Applications of Circle Packing Moduli

Brock WilliamsTexas Tech University

Roger W. Barnard

John Calhoun

Levi Johnson

Kent Pearce

Hamed Sari-Sarraf

Arun Shammanna

Including Joint Work With

Grants727944742402745472

10350960930257

Work Supported By

GrantT365Z110259

A circle packing is a configuration of circles with a specified pattern of tangencies.

Circle Packings

Circle Packings

Circle Packings

Realization of the Triangulation

Discrete Conformal Maps

About 900 Circles in Each Packing

Discrete Conformal Maps

About 12000 Circles in Each Packing

Modulus of Ring Domain

2mod(D)

logRr

R

r

D

1

Circle Packing Modulus

Circle Packing Modulus

Circle Packing Modulus

0.61456...

2modulus .

1l

12.9058..og

r

r

Cotton Maturity

Image from Google Maps.

Immature cotton fibers are round, thick, and hollow.

As they mature, the lumen collapses and the cellulose thins.

Cotton Fibers

Cross section of fiber

“Rounding the Fiber Out”

0.67277...

2modulus .

1l

15.8525..og

r

r

“Rounding the Fiber Out”

0.91595...

2modulus .

1l

71.5678..og

r

r

In work with John Calhoun, we’ve used random walk ideas to model disease

outbreaks.

Disease Modeling

Zombie Virus

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

Incorporates additional parameters such as◦ Geography of the City◦ Hospitals◦ Human Offensive Capability

Predicts human population survival under some circumstances

Our Model

Conformal Landscape

Conformal Landscape

0.7292...

modu 19.89lus ...65

r

Effect on the Disease

Human Population Zombie Population

A Different Conformal Landscape

A Different Conformal Landscape

0.2387...

mod 4.386..ulus .

r

Effect on the Disease

Human Population Zombie Population

Questions?

Thanks!