Timeline for Soddy-type relation for Steiner chains
Current License: CC BY-SA 3.0
13 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| S Oct 17, 2013 at 21:31 | history | bounty ended | James Propp | ||
| S Oct 17, 2013 at 21:31 | history | notice removed | James Propp | ||
| Oct 17, 2013 at 21:29 | vote | accept | James Propp | ||
| Oct 16, 2013 at 14:59 | answer | added | Dimitri | timeline score: 4 | |
| Oct 16, 2013 at 10:48 | comment | added | user25199 | Maybe not generic, but $n=3$ is just given by the usual Soddy formula. Note that there are two relations - both $r_−$ and $r_+$ are determined by the others, which are free. A starting point might be to look at $n=4$, for which the middle circles are now constrained in some way (need to allow an exact incircle and circumcircle). The number of constraints appears to be $n-1$. It might be interesting to study the allowed $(r_-,r_+)$ for a given $n$. | |
| Oct 13, 2013 at 16:34 | comment | added | Gil Kalai | Great question! | |
| Oct 10, 2013 at 22:19 | comment | added | Gerry Myerson | I haven't looked at the paper, but the review of Paul Yiu, Rational Steiner porism, Forum Geom. 11 (2011) 237–249, MR2877262 suggests there might be something of interest there. | |
| S Oct 10, 2013 at 21:48 | history | bounty started | James Propp | ||
| S Oct 10, 2013 at 21:48 | history | notice added | James Propp | Draw attention | |
| Oct 4, 2013 at 3:19 | comment | added | James Propp | Presumably the answer would be of the form $P_n(r_1,\dots,r_n,r_-,r_+)=0$ where $P_n$ is a homogeneous polynomial that is cyclically symmetric (mod $n$) in $r_1,\dots,r_n$. Perhaps an algebraic geometry argument tells us ahead of time that such a polynomial must exist? To get a sense of how big the degree of $P_n$ might need to be, one could look at the situation where $r_1=r_2=\dots=r_n=1$; then $r_- = (\csc \pi/n) - 1$ and $r_+ = (\csc \pi/n) + 1$ (or something like that). Does this give information about the degree of $P_n$? | |
| Oct 3, 2013 at 18:37 | history | edited | Ricardo Andrade |
replaced tags
|
|
| Oct 3, 2013 at 16:03 | history | edited | Joseph O'Rourke | CC BY-SA 3.0 |
Added image.
|
| Oct 3, 2013 at 15:58 | history | asked | James Propp | CC BY-SA 3.0 |