On Francis Bonahan's page, which Jim linked for us, the blowups of different parts of the tesselation are labeled misleadingly (at least in my browser). The labels for various points on the real number line are arrayed uniformly at equal intervals. This happens to work in the first two images, but for the subsequent magnifications the labels wind up nowhere near their proper positions. I separately emailed Bonahan with this observation, but short of incorporating the labels in the images I can't think of a way for him to fix it. On Fri, Jan 27, 2017 at 6:01 PM, James Propp <jamespropp@gmail.com> wrote:
Oops: they're ALL horocycles (thanks Dan). Only one of them is a horizontal line.
Jim
On Fri, Jan 27, 2017 at 4:14 PM, James Propp <jamespropp@gmail.com> wrote:
See http://www-bcf.usc.edu/~fbonahon/STML49/FareyFord.html .
Note that one of the "circles" is a horocycle (horizontal line). The other circles look like circles.
I believe there's a theorem that says that you can get the convergents to an irrational number x by looking at the line Re z = x and seeing which of the circles it pierces.
Jim Propp
On Fri, Jan 27, 2017 at 3:09 PM, Dan Asimov <dasimov@earthlink.net> wrote:
What do you mean by the "Ford circle packing" of the upper half plane?
—Dan
----- From: James Propp <jamespropp@gmail.com> Sent: Jan 27, 2017 10:08 AM
Anyone care to make a short video (or GIF) that zooms in on the Ford circle packing of the upper half plane, converging on the point pi on the real axis, highlighting the circles we pass through (whose points of tangency to the real axis are the continued-fraction convergents to pi)?
I searched YouTube but nothing like this seems to currently exist. -----
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