Timeline for Arrangements of points in the plane
Current License: CC BY-SA 4.0
5 events
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| Apr 21, 2020 at 23:43 | history | edited | Joseph O'Rourke | CC BY-SA 4.0 |
Updated to Handbook 3rd Ed.
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| Dec 31, 2010 at 6:53 | comment | added | Aaron Meyerowitz | These are great comments and answers BUT isn't the question of the OP when can a list of n circular permutations (each omitting one of n symbols) be realized by an arrangement of points (or if I understand your remark, by an arrangement of n actual lines)? Or am I missing something? | |
| Dec 30, 2010 at 18:15 | comment | added | Joseph O'Rourke | @Bill: Perhaps Mnev's Universality Theorem? en.wikipedia.org/wiki/Mnev%27s_universality_theorem . Here is one statement: If $V$ is a basic semialgebraic set defined over $\mathbb{Q}$, there is a configuration of points in the plane such that the space of all configurations of the same order type as the points is stably equivalent to $V$. (Here I am relying on the same Goodman chapter cited above.) | |
| Dec 30, 2010 at 15:30 | comment | added | Bill Thurston | As I understand, the homotopy type of the set of arrangements of lines in $\mathbb R^2$ can be that of any real algebraic variety. In particular, it need not be connected, which in a certain sense says you can't reconstruct the arrangement from these order types. I'm having trouble finding the correct reference, though --- but perhaps Joseph or someone else can fill it in. | |
| Dec 30, 2010 at 13:21 | history | answered | Joseph O'Rourke | CC BY-SA 2.5 |