Timeline for Can $n$ circles on a plane generate $m$ intersection points where at least $k$ circles intersect?
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
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| Feb 25, 2017 at 22:13 | vote | accept | myro | ||
| Feb 25, 2017 at 20:26 | comment | added | Gerhard Paseman | I lost count. (Oops.) I thought we were going for 7 pairs of triple concurrency. (Actually I suspect a deeper level of confusion. Let's go instead with my losing count.) Gerhard "Gets Excited Around Projective Planes" Paseman, 2017.02.25. | |
| Feb 25, 2017 at 20:15 | comment | added | myro | I don't understand why we need the Fano plane. In Gerhard's construction after we have constructed the 5 circles on the sphere and then thread the two remaining circles we already get 12 intersection points. So why do we need the concurrency between the last two circles? | |
| Feb 25, 2017 at 20:10 | comment | added | myro | Gerhard Paseman, thanks for your construction. | |
| Feb 25, 2017 at 20:09 | comment | added | myro | Adam P. Goucher, thanks for your answer. | |
| Feb 25, 2017 at 17:45 | comment | added | Gerhard Paseman | Actually, you may have to coordinate the choices of the last four circles, say two assigned to each of two of the original three circles, arranged so that they intersect in pairs on the third original. In any case, the choice can be motivated by the Fano plane design without using projective space as an intermediary. Gerhard "Saves Projecting For Other Domains" Paseman, 2017.02.25. | |
| Feb 25, 2017 at 17:34 | comment | added | Gerhard Paseman | A faster route: choose three concurrent great circles on the sphere. Pick "a nice point" on one of the circles far from the two points of concurrency, and make that new point (and antipode) another two points of concurrency with two more great circles going through the point. This gives four more pairs of antipodal points through which one threads the two remaining circles. Now use the Fano plane or other means to show that the concurrency between these last two circles is (or can be, with judicious choices) a triple concurrency. Gerhard "Geodesic Routes Are Often Faster" Paseman, 2017.02.25. | |
| Feb 25, 2017 at 16:26 | history | answered | Adam P. Goucher | CC BY-SA 3.0 |