Timeline for Can a fixed finite-length straightedge and finite-size compass still construct all constructible points in the plane?
Current License: CC BY-SA 4.0
9 events
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| Jul 13, 2020 at 14:49 | comment | added | Joel David Hamkins | @AlbertvanderHorst Doesn't Noam's observation (comment on the OP) enable you to pick the right directions? You can tile the whole region between the points before choosing the lines. | |
| Jul 12, 2020 at 8:57 | comment | added | Albert van der Horst | @Goucher. Note that one can do an arbitrary line of lenght n by n applications of the ruler. So the arbitrary lines can solve the problem in one step, if you're lucky enough. That makes it seem like your argument is way to pessimistic. On the other hand, it seems that there is a carpenters way, but no mathematicians way to select the directions. Next thing you know the problem is NP complete. Groetjes Albert | |
| Jul 12, 2020 at 8:32 | comment | added | Joel David Hamkins | See also this 2008 article (in French) by Xavier Caruso xavier.toonywood.org/papers/publis/troppetit.pdf, which seems to use the same method. | |
| Jul 12, 2020 at 8:30 | comment | added | Joel David Hamkins | Thanks for your excellent answer. This is great! | |
| Jul 12, 2020 at 8:28 | vote | accept | Joel David Hamkins | ||
| Jul 11, 2020 at 17:30 | comment | added | Adam P. Goucher | Note that this naive recursive approach means that, in particular, joining two points separated by a distance of $(1 + \varepsilon)^n$ would take $10^n$ applications of the original straightedge! (It's still polynomial in the length, though, so not too bad.) | |
| Jul 11, 2020 at 17:27 | comment | added | Adam P. Goucher | The construction means that you can henceforth proceed as though you have a length-$(1 + \varepsilon)$ straightedge instead of a length-$1$ straightedge. By the same argument, you can proceed as though you have a length-$(1 + \varepsilon)^2$ straightedge (by using the length-$(1 + \varepsilon)$ straightedge to draw each line in the diagram). | |
| Jul 11, 2020 at 17:12 | history | edited | Adam P. Goucher | CC BY-SA 4.0 |
visual demonstration
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| Jul 11, 2020 at 17:00 | history | answered | Adam P. Goucher | CC BY-SA 4.0 |