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I am interested in the Hausdorff dimension of the Apollonian circle packing. There seem to be two numerical calculations of the value:

1.305686729(10)

from P.B Thomas and D.Dhar, The Hausdorf[sic!] dimension of the Apollonian packing of circles Journal of Physics A: Mathematical and General, Volume 27, Number 7, April 1994

and later from Curt McMullen (he does not cite Thomas and Dhar):

1.305688

McMullen, Curtis T. Hausdorff dimension and conformal dynamics. III. Computation of dimension. Amer. J. Math. 120, no. 4, 691–721. 1998

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

Running

./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?
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    $\begingroup$ Doesn't that calculation mean the answer is within .00005 of the correct value? Both of the values cited above satisfy this. $\endgroup$
    – Gerald Edgar
    Commented Feb 16, 2016 at 17:42
  • $\begingroup$ @GeraldEdgar "-e" is the pruning epsilon. From the README: "-e: prunes Markov partition by stopping subdivision of blocks of size less than epsilon." $\endgroup$
    – Moritz Firsching
    Commented Feb 16, 2016 at 18:53

2 Answers 2

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Roberto De Leo, A conjecture on the Hausdorff dimension of attractors of real self-projective Iterated Function Systems, Experimental Mathematics 24, 270 (2015), doi:10.1080/10586458.2014.987884, using a different method, confirms the figure by Thomas & Dhar (at least the digits 1.3056867) and suggests an error in McMullen's result.

I will add this reference to the OEIS.

2024 Update: Polina Vytnova and Caroline Wormell, Hausdorff dimension of the Apollonian gasket, arXiv:2406.04922 [math.DS], provide rigorously proven 128 decimal digits of the constant. In particular, they confirm the digits found by Bai & Finch (whose method was partly empirical and apparently can't be made rigorous).

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It seems in the meantime, there are new results:

Bai, Zai-Qiao; Finch, Steven R. Precise calculation of Hausdorff dimension of Apollonian gasket. Fractals 26 (2018), no. 4, 9 pp. claims a better approximation is

$$1.3056867280498771846459862068510\dots$$

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