Approximating the dimension of circle packings

Jason Liu1

Mentor: Sergiy Merenkov2

1Davidson Academy Student, PRIMES-USA Student

2Professor of Mathematics at CCNYFaculty of Mathematics at CUNY

PRIMES-MIT Conference, May 18-19 2019

Liu, Merenkov Approximating the dimension of circle packings

Round Packings

Apollonian Gaskets: Other packings:

Conjecture (Chris Bishop):

The Apollonian Gasket has the lowest Hausdorff dimension of allround packings.

Liu, Merenkov Approximating the dimension of circle packings

Fractional dimension

Box-counting dimension (Minkowski–Bouligand):

dimbox(S) := limε→0

logN(ε)

log(1/ε)

Upper box dimension

Lower box dimension

Liu, Merenkov Approximating the dimension of circle packings

Fractional dimension (cont.)

Hausdorff measure:

Hd(S) = limδ→0

inf

{ ∞∑i=1

(diamUi )d :

∞⋃i=1

Ui ⊇ S , diamUi < δ

}

Hausdorff dimension:

dimH(S) := inf{d ≥ 0 : Hd(S) = 0}

Liu, Merenkov Approximating the dimension of circle packings

Markov Partition of Gasket

Gaskets: Markov Partitions:

Liu, Merenkov Approximating the dimension of circle packings

Current Results

Approximations of the Hausdorff dimension of gasket:

1.239652822.638258241.359014291.359014291.289410771.302511931.29058691.298642291.294388651.299169151.297135211.300156331.299006861.30103601

McMullen: 1.305688Boyd: Between 1.300197 and 1.314534

Liu, Merenkov Approximating the dimension of circle packings

Future Research

Implementing further practical considerations

Approximating dimensions of other gaskets

Looking for patterns

Liu, Merenkov Approximating the dimension of circle packings

Acknowledgments

I would like to thank my mentor, Professor Sergiy Merenkov, alongwith the PRIMES-USA program and its director, Doctor SlavaGerovitch. I would also like to thank the Head Mentor DoctorTanya Khovanova, and the MIT math department.

Liu, Merenkov Approximating the dimension of circle packings

Bibliography

1. C. McMullen. Hausdorff dimension and conformal dynamicsIII: Computation of dimension. American Journal ofMathematics 120(1998), 691-721

2. O. Jenkinson and M. Pollicott. Calculating HausdorffDimension of Julia Sets and Kleinian Limit Sets. AmericanJournal of Mathematics 124(2002), 495-545

3. D. Boyd. The residual set dimension of the Apollonianpacking. Mathematika 20(1973), 170–174.

4.https://en.wikipedia.org/wiki/Minkowski–Bouligand dimension

Liu, Merenkov Approximating the dimension of circle packings