I am interested in the Hausdorff dimension of the Apollonian circle packing. There seem to be two numerical calculations of the value:
and later from Curt McMullen (he does not cite Thomas and Dhar):
His algorithm is also explained in Section 5.5 on page 156 in "Indra's Pearls" by Mumford and Series (beware of some typos on that page!).
It is somewhat unsatisfying to have two contradicting numerical approximations. The OEIS gives 1.3056867 and does not cite McMullen, see A052483 (as of today).
With modern computers it might be worthwhile trying to settle this question. Curt McMullen has a C-program on his website that can calculate the Hausdorff dimension, that I am interested in: hdim.tar on http://abel.math.harvard.edu/~ctm/programs/index.html
./hdim -a -e .00005
gives as output
Apollonian gasket Epsilon Dimension Cover Matrix Steps 5.00e-05 1.305687542911558287346746 76
So I guess after 76 steps (and about a week of computation time) we get 1.30568754291 as an numerical approximation.
- Does this mean that actually the first few digits are 1.305687?
- What are the correct first few decimal digits of this number?
- What are the best proven exact bounds and what are the most promising numerical experiments to get a good approximation?