Read The Article in blue letters

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    thanks! i'm gonna try to prove the equivalence you said. – 2017-01-06
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    It is also described and explained in "Introduction to Hyperbolic geometry " by Ramsay and Richtmeyer ISBN 0-387-94339-0 pages 255 - 275,page 259 says a length $y$ can be constructed if $ e^{-y} $ is in a quadratic-surd field. I think the definitions are equivalent. – 2017-01-10
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    @Willemien yes, they are the same. In his book, Hartshorne goes so far as to make a "multiplicative" length, which would be the same as $e^x$ for length $x$ in the ordinary hyperbolic plane. The length of two consecutive line segments in a line becomes the product, the length of a single point is $1,$ and so on. This way he avoids any transcendental functions or limiting processes. – 2017-01-10
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    I wonder if a tool for drawing equidistants would be permissible for the constructions? Technically, it would be a simple device -- a ruler you move along the line, with a perpendicular arm where you can set a pencil at a certain distance to draw an equidistant when the ruler moves. – 2017-08-11
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    @Marek14 that is actually how Mordukhai-Boltovskoi originally proved the result, maybe about 1930. Later, Nesterovich showed that the extra instruments did not change the set of points that could be constructed. I give a very brief discussion in my article. I don't think any of this has been translated. i had a Russian friend confirm the contents of one Nesterovich article from about 1949. – 2017-08-11