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The Mathematics of BilliardsWashington University Math Circle
Chris Cox
March 6, 2016
Chris Cox Wash U Stl
The Mathematics of Billiards
One thing you could do but wewon’t: play real billiards! /
Chris Cox Wash U Stl
The Mathematics of Billiards
Instead, focus on one key idea:“specular reflection”
Chris Cox Wash U Stl
The Mathematics of Billiards
Warm up problem:Draw the path of ONE billiard on a
rectangular table through several collisions.
(1)
Chris Cox Wash U Stl
The Mathematics of Billiards
Warm up problem:Draw the path of a billiard on a rectangular
table through several collisions.
Chris Cox Wash U Stl
The Mathematics of Billiards
Warm up problem:Draw the path of a billiard on a rectangular
table through several collisions.
Chris Cox Wash U Stl
The Mathematics of Billiards
Warm up problem:Draw the path of a billiard on a rectangular
table through several collisions.
Chris Cox Wash U Stl
The Mathematics of Billiards
Rectangle billiardsOne type of behavior: 41 collisions
Chris Cox Wash U Stl
The Mathematics of Billiards
Rectangle billiardsOne type of behavior: 221 collisions
Chris Cox Wash U Stl
The Mathematics of Billiards
Rectangle billiardsOne type of behavior: 362 collisions
Chris Cox Wash U Stl
The Mathematics of Billiards
Rectangle billiardsAnother type: “periodic orbit” of period 10
Chris Cox Wash U Stl
The Mathematics of Billiards
Specular collisions
Next problem: somemore interestingshapes!
Chris Cox Wash U Stl
The Mathematics of Billiards
Rectangle billiardsActually the first one might be periodic too:
Chris Cox Wash U Stl
The Mathematics of Billiards
Specular collisions
Next problem: somemore interestingshapes!
Chris Cox Wash U Stl
The Mathematics of Billiards
Triangle billiards
Chris Cox Wash U Stl
The Mathematics of Billiards
Specular collisions
Things to notice:▸ The model works for curved
boundaries, not just lines
▸ This model assumes there is nofriction, no loss of energy, and nospinning
Chris Cox Wash U Stl
The Mathematics of Billiards
Specular collisionsFor curves, we use the tangent line:
Chris Cox Wash U Stl
The Mathematics of Billiards
Moon billiards
Chris Cox Wash U Stl
The Mathematics of Billiards
Worksheet Question 4: What dobilliards look like in a circular table?
Chris Cox Wash U Stl
The Mathematics of Billiards
Worksheet Question 4: What dobilliards look like in a circular table?
Chris Cox Wash U Stl
The Mathematics of Billiards
Question 1: What do billiards look like in a circular table?
That one looks like this:
Chris Cox Wash U Stl
The Mathematics of Billiards
Question 1: What do billiards look like in a circular table?
With different direction:
Chris Cox Wash U Stl
The Mathematics of Billiards
Question 1: What do billiards look like in a circular table?
With different direction:
Chris Cox Wash U Stl
The Mathematics of Billiards
Question 1: What do billiards look like in a circular table?
With different direction:
Chris Cox Wash U Stl
The Mathematics of Billiards
Question 1: What do billiards look like in a circular table?
With different direction:
Chris Cox Wash U Stl
The Mathematics of Billiards
Question 1: What do billiards look like in a circular table?
With different direction:
Chris Cox Wash U Stl
The Mathematics of Billiards
Question 1: What do billiards look like in a circular table?
With different direction:
Chris Cox Wash U Stl
The Mathematics of Billiards
Question 2: What is wrong withmy circle?
Chris Cox Wash U Stl
The Mathematics of Billiards
Three types of behavior
▸ Periodic
▸ Nice but non-periodic
▸ Chaotic
Chris Cox Wash U Stl
The Mathematics of Billiards
Undergraduates Research Billiard Dynamics
Chris Cox Wash U Stl
The Mathematics of Billiards
Yakov Sinai awarded the Abel Prize
Chris Cox Wash U Stl
The Mathematics of Billiards
Can you make aperiodic circlebilliard?
Chris Cox Wash U Stl
The Mathematics of Billiards
Question: What is wrong with mycircle?
Chris Cox Wash U Stl
The Mathematics of Billiards
Question: What is wrong with mycircle?
Chris Cox Wash U Stl
The Mathematics of Billiards
Question: What is wrong with mycircle?
Chris Cox Wash U Stl
The Mathematics of Billiards
Question: What is wrong with mycircle?
Chris Cox Wash U Stl
The Mathematics of Billiards
Question: What is wrong with mycircle?
Chris Cox Wash U Stl
The Mathematics of Billiards
Question: What is wrong with mycircle?
Chris Cox Wash U Stl
The Mathematics of Billiards
Question: What is wrong with mycircle?
Chris Cox Wash U Stl
The Mathematics of Billiards
Some really interesting tables
Worksheet question 6: ellipses and moons
and stadia and mushrooms
Chris Cox Wash U Stl
The Mathematics of Billiards
The mushroom billiard
Figure by Carl Dettmann
Chris Cox Wash U Stl
The Mathematics of Billiards
The ergodic game
How does our sampling rule change the
“average”?
Chris Cox Wash U Stl
The Mathematics of Billiards
A new rule:“no-slip” circles
Chris Cox Wash U Stl
The Mathematics of Billiards
Applications of specular collisions
This simple model has many applications:
▸ Modeling fluids (Lorentz gases)
Chris Cox Wash U Stl
The Mathematics of Billiards
Applications of specular collisions
This simple model has many applications:
▸ Modeling fluids (Lorentz gases)
▸ Brownian motion
Chris Cox Wash U Stl
The Mathematics of Billiards
Applications of specular collisions
This simple model has many applications:
▸ Modeling fluids (Lorentz gases)
▸ Brownian motion
▸ Heat transfer (How do things cool off?)
Chris Cox Wash U Stl
The Mathematics of Billiards
Applications of specular collisions
This simple model has many applications:
▸ Modeling fluids (Lorentz gases)
▸ Brownian motion
▸ Heat transfer (How do things cool off?)
▸ Diffusion (How do mixtures spread out?)
Chris Cox Wash U Stl
The Mathematics of Billiards
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