This configuration is known as the "Apollonian packing / gasket", see https://en.wikipedia.org/wiki/Apollonian_gasket --- the classical associated problem of estimating its fractal dimension has a fair amount of literature, and I am sure that Dan's question has received collateral attention there. The analogous problem in 3-space has apparently industrial applications; and in 4-space presents an unexpected and inconvenient feature ... For a more recent pair of papers by Ron Graham, Jeff Lagarias et al (which I have not read) see https://dl.acm.org/citation.cfm?id=3116525 Fred Lunnon On 3/21/19, Dan Asimov <dasimov@earthlink.net> wrote:
Suppose that inside a unit circle C (the edge the unit disk D) lie three smaller disks tangent to each other and to C. Suppose their radii are 0 < R, S, T < 1.*
Call these radii "stage 1".
The exterior of these disks now fall into four curvilinear triangles, inside each of which there is a unique largest disk, which will be tangent to all three sides.
The radii of the three new disks are "stage 2".
Iterating, we get an infinite tree of disks, each of maximal size in some curvilinear triangle lying between three previous disks.
The nth stage will add 3^n new (disks and) radii.
Now let X(n) denote the multiset of all radii
1, R, S, T, ...,
through the nth stage, for a total of
|X(n)| = 1 + 3 + ... + 3^n = (3^(n+1) - 1)/2
radii in X(n).
Define the usual cumulative distribution function (cdf) for the data of X(n) via
F_n : [0, 1] —> [0, 1]
via
F_n(x) = |{r in X(n) | r <= x}| / |X(n)|.
***> Question: What is the limiting distribution of F_n as n —> oo and how does it depend on R, S, T ???
—Dan ————— * From Frederick Soddy's poem "The Kiss Precise" we know that R, S, T can be any numbers satisfying
(1 - (1/R + 1/S + 1/T)^2 = 2(1 + 1/R^2 + 1/S^2 + 1/T^2)
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun