Timeline for Is there a straightedge and compass construction of incommensurables in the hyperbolic plane?
Current License: CC BY-SA 3.0
18 events
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| Sep 3, 2014 at 3:49 | history | edited | Will Jagy | CC BY-SA 3.0 |
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| Sep 2, 2014 at 20:57 | vote | accept | Conifold | ||
| Sep 2, 2014 at 20:55 | comment | added | Will Jagy | @Conifold I guess you had better email me for a pdf, Look me up at ams.org/cml Note that i quite like Emil's translation of my definition, "that you consider x constructible if there exists a construction which always produces a length x starting from two given distinct points, no matter what these points are, or something like that? " | |
| Sep 2, 2014 at 20:54 | comment | added | Will Jagy | Note to both: the last time I argued with Marvin he accused me of thinking like a geometer, which seemed to be his ultimate insult. I decided i would leave the foundational issues to others. | |
| Sep 2, 2014 at 20:54 | comment | added | Conifold | @Will Jagy Sorry, I just didn't understand what "random lengths, which are discarded once no longer in use" means (couldn't access the Intelligencer article). Can the original segment serve as "random length" to generate the triangle, or does it require some other segments of no particular length to be available. | |
| Sep 2, 2014 at 20:48 | comment | added | Emil Jeřábek | @Conifold: You can’t construct an equilateral triangle from nothing. Straightedge can draw a line through a given pair of distinct points, a compass can draw a circle with a given point as a centre going through another given point, and new points are only constructed as intersections of lines or circles. So, unless you are given two points (or enough lines or circles to get them as intersection points), you are stuck. | |
| Sep 2, 2014 at 20:48 | comment | added | Will Jagy | @Conifold, don't know what to tell you. I grew up doing constructions, compass, paper, cardboard, Platonic solids. i have a view of constructions that reflects actually doing them. As far as I am concerned, Lambert's length is constructed from nothing. Similar for Schweikart's constant. | |
| Sep 2, 2014 at 20:43 | comment | added | Will Jagy | @EmilJeřábek, had it backwards, $\log 2$ and $\log 3$ should work. In Hartshorne's book, he cleverly worked with $e^x$ and called it a "multiplicative length," clever I thought. I came to think that $\sinh x$ was the better bet as there are then right triangles with three integer sides | |
| Sep 2, 2014 at 20:43 | comment | added | Conifold | @Will Jagy Why do we need "random lengths, which are discarded once no longer in use"? If we construct that equilateral triangle from "nothing", then either its side is already incommensurable with the original segment, or they are commensurable and there is further construction that produces something incommensurable with both. Is that right? | |
| Sep 2, 2014 at 20:37 | comment | added | Emil Jeřábek | Alright, thanks. Now one more thing: the question didn’t ask for a nonconstructible length, but for a pair of constructible lengths with irrational ratio. | |
| Sep 2, 2014 at 20:36 | comment | added | Will Jagy | @EmilJeřábek, yes, that sounds perfect. | |
| Sep 2, 2014 at 20:34 | comment | added | Emil Jeřábek | I can’t find a definition of random lengths in your article. Does it mean that you consider $x$ constructible if there exists a construction which always produces a length $x$ starting from two given distinct points, no matter what these points are, or something like that? | |
| Sep 2, 2014 at 20:22 | comment | added | Will Jagy | Anyway, to me, you can always erect a perpendicular line to any line, always bisect an angle. Several steps to make that equilateral triangle. Other fundamental lengths include the finite edge of a right triangle with a $45^\circ$ angle and then two infinite edges. Oh, where was I , similar triangles are congruent here, strange. So, for example, you can bisect a segment but not trisect it. | |
| Sep 2, 2014 at 20:21 | comment | added | Will Jagy | @Emil, that was Marvin's argument with me; I allow constructions with random lengths, which are discarded once no longer in use, as being unknown length for one thing. Marvin never liked that; you might enjoy his article maa.org/programs/maa-awards/writing-awards/… Meanwhile, in the Euclidean plane you really do need to be given a segment of length $1,$ but not here. For example, one "fundamental" length is the side of an equilateral triangle with three $45^\circ$ angles. | |
| Sep 2, 2014 at 20:16 | comment | added | Emil Jeřábek | Any way I look at it, you can’t construct anything with a straightedge and compass unless you are given at least two distinct points. Surely the set of constructible lengths will then depend on the distance between these two points (what if they are already at distance 1?). | |
| Sep 2, 2014 at 20:14 | history | edited | Will Jagy | CC BY-SA 3.0 |
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| Sep 2, 2014 at 19:50 | comment | added | Will Jagy | I remembered it correctly, $\sinh x$ | |
| Sep 2, 2014 at 19:33 | history | answered | Will Jagy | CC BY-SA 3.0 |