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
11 events
| when toggle format | what | by | license | comment | |
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| Jul 24, 2020 at 16:05 | answer | added | Albert van der Horst | timeline score: 0 | |
| Jul 12, 2020 at 8:28 | vote | accept | Joel David Hamkins | ||
| Jul 12, 2020 at 0:34 | history | became hot network question | |||
| Jul 11, 2020 at 17:00 | answer | added | Adam P. Goucher | timeline score: 32 | |
| Jul 11, 2020 at 16:58 | comment | added | Noam D. Elkies | "approximately in the direction of" is a tall order if P,Q are far away and you must get within one unit. A more robust solution is to tile the plane with squares (or even parallelograms) and count to determine the coordinates $(a,b)$, and thus also $(a/n,b/n)$, within one unit. (We can construct the tiling in square-spiral order to make sure we get from P to Q in a finite number of steps.) | |
| Jul 11, 2020 at 16:54 | comment | added | Joel David Hamkins | @WillSawin That is what I was trying to do, but didn't see how to carry out the scaling part. If you could post an answer, I'd be grateful. | |
| Jul 11, 2020 at 16:52 | comment | added | Will Sawin | For the difficult case you mention, suppose you have two points P and Q that are very far apart. Suppose you pick a line leaving $P$, approximately in the direction of $Q$, and count along it, one unit interval at a time, until you reach a point $R$ within one unit interval of $Q$, after $n$ unit intervals in total. Then to construct the line from $P$ to $Q$ we need to take $Q-R$, shrink my a factor of $n$, and then translate it to the point one unit interval off from $P$. These steps all seem like they can probably be done with a bounded straightedge but I didn't check. | |
| Jul 11, 2020 at 16:47 | history | edited | Joel David Hamkins | CC BY-SA 4.0 |
added 375 characters in body
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| Jul 11, 2020 at 16:45 | comment | added | Joel David Hamkins | Ah, I've realized that this answers the question I asked, but not the question I was thinking of, which was the relative constructibility question. Given points, can you still construct the same points relative to them? In other words, is the tool set fully equivalent? (Meanwhile, Will, please post your comment as an answer.) | |
| Jul 11, 2020 at 16:42 | comment | added | Will Sawin | If $(a,b)$ is a constructible point, can't we construct $( \frac{a}{n}, \frac{b}{n})$ for $n$ sufficiently large by pretending my straightedge is infinite and then shrinking my construction? Then can I walk along that line in steps of size $\frac{a}{n}, \frac{b}{n}$ by using a special case of the rusty compass theorem construction? | |
| Jul 11, 2020 at 16:32 | history | asked | Joel David Hamkins | CC BY-SA 4.0 |