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](http://dx.doi.org/10.1088/0305-4470/27/7/007)


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](https://dash.harvard.edu/handle/1/3445973)


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](http://oeis.org/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?



Hausdorff dimension of Apollonian circle packing, 1.305686729, 1.305688 or yet something else?
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.](https://www.worldscientific.com/doi/pdf/10.1142/S0218348X18500500)
claims a better approximation is


$$1.3056867280498771846459862068510\dots$$


