the mathematics of billiards - washington university...
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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
![Page 2: The Mathematics of Billiards - Washington University …mathcircle.wustl.edu/uploads/4/9/7/9/49791831/20160306-billiards...The Mathematics of Billiards Washington University Math Circle](https://reader031.vdocuments.us/reader031/viewer/2022021616/5b32c9937f8b9ab5728d62c3/html5/thumbnails/2.jpg)
One thing you could do but wewon’t: play real billiards! /
Chris Cox Wash U Stl
The Mathematics of Billiards
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Instead, focus on one key idea:“specular reflection”
Chris Cox Wash U Stl
The Mathematics of Billiards
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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
![Page 5: The Mathematics of Billiards - Washington University …mathcircle.wustl.edu/uploads/4/9/7/9/49791831/20160306-billiards...The Mathematics of Billiards Washington University Math Circle](https://reader031.vdocuments.us/reader031/viewer/2022021616/5b32c9937f8b9ab5728d62c3/html5/thumbnails/5.jpg)
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
![Page 6: The Mathematics of Billiards - Washington University …mathcircle.wustl.edu/uploads/4/9/7/9/49791831/20160306-billiards...The Mathematics of Billiards Washington University Math Circle](https://reader031.vdocuments.us/reader031/viewer/2022021616/5b32c9937f8b9ab5728d62c3/html5/thumbnails/6.jpg)
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
![Page 7: The Mathematics of Billiards - Washington University …mathcircle.wustl.edu/uploads/4/9/7/9/49791831/20160306-billiards...The Mathematics of Billiards Washington University Math Circle](https://reader031.vdocuments.us/reader031/viewer/2022021616/5b32c9937f8b9ab5728d62c3/html5/thumbnails/7.jpg)
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
![Page 8: The Mathematics of Billiards - Washington University …mathcircle.wustl.edu/uploads/4/9/7/9/49791831/20160306-billiards...The Mathematics of Billiards Washington University Math Circle](https://reader031.vdocuments.us/reader031/viewer/2022021616/5b32c9937f8b9ab5728d62c3/html5/thumbnails/8.jpg)
Rectangle billiardsOne type of behavior: 41 collisions
Chris Cox Wash U Stl
The Mathematics of Billiards
![Page 9: The Mathematics of Billiards - Washington University …mathcircle.wustl.edu/uploads/4/9/7/9/49791831/20160306-billiards...The Mathematics of Billiards Washington University Math Circle](https://reader031.vdocuments.us/reader031/viewer/2022021616/5b32c9937f8b9ab5728d62c3/html5/thumbnails/9.jpg)
Rectangle billiardsOne type of behavior: 221 collisions
Chris Cox Wash U Stl
The Mathematics of Billiards
![Page 10: The Mathematics of Billiards - Washington University …mathcircle.wustl.edu/uploads/4/9/7/9/49791831/20160306-billiards...The Mathematics of Billiards Washington University Math Circle](https://reader031.vdocuments.us/reader031/viewer/2022021616/5b32c9937f8b9ab5728d62c3/html5/thumbnails/10.jpg)
Rectangle billiardsOne type of behavior: 362 collisions
Chris Cox Wash U Stl
The Mathematics of Billiards
![Page 11: The Mathematics of Billiards - Washington University …mathcircle.wustl.edu/uploads/4/9/7/9/49791831/20160306-billiards...The Mathematics of Billiards Washington University Math Circle](https://reader031.vdocuments.us/reader031/viewer/2022021616/5b32c9937f8b9ab5728d62c3/html5/thumbnails/11.jpg)
Rectangle billiardsAnother type: “periodic orbit” of period 10
Chris Cox Wash U Stl
The Mathematics of Billiards
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Specular collisions
Next problem: somemore interestingshapes!
Chris Cox Wash U Stl
The Mathematics of Billiards
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Rectangle billiardsActually the first one might be periodic too:
Chris Cox Wash U Stl
The Mathematics of Billiards
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Specular collisions
Next problem: somemore interestingshapes!
Chris Cox Wash U Stl
The Mathematics of Billiards
![Page 15: The Mathematics of Billiards - Washington University …mathcircle.wustl.edu/uploads/4/9/7/9/49791831/20160306-billiards...The Mathematics of Billiards Washington University Math Circle](https://reader031.vdocuments.us/reader031/viewer/2022021616/5b32c9937f8b9ab5728d62c3/html5/thumbnails/15.jpg)
Triangle billiards
Chris Cox Wash U Stl
The Mathematics of Billiards
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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
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Specular collisionsFor curves, we use the tangent line:
Chris Cox Wash U Stl
The Mathematics of Billiards
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Moon billiards
Chris Cox Wash U Stl
The Mathematics of Billiards
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Worksheet Question 4: What dobilliards look like in a circular table?
Chris Cox Wash U Stl
The Mathematics of Billiards
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Worksheet Question 4: What dobilliards look like in a circular table?
Chris Cox Wash U Stl
The Mathematics of Billiards
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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
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Question 1: What do billiards look like in a circular table?
With different direction:
Chris Cox Wash U Stl
The Mathematics of Billiards
![Page 23: The Mathematics of Billiards - Washington University …mathcircle.wustl.edu/uploads/4/9/7/9/49791831/20160306-billiards...The Mathematics of Billiards Washington University Math Circle](https://reader031.vdocuments.us/reader031/viewer/2022021616/5b32c9937f8b9ab5728d62c3/html5/thumbnails/23.jpg)
Question 1: What do billiards look like in a circular table?
With different direction:
Chris Cox Wash U Stl
The Mathematics of Billiards
![Page 24: The Mathematics of Billiards - Washington University …mathcircle.wustl.edu/uploads/4/9/7/9/49791831/20160306-billiards...The Mathematics of Billiards Washington University Math Circle](https://reader031.vdocuments.us/reader031/viewer/2022021616/5b32c9937f8b9ab5728d62c3/html5/thumbnails/24.jpg)
Question 1: What do billiards look like in a circular table?
With different direction:
Chris Cox Wash U Stl
The Mathematics of Billiards
![Page 25: The Mathematics of Billiards - Washington University …mathcircle.wustl.edu/uploads/4/9/7/9/49791831/20160306-billiards...The Mathematics of Billiards Washington University Math Circle](https://reader031.vdocuments.us/reader031/viewer/2022021616/5b32c9937f8b9ab5728d62c3/html5/thumbnails/25.jpg)
Question 1: What do billiards look like in a circular table?
With different direction:
Chris Cox Wash U Stl
The Mathematics of Billiards
![Page 26: The Mathematics of Billiards - Washington University …mathcircle.wustl.edu/uploads/4/9/7/9/49791831/20160306-billiards...The Mathematics of Billiards Washington University Math Circle](https://reader031.vdocuments.us/reader031/viewer/2022021616/5b32c9937f8b9ab5728d62c3/html5/thumbnails/26.jpg)
Question 1: What do billiards look like in a circular table?
With different direction:
Chris Cox Wash U Stl
The Mathematics of Billiards
![Page 27: The Mathematics of Billiards - Washington University …mathcircle.wustl.edu/uploads/4/9/7/9/49791831/20160306-billiards...The Mathematics of Billiards Washington University Math Circle](https://reader031.vdocuments.us/reader031/viewer/2022021616/5b32c9937f8b9ab5728d62c3/html5/thumbnails/27.jpg)
Question 1: What do billiards look like in a circular table?
With different direction:
Chris Cox Wash U Stl
The Mathematics of Billiards
![Page 28: The Mathematics of Billiards - Washington University …mathcircle.wustl.edu/uploads/4/9/7/9/49791831/20160306-billiards...The Mathematics of Billiards Washington University Math Circle](https://reader031.vdocuments.us/reader031/viewer/2022021616/5b32c9937f8b9ab5728d62c3/html5/thumbnails/28.jpg)
Question 2: What is wrong withmy circle?
Chris Cox Wash U Stl
The Mathematics of Billiards
![Page 29: The Mathematics of Billiards - Washington University …mathcircle.wustl.edu/uploads/4/9/7/9/49791831/20160306-billiards...The Mathematics of Billiards Washington University Math Circle](https://reader031.vdocuments.us/reader031/viewer/2022021616/5b32c9937f8b9ab5728d62c3/html5/thumbnails/29.jpg)
Three types of behavior
▸ Periodic
▸ Nice but non-periodic
▸ Chaotic
Chris Cox Wash U Stl
The Mathematics of Billiards
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Undergraduates Research Billiard Dynamics
Chris Cox Wash U Stl
The Mathematics of Billiards
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Yakov Sinai awarded the Abel Prize
Chris Cox Wash U Stl
The Mathematics of Billiards
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Can you make aperiodic circlebilliard?
Chris Cox Wash U Stl
The Mathematics of Billiards
![Page 33: The Mathematics of Billiards - Washington University …mathcircle.wustl.edu/uploads/4/9/7/9/49791831/20160306-billiards...The Mathematics of Billiards Washington University Math Circle](https://reader031.vdocuments.us/reader031/viewer/2022021616/5b32c9937f8b9ab5728d62c3/html5/thumbnails/33.jpg)
Question: What is wrong with mycircle?
Chris Cox Wash U Stl
The Mathematics of Billiards
![Page 34: The Mathematics of Billiards - Washington University …mathcircle.wustl.edu/uploads/4/9/7/9/49791831/20160306-billiards...The Mathematics of Billiards Washington University Math Circle](https://reader031.vdocuments.us/reader031/viewer/2022021616/5b32c9937f8b9ab5728d62c3/html5/thumbnails/34.jpg)
Question: What is wrong with mycircle?
Chris Cox Wash U Stl
The Mathematics of Billiards
![Page 35: The Mathematics of Billiards - Washington University …mathcircle.wustl.edu/uploads/4/9/7/9/49791831/20160306-billiards...The Mathematics of Billiards Washington University Math Circle](https://reader031.vdocuments.us/reader031/viewer/2022021616/5b32c9937f8b9ab5728d62c3/html5/thumbnails/35.jpg)
Question: What is wrong with mycircle?
Chris Cox Wash U Stl
The Mathematics of Billiards
![Page 36: The Mathematics of Billiards - Washington University …mathcircle.wustl.edu/uploads/4/9/7/9/49791831/20160306-billiards...The Mathematics of Billiards Washington University Math Circle](https://reader031.vdocuments.us/reader031/viewer/2022021616/5b32c9937f8b9ab5728d62c3/html5/thumbnails/36.jpg)
Question: What is wrong with mycircle?
Chris Cox Wash U Stl
The Mathematics of Billiards
![Page 37: The Mathematics of Billiards - Washington University …mathcircle.wustl.edu/uploads/4/9/7/9/49791831/20160306-billiards...The Mathematics of Billiards Washington University Math Circle](https://reader031.vdocuments.us/reader031/viewer/2022021616/5b32c9937f8b9ab5728d62c3/html5/thumbnails/37.jpg)
Question: What is wrong with mycircle?
Chris Cox Wash U Stl
The Mathematics of Billiards
![Page 38: The Mathematics of Billiards - Washington University …mathcircle.wustl.edu/uploads/4/9/7/9/49791831/20160306-billiards...The Mathematics of Billiards Washington University Math Circle](https://reader031.vdocuments.us/reader031/viewer/2022021616/5b32c9937f8b9ab5728d62c3/html5/thumbnails/38.jpg)
Question: What is wrong with mycircle?
Chris Cox Wash U Stl
The Mathematics of Billiards
![Page 39: The Mathematics of Billiards - Washington University …mathcircle.wustl.edu/uploads/4/9/7/9/49791831/20160306-billiards...The Mathematics of Billiards Washington University Math Circle](https://reader031.vdocuments.us/reader031/viewer/2022021616/5b32c9937f8b9ab5728d62c3/html5/thumbnails/39.jpg)
Question: What is wrong with mycircle?
Chris Cox Wash U Stl
The Mathematics of Billiards
![Page 40: The Mathematics of Billiards - Washington University …mathcircle.wustl.edu/uploads/4/9/7/9/49791831/20160306-billiards...The Mathematics of Billiards Washington University Math Circle](https://reader031.vdocuments.us/reader031/viewer/2022021616/5b32c9937f8b9ab5728d62c3/html5/thumbnails/40.jpg)
Some really interesting tables
Worksheet question 6: ellipses and moons
and stadia and mushrooms
Chris Cox Wash U Stl
The Mathematics of Billiards
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The mushroom billiard
Figure by Carl Dettmann
Chris Cox Wash U Stl
The Mathematics of Billiards
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The ergodic game
How does our sampling rule change the
“average”?
Chris Cox Wash U Stl
The Mathematics of Billiards
![Page 43: The Mathematics of Billiards - Washington University …mathcircle.wustl.edu/uploads/4/9/7/9/49791831/20160306-billiards...The Mathematics of Billiards Washington University Math Circle](https://reader031.vdocuments.us/reader031/viewer/2022021616/5b32c9937f8b9ab5728d62c3/html5/thumbnails/43.jpg)
A new rule:“no-slip” circles
Chris Cox Wash U Stl
The Mathematics of Billiards
![Page 44: The Mathematics of Billiards - Washington University …mathcircle.wustl.edu/uploads/4/9/7/9/49791831/20160306-billiards...The Mathematics of Billiards Washington University Math Circle](https://reader031.vdocuments.us/reader031/viewer/2022021616/5b32c9937f8b9ab5728d62c3/html5/thumbnails/44.jpg)
Applications of specular collisions
This simple model has many applications:
▸ Modeling fluids (Lorentz gases)
Chris Cox Wash U Stl
The Mathematics of Billiards
![Page 45: The Mathematics of Billiards - Washington University …mathcircle.wustl.edu/uploads/4/9/7/9/49791831/20160306-billiards...The Mathematics of Billiards Washington University Math Circle](https://reader031.vdocuments.us/reader031/viewer/2022021616/5b32c9937f8b9ab5728d62c3/html5/thumbnails/45.jpg)
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
![Page 46: The Mathematics of Billiards - Washington University …mathcircle.wustl.edu/uploads/4/9/7/9/49791831/20160306-billiards...The Mathematics of Billiards Washington University Math Circle](https://reader031.vdocuments.us/reader031/viewer/2022021616/5b32c9937f8b9ab5728d62c3/html5/thumbnails/46.jpg)
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
![Page 47: The Mathematics of Billiards - Washington University …mathcircle.wustl.edu/uploads/4/9/7/9/49791831/20160306-billiards...The Mathematics of Billiards Washington University Math Circle](https://reader031.vdocuments.us/reader031/viewer/2022021616/5b32c9937f8b9ab5728d62c3/html5/thumbnails/47.jpg)
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