Timeline for Is there a universal straightedge and compass construction of a segment incommensurable to a given one in the hyperbolic plane?
Current License: CC BY-SA 3.0
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| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
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| Mar 27, 2015 at 22:50 | comment | added | Conifold | @Will Jagy Commensurability of standard lengths makes sense from measurement in neutral geometry point of view. Taking the hypotenuse won't work universally, there is a countable set of leg lengths that are commensurable with their hypotenuses. Asking what happens "universally" might have the effect of weeding out constructions that are specific either to Euclidean or to hyperbolic plane, and leaving what is reasonable for both. | |
| Mar 27, 2015 at 18:28 | comment | added | Will Jagy | Anyway, in case what you want is to be given an unknown segment and produce something else, how about taking the given length and making that two legs of a right triangle? The hypotenuse can be solved for explicitly. | |
| Mar 27, 2015 at 18:12 | comment | added | Will Jagy | It appears that you do understand that a length $x$ in the hyperbolic plane, take curvature as $-1,$ is constructible if and only if $e^x,$ $\cosh x,$ $\sinh x,$ $\tanh x$ are in the field of lengths constructible in the Euclidean plane. As that does not answer your question, I don't think I can help you at a distance. Commensurability is not a reasonable condition in the hyperbolic plane, there are explicit transcendental functions involved. | |
| Mar 27, 2015 at 18:02 | comment | added | Conifold | @Will Jagy I tried to explain in the edit, but not sure if this addresses your question. Could you explain more how to use angles? | |
| Mar 27, 2015 at 18:00 | history | edited | Conifold | CC BY-SA 3.0 |
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| Mar 26, 2015 at 22:52 | comment | added | Will Jagy | This is probably why Hartshorne switched to a "multiplicative length" for his (book) axiomatic treatment of the hyperbolic plane. For a length most of us would call $x,$ his multiplicative length is just $e^x,$ which is then in the "constructible field." | |
| Mar 26, 2015 at 22:47 | comment | added | Will Jagy | not sure what you are getting at, but the natural thing to consider is angles, between pairs of lines or two circles that meet or a circle and a line. The fundamental theorem is that the constructible angles in the hyperbolic plane are exactly the same as the constructible angles in the Euclidean plane. | |
| Mar 26, 2015 at 20:02 | history | asked | Conifold | CC BY-SA 3.0 |