Timeline for "Fractally self-similar" numbers
Current License: CC BY-SA 3.0
5 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 1, 2015 at 6:09 | comment | added | მამუკა ჯიბლაძე | Hm on the afterthought the first part is obvious - ratios of rational radii cannot become constant if they must gradually approach an irrational number. But what about the second? | |
| Dec 1, 2015 at 6:03 | comment | added | მამუკა ჯიბლაძე | So actually no pair of touching circles produces exactly homothety-periodic sequence? Somehow I feel there must be a natural metric enhancing strict periodicity of continued fractions / Stern-Brocot paths to strict periodicity in the metric homothety sense | |
| Dec 1, 2015 at 5:57 | comment | added | David Eppstein | Because the first two Ford circles you start with have no reason to have the correct proportions to be part of an exactly periodic sequence of circle shapes. And you can't suddenly jump into a periodic sequence of circle shapes from a sequence that does not have the same shapes, because the system is reversible. But the more times you repeat, the closer it converges to that exact periodicity. | |
| Dec 1, 2015 at 5:44 | comment | added | მამუკა ჯიბლაძე | This is what confuses me, and it also appeared in the another answer: the continued fraction expansion is just periodic; both it and the succession of circles corresponds to a path in the Stern-Brocot tree and its turns are periodic too; yet, the picture itself turns out to be only "asymptotically periodic". Why? Is there a modification of the metric in which it is also strictly periodic? | |
| Dec 1, 2015 at 1:42 | history | answered | David Eppstein | CC BY-SA 3.0 |