Timeline for Numbers of intersection points and lines
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
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| Dec 11, 2011 at 13:19 | vote | accept | Nekochan | ||
| Dec 8, 2011 at 12:57 | answer | added | Joseph O'Rourke | timeline score: 6 | |
| Dec 5, 2011 at 17:45 | comment | added | Joseph O'Rourke | I need to correct myself above. Although Grünbaum is surely the expert, the book I cited uses a very particular meaning of "configuration": An $n_k$ configuration is a set of $n$ points and a set of $n$ lines so that every point lies on precisely $k$ of the lines and every line contains precisely $k$ of the points. There can be many line intersections that are superfluous for these configurations. For example, the Pappus configuration is a $9_3$ configuration although there are more than 9 line-line intersections. | |
| Dec 5, 2011 at 12:28 | comment | added | Nekochan | Thanks Joseph for the book however it won't be possible for me to read it :/ You're right Will, that's l(l-1)/2, not l(l+1)/2 Thanks Gerry for this discussion, I will take a look at this, maybe this can help | |
| Dec 4, 2011 at 22:23 | comment | added | Gerry Myerson | A related question is to characterize the pairs $(\ell,r)$ such that one can draw $\ell$ lines in the plane dividing the plane into exactly $r$ regions. This was discussed at math.stackexchange.com/questions/38350/… and, given Euler's formula, perhaps some of the discussion and references there would be relevant. | |
| Dec 4, 2011 at 19:49 | comment | added | Igor Rivin | @Will: Easier said than done. | |
| Dec 4, 2011 at 19:16 | comment | added | Will Sawin | So if you count points of infinity, each pair of lines must intersect one. The intersection of $k$ lines has multiplicity $k(k-1)/2$. Also your upper bound for $l$ lines should probably be $l(l-1)/2$. So you take $l(l-1)/2$, partition it into smaller triangular numbers, throw out the ones at infinity, and count the remaining ones. | |
| Dec 4, 2011 at 14:47 | comment | added | Joseph O'Rourke | The expert on this topic is Branko Grünbaum, who recently summarized his extensive knowledge in the book, Configurations of Points and Lines: ams.org/bookstore-getitem/item=GSM-103 . | |
| Dec 4, 2011 at 14:09 | history | edited | Edmund Harriss |
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| Dec 4, 2011 at 13:58 | history | asked | Nekochan | CC BY-SA 3.0 |