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Applications of Circle Packing Moduli
Brock WilliamsTexas Tech University
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Roger W. Barnard
John Calhoun
Levi Johnson
Kent Pearce
Hamed Sari-Sarraf
Arun Shammanna
Including Joint Work With
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Grants727944742402745472
10350960930257
Work Supported By
GrantT365Z110259
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A circle packing is a configuration of circles with a specified pattern of tangencies.
Circle Packings
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Circle Packings
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Circle Packings
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Realization of the Triangulation
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Discrete Conformal Maps
About 900 Circles in Each Packing
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Discrete Conformal Maps
About 12000 Circles in Each Packing
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Modulus of Ring Domain
2mod(D)
logRr
R
r
D
1
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Circle Packing Modulus
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Circle Packing Modulus
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Circle Packing Modulus
0.61456...
2modulus .
1l
12.9058..og
r
r
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Cotton Maturity
Image from Google Maps.
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Immature cotton fibers are round, thick, and hollow.
As they mature, the lumen collapses and the cellulose thins.
Cotton Fibers
Cross section of fiber
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“Rounding the Fiber Out”
0.67277...
2modulus .
1l
15.8525..og
r
r
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“Rounding the Fiber Out”
0.91595...
2modulus .
1l
71.5678..og
r
r
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In work with John Calhoun, we’ve used random walk ideas to model disease
outbreaks.
Disease Modeling
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Zombie Virus
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Munz, Hudea, Imad, and Smith published a PDE model of a zombie outbreak in 2009.
They found logistic growth of the zombie population (and corresponding logistic decay of the human population).
Previous Zombie Work
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Incorporates additional parameters such as◦ Geography of the City◦ Hospitals◦ Human Offensive Capability
Predicts human population survival under some circumstances
Our Model
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Conformal Landscape
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Conformal Landscape
0.7292...
modu 19.89lus ...65
r
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Effect on the Disease
Human Population Zombie Population
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A Different Conformal Landscape
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A Different Conformal Landscape
0.2387...
mod 4.386..ulus .
r
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Effect on the Disease
Human Population Zombie Population
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Questions?
Thanks!