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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
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