Fractals - SFU.carpyke/mathcamp/Fractals presentation.pdfAll fractals have an address system you can...
Transcript of Fractals - SFU.carpyke/mathcamp/Fractals presentation.pdfAll fractals have an address system you can...
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Fractals
SFU-CMS Math Camp 2008Randall Pyke; www.sfu.ca/~rpyke/mathcamp
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Benoit Mandelbrot, 1977
How long is the coast of Britain?
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How long is the coast of Britain?
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How long is the coast of Britain?
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How long is the coast of Britain?
Coast gets longer and longer as ruler shrinks. . . .
How long is the coast of Britain (or any coast)?
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Regular curves
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coast linem
ea
su
red
le
ng
th
circle
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Zooming in on regular curves
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Zooming in on regular curves
All regular curves look like straight lines if you zoom in enough
(and that's why their measured length does not get arbitrarily large)
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Zooming in on regular curves
All regular curves look like straight lines if you zoom in enough
(and that's why their measured length does not get arbitrarily large)
But there are curves that don‟t „straighten up‟ as you get closer....
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Zooming in on fractal curves
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The measured length of fractal curves gets
longer and longer because they never
straighten out.
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Regular (Euclidean) geometry
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Fractal geometry. . . .
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Self-Similarity
The whole fractal
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Self-Similarity
Pick a small copy of it
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Self-Similarity
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Self-Similarity
Rotate it
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Self-Similarity
Blow it up: You get the same thing!
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Self-Similarity
Now continue with that piece
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Self-Similarity
You can find the same small piece on
this small piece…
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Self-Similarity
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Self-Similarity
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Four self-similar pieces Infinitely many self-similar pieces!
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Self-similar : Made up of smaller copies of itself
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Other self-similar objects:
These are self-similar, but not “complicated”
Fractals are self-similar and complicated
Solid square Solid triangle
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Fractal curves Fractal areas Fractal volumesFractal dust
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How to draw fractals?
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How to draw fractals?
• Here are three ways:
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How to draw fractals?
• Here are three ways:
- Removing pieces
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How to draw fractals?
• Here are three ways:
- Removing pieces
- Adding pieces
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How to draw fractals?
• Here are three ways:
- Removing pieces
- Adding pieces
- The „Chaos Game‟
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Removing pieces
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Removing pieces
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Or in 3-D
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Now you try
.
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Now you try
Start with a triangle:
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Now you try
Start with a triangle: Partition it into 4
smaller triangles:
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Now you try
Remove sub-triangle 3:
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Now you try
Remove sub-triangle 3:
Now continue . . .
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Partition each remaining
triangle
into 4 smaller triangles
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Partition each remaining
triangle
into 4 smaller triangles
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Partition each remaining
triangle
into 4 smaller triangles
Now remove each „3‟
triangle
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Another type: Remove triangles 1 and 2
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Shapes produced with
the 4 triangle partition;
![Page 68: Fractals - SFU.carpyke/mathcamp/Fractals presentation.pdfAll fractals have an address system you can use to label parts of the fractal Let‟s begin the Sierpinski Chaos game at the](https://reader034.fdocuments.in/reader034/viewer/2022042601/5f964e5c05aed62d6f1f2a11/html5/thumbnails/68.jpg)
Shapes produced with
the 4 triangle partition;
Sierpinski triangle (remove centre triangle):
Sierpinski variation (remove corner triangle):
![Page 69: Fractals - SFU.carpyke/mathcamp/Fractals presentation.pdfAll fractals have an address system you can use to label parts of the fractal Let‟s begin the Sierpinski Chaos game at the](https://reader034.fdocuments.in/reader034/viewer/2022042601/5f964e5c05aed62d6f1f2a11/html5/thumbnails/69.jpg)
Shapes produced with
the 4 triangle partition;
Sierpinski triangle (remove centre triangle):
Sierpinski variation (remove corner triangle):
![Page 70: Fractals - SFU.carpyke/mathcamp/Fractals presentation.pdfAll fractals have an address system you can use to label parts of the fractal Let‟s begin the Sierpinski Chaos game at the](https://reader034.fdocuments.in/reader034/viewer/2022042601/5f964e5c05aed62d6f1f2a11/html5/thumbnails/70.jpg)
For more variety, begin with the partition of the
triangle into 16 smaller triangles;
How many different shapes can you make?
![Page 71: Fractals - SFU.carpyke/mathcamp/Fractals presentation.pdfAll fractals have an address system you can use to label parts of the fractal Let‟s begin the Sierpinski Chaos game at the](https://reader034.fdocuments.in/reader034/viewer/2022042601/5f964e5c05aed62d6f1f2a11/html5/thumbnails/71.jpg)
Can do the same with a square
![Page 72: Fractals - SFU.carpyke/mathcamp/Fractals presentation.pdfAll fractals have an address system you can use to label parts of the fractal Let‟s begin the Sierpinski Chaos game at the](https://reader034.fdocuments.in/reader034/viewer/2022042601/5f964e5c05aed62d6f1f2a11/html5/thumbnails/72.jpg)
What is the „recipe‟ for this one?
![Page 73: Fractals - SFU.carpyke/mathcamp/Fractals presentation.pdfAll fractals have an address system you can use to label parts of the fractal Let‟s begin the Sierpinski Chaos game at the](https://reader034.fdocuments.in/reader034/viewer/2022042601/5f964e5c05aed62d6f1f2a11/html5/thumbnails/73.jpg)
“Remove all number 5 squares”
![Page 74: Fractals - SFU.carpyke/mathcamp/Fractals presentation.pdfAll fractals have an address system you can use to label parts of the fractal Let‟s begin the Sierpinski Chaos game at the](https://reader034.fdocuments.in/reader034/viewer/2022042601/5f964e5c05aed62d6f1f2a11/html5/thumbnails/74.jpg)
What is the „recipe‟ for this image?
![Page 75: Fractals - SFU.carpyke/mathcamp/Fractals presentation.pdfAll fractals have an address system you can use to label parts of the fractal Let‟s begin the Sierpinski Chaos game at the](https://reader034.fdocuments.in/reader034/viewer/2022042601/5f964e5c05aed62d6f1f2a11/html5/thumbnails/75.jpg)
![Page 76: Fractals - SFU.carpyke/mathcamp/Fractals presentation.pdfAll fractals have an address system you can use to label parts of the fractal Let‟s begin the Sierpinski Chaos game at the](https://reader034.fdocuments.in/reader034/viewer/2022042601/5f964e5c05aed62d6f1f2a11/html5/thumbnails/76.jpg)
![Page 77: Fractals - SFU.carpyke/mathcamp/Fractals presentation.pdfAll fractals have an address system you can use to label parts of the fractal Let‟s begin the Sierpinski Chaos game at the](https://reader034.fdocuments.in/reader034/viewer/2022042601/5f964e5c05aed62d6f1f2a11/html5/thumbnails/77.jpg)
Remove all 4, 6, and 8 squares
![Page 78: Fractals - SFU.carpyke/mathcamp/Fractals presentation.pdfAll fractals have an address system you can use to label parts of the fractal Let‟s begin the Sierpinski Chaos game at the](https://reader034.fdocuments.in/reader034/viewer/2022042601/5f964e5c05aed62d6f1f2a11/html5/thumbnails/78.jpg)
Adding pieces
![Page 79: Fractals - SFU.carpyke/mathcamp/Fractals presentation.pdfAll fractals have an address system you can use to label parts of the fractal Let‟s begin the Sierpinski Chaos game at the](https://reader034.fdocuments.in/reader034/viewer/2022042601/5f964e5c05aed62d6f1f2a11/html5/thumbnails/79.jpg)
Finally…..
![Page 80: Fractals - SFU.carpyke/mathcamp/Fractals presentation.pdfAll fractals have an address system you can use to label parts of the fractal Let‟s begin the Sierpinski Chaos game at the](https://reader034.fdocuments.in/reader034/viewer/2022042601/5f964e5c05aed62d6f1f2a11/html5/thumbnails/80.jpg)
Try this:
Instead of this generator
Use this one
(all sides 1/3 long)
![Page 81: Fractals - SFU.carpyke/mathcamp/Fractals presentation.pdfAll fractals have an address system you can use to label parts of the fractal Let‟s begin the Sierpinski Chaos game at the](https://reader034.fdocuments.in/reader034/viewer/2022042601/5f964e5c05aed62d6f1f2a11/html5/thumbnails/81.jpg)
![Page 82: Fractals - SFU.carpyke/mathcamp/Fractals presentation.pdfAll fractals have an address system you can use to label parts of the fractal Let‟s begin the Sierpinski Chaos game at the](https://reader034.fdocuments.in/reader034/viewer/2022042601/5f964e5c05aed62d6f1f2a11/html5/thumbnails/82.jpg)
The Chaos Game
![Page 83: Fractals - SFU.carpyke/mathcamp/Fractals presentation.pdfAll fractals have an address system you can use to label parts of the fractal Let‟s begin the Sierpinski Chaos game at the](https://reader034.fdocuments.in/reader034/viewer/2022042601/5f964e5c05aed62d6f1f2a11/html5/thumbnails/83.jpg)
Playing the Chaos game to draw fractals
![Page 84: Fractals - SFU.carpyke/mathcamp/Fractals presentation.pdfAll fractals have an address system you can use to label parts of the fractal Let‟s begin the Sierpinski Chaos game at the](https://reader034.fdocuments.in/reader034/viewer/2022042601/5f964e5c05aed62d6f1f2a11/html5/thumbnails/84.jpg)
Playing the Chaos game to draw fractals
![Page 85: Fractals - SFU.carpyke/mathcamp/Fractals presentation.pdfAll fractals have an address system you can use to label parts of the fractal Let‟s begin the Sierpinski Chaos game at the](https://reader034.fdocuments.in/reader034/viewer/2022042601/5f964e5c05aed62d6f1f2a11/html5/thumbnails/85.jpg)
Playing the Chaos game to draw fractals
![Page 86: Fractals - SFU.carpyke/mathcamp/Fractals presentation.pdfAll fractals have an address system you can use to label parts of the fractal Let‟s begin the Sierpinski Chaos game at the](https://reader034.fdocuments.in/reader034/viewer/2022042601/5f964e5c05aed62d6f1f2a11/html5/thumbnails/86.jpg)
Playing the Chaos game to draw fractals
![Page 87: Fractals - SFU.carpyke/mathcamp/Fractals presentation.pdfAll fractals have an address system you can use to label parts of the fractal Let‟s begin the Sierpinski Chaos game at the](https://reader034.fdocuments.in/reader034/viewer/2022042601/5f964e5c05aed62d6f1f2a11/html5/thumbnails/87.jpg)
Playing the Chaos game to draw fractals
![Page 88: Fractals - SFU.carpyke/mathcamp/Fractals presentation.pdfAll fractals have an address system you can use to label parts of the fractal Let‟s begin the Sierpinski Chaos game at the](https://reader034.fdocuments.in/reader034/viewer/2022042601/5f964e5c05aed62d6f1f2a11/html5/thumbnails/88.jpg)
Playing the Chaos game to draw fractals
![Page 89: Fractals - SFU.carpyke/mathcamp/Fractals presentation.pdfAll fractals have an address system you can use to label parts of the fractal Let‟s begin the Sierpinski Chaos game at the](https://reader034.fdocuments.in/reader034/viewer/2022042601/5f964e5c05aed62d6f1f2a11/html5/thumbnails/89.jpg)
Playing the Chaos game to draw fractals
![Page 90: Fractals - SFU.carpyke/mathcamp/Fractals presentation.pdfAll fractals have an address system you can use to label parts of the fractal Let‟s begin the Sierpinski Chaos game at the](https://reader034.fdocuments.in/reader034/viewer/2022042601/5f964e5c05aed62d6f1f2a11/html5/thumbnails/90.jpg)
![Page 91: Fractals - SFU.carpyke/mathcamp/Fractals presentation.pdfAll fractals have an address system you can use to label parts of the fractal Let‟s begin the Sierpinski Chaos game at the](https://reader034.fdocuments.in/reader034/viewer/2022042601/5f964e5c05aed62d6f1f2a11/html5/thumbnails/91.jpg)
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![Page 109: Fractals - SFU.carpyke/mathcamp/Fractals presentation.pdfAll fractals have an address system you can use to label parts of the fractal Let‟s begin the Sierpinski Chaos game at the](https://reader034.fdocuments.in/reader034/viewer/2022042601/5f964e5c05aed62d6f1f2a11/html5/thumbnails/109.jpg)
![Page 110: Fractals - SFU.carpyke/mathcamp/Fractals presentation.pdfAll fractals have an address system you can use to label parts of the fractal Let‟s begin the Sierpinski Chaos game at the](https://reader034.fdocuments.in/reader034/viewer/2022042601/5f964e5c05aed62d6f1f2a11/html5/thumbnails/110.jpg)
![Page 111: Fractals - SFU.carpyke/mathcamp/Fractals presentation.pdfAll fractals have an address system you can use to label parts of the fractal Let‟s begin the Sierpinski Chaos game at the](https://reader034.fdocuments.in/reader034/viewer/2022042601/5f964e5c05aed62d6f1f2a11/html5/thumbnails/111.jpg)
Why the Chaos Game Works
![Page 112: Fractals - SFU.carpyke/mathcamp/Fractals presentation.pdfAll fractals have an address system you can use to label parts of the fractal Let‟s begin the Sierpinski Chaos game at the](https://reader034.fdocuments.in/reader034/viewer/2022042601/5f964e5c05aed62d6f1f2a11/html5/thumbnails/112.jpg)
Why the Chaos Game Works
• Addresses:
![Page 113: Fractals - SFU.carpyke/mathcamp/Fractals presentation.pdfAll fractals have an address system you can use to label parts of the fractal Let‟s begin the Sierpinski Chaos game at the](https://reader034.fdocuments.in/reader034/viewer/2022042601/5f964e5c05aed62d6f1f2a11/html5/thumbnails/113.jpg)
Why the Chaos Game Works
• Addresses:
![Page 114: Fractals - SFU.carpyke/mathcamp/Fractals presentation.pdfAll fractals have an address system you can use to label parts of the fractal Let‟s begin the Sierpinski Chaos game at the](https://reader034.fdocuments.in/reader034/viewer/2022042601/5f964e5c05aed62d6f1f2a11/html5/thumbnails/114.jpg)
Why the Chaos Game Works
• Addresses:
![Page 115: Fractals - SFU.carpyke/mathcamp/Fractals presentation.pdfAll fractals have an address system you can use to label parts of the fractal Let‟s begin the Sierpinski Chaos game at the](https://reader034.fdocuments.in/reader034/viewer/2022042601/5f964e5c05aed62d6f1f2a11/html5/thumbnails/115.jpg)
Address length 1
Address length 2
![Page 116: Fractals - SFU.carpyke/mathcamp/Fractals presentation.pdfAll fractals have an address system you can use to label parts of the fractal Let‟s begin the Sierpinski Chaos game at the](https://reader034.fdocuments.in/reader034/viewer/2022042601/5f964e5c05aed62d6f1f2a11/html5/thumbnails/116.jpg)
Address length 3
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Address length 3
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Address length 3
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Address length 3
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Address length 3
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Address length 3
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Address length 3
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Address length 3
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Address length 3
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Address length 3
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Address length 3
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Address length 3
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Address length 4
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Address length 4
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Address length 4
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Address length 4
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Address length 4
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Address length 4
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Address length 4
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Address length 4
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Address length 4
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Address length 4
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What is Sierpinski‟s Triangle?
![Page 139: Fractals - SFU.carpyke/mathcamp/Fractals presentation.pdfAll fractals have an address system you can use to label parts of the fractal Let‟s begin the Sierpinski Chaos game at the](https://reader034.fdocuments.in/reader034/viewer/2022042601/5f964e5c05aed62d6f1f2a11/html5/thumbnails/139.jpg)
What is Sierpinski‟s Triangle?
All regions without a „4‟ in their address:
![Page 140: Fractals - SFU.carpyke/mathcamp/Fractals presentation.pdfAll fractals have an address system you can use to label parts of the fractal Let‟s begin the Sierpinski Chaos game at the](https://reader034.fdocuments.in/reader034/viewer/2022042601/5f964e5c05aed62d6f1f2a11/html5/thumbnails/140.jpg)
What is Sierpinski‟s Triangle?
So we can draw the Sierpinski triangle if we put a dot
in every address region that doesn’t have a 4
All regions without a „4‟ in their address:
![Page 141: Fractals - SFU.carpyke/mathcamp/Fractals presentation.pdfAll fractals have an address system you can use to label parts of the fractal Let‟s begin the Sierpinski Chaos game at the](https://reader034.fdocuments.in/reader034/viewer/2022042601/5f964e5c05aed62d6f1f2a11/html5/thumbnails/141.jpg)
All fractals have an address system you can
use to label parts of the fractal
![Page 142: Fractals - SFU.carpyke/mathcamp/Fractals presentation.pdfAll fractals have an address system you can use to label parts of the fractal Let‟s begin the Sierpinski Chaos game at the](https://reader034.fdocuments.in/reader034/viewer/2022042601/5f964e5c05aed62d6f1f2a11/html5/thumbnails/142.jpg)
All fractals have an address system you can
use to label parts of the fractal
![Page 143: Fractals - SFU.carpyke/mathcamp/Fractals presentation.pdfAll fractals have an address system you can use to label parts of the fractal Let‟s begin the Sierpinski Chaos game at the](https://reader034.fdocuments.in/reader034/viewer/2022042601/5f964e5c05aed62d6f1f2a11/html5/thumbnails/143.jpg)
All fractals have an address system you can
use to label parts of the fractal
![Page 144: Fractals - SFU.carpyke/mathcamp/Fractals presentation.pdfAll fractals have an address system you can use to label parts of the fractal Let‟s begin the Sierpinski Chaos game at the](https://reader034.fdocuments.in/reader034/viewer/2022042601/5f964e5c05aed62d6f1f2a11/html5/thumbnails/144.jpg)
All fractals have an address system you can
use to label parts of the fractal
![Page 145: Fractals - SFU.carpyke/mathcamp/Fractals presentation.pdfAll fractals have an address system you can use to label parts of the fractal Let‟s begin the Sierpinski Chaos game at the](https://reader034.fdocuments.in/reader034/viewer/2022042601/5f964e5c05aed62d6f1f2a11/html5/thumbnails/145.jpg)
Let‟s begin the Sierpinski Chaos game at
the bottom left corner
![Page 146: Fractals - SFU.carpyke/mathcamp/Fractals presentation.pdfAll fractals have an address system you can use to label parts of the fractal Let‟s begin the Sierpinski Chaos game at the](https://reader034.fdocuments.in/reader034/viewer/2022042601/5f964e5c05aed62d6f1f2a11/html5/thumbnails/146.jpg)
Let‟s begin the Sierpinski Chaos game at
the bottom left corner
Address of this point is
![Page 147: Fractals - SFU.carpyke/mathcamp/Fractals presentation.pdfAll fractals have an address system you can use to label parts of the fractal Let‟s begin the Sierpinski Chaos game at the](https://reader034.fdocuments.in/reader034/viewer/2022042601/5f964e5c05aed62d6f1f2a11/html5/thumbnails/147.jpg)
Let‟s begin the Sierpinski Chaos game at
the bottom left corner
Address of this point is
![Page 148: Fractals - SFU.carpyke/mathcamp/Fractals presentation.pdfAll fractals have an address system you can use to label parts of the fractal Let‟s begin the Sierpinski Chaos game at the](https://reader034.fdocuments.in/reader034/viewer/2022042601/5f964e5c05aed62d6f1f2a11/html5/thumbnails/148.jpg)
Let‟s begin the Sierpinski Chaos game at
the bottom left corner
Address of this point is
111111111111111……
![Page 149: Fractals - SFU.carpyke/mathcamp/Fractals presentation.pdfAll fractals have an address system you can use to label parts of the fractal Let‟s begin the Sierpinski Chaos game at the](https://reader034.fdocuments.in/reader034/viewer/2022042601/5f964e5c05aed62d6f1f2a11/html5/thumbnails/149.jpg)
Suppose first game number is a 2;
![Page 150: Fractals - SFU.carpyke/mathcamp/Fractals presentation.pdfAll fractals have an address system you can use to label parts of the fractal Let‟s begin the Sierpinski Chaos game at the](https://reader034.fdocuments.in/reader034/viewer/2022042601/5f964e5c05aed62d6f1f2a11/html5/thumbnails/150.jpg)
Suppose first game number is a 2;
Address of this game point is
![Page 151: Fractals - SFU.carpyke/mathcamp/Fractals presentation.pdfAll fractals have an address system you can use to label parts of the fractal Let‟s begin the Sierpinski Chaos game at the](https://reader034.fdocuments.in/reader034/viewer/2022042601/5f964e5c05aed62d6f1f2a11/html5/thumbnails/151.jpg)
Suppose first game number is a 2;
Address of this game point is
211111111111……..
![Page 152: Fractals - SFU.carpyke/mathcamp/Fractals presentation.pdfAll fractals have an address system you can use to label parts of the fractal Let‟s begin the Sierpinski Chaos game at the](https://reader034.fdocuments.in/reader034/viewer/2022042601/5f964e5c05aed62d6f1f2a11/html5/thumbnails/152.jpg)
And if the next game number is a 3;
![Page 153: Fractals - SFU.carpyke/mathcamp/Fractals presentation.pdfAll fractals have an address system you can use to label parts of the fractal Let‟s begin the Sierpinski Chaos game at the](https://reader034.fdocuments.in/reader034/viewer/2022042601/5f964e5c05aed62d6f1f2a11/html5/thumbnails/153.jpg)
And if the next game number is a 3;
Address of this game point is
![Page 154: Fractals - SFU.carpyke/mathcamp/Fractals presentation.pdfAll fractals have an address system you can use to label parts of the fractal Let‟s begin the Sierpinski Chaos game at the](https://reader034.fdocuments.in/reader034/viewer/2022042601/5f964e5c05aed62d6f1f2a11/html5/thumbnails/154.jpg)
And if the next game number is a 3;
Address of this game point is
32111111111……..
![Page 155: Fractals - SFU.carpyke/mathcamp/Fractals presentation.pdfAll fractals have an address system you can use to label parts of the fractal Let‟s begin the Sierpinski Chaos game at the](https://reader034.fdocuments.in/reader034/viewer/2022042601/5f964e5c05aed62d6f1f2a11/html5/thumbnails/155.jpg)
And if the next game number is a 3;
Address of this game point is
32111111111……..
![Page 156: Fractals - SFU.carpyke/mathcamp/Fractals presentation.pdfAll fractals have an address system you can use to label parts of the fractal Let‟s begin the Sierpinski Chaos game at the](https://reader034.fdocuments.in/reader034/viewer/2022042601/5f964e5c05aed62d6f1f2a11/html5/thumbnails/156.jpg)
Next game number is a 2;
![Page 157: Fractals - SFU.carpyke/mathcamp/Fractals presentation.pdfAll fractals have an address system you can use to label parts of the fractal Let‟s begin the Sierpinski Chaos game at the](https://reader034.fdocuments.in/reader034/viewer/2022042601/5f964e5c05aed62d6f1f2a11/html5/thumbnails/157.jpg)
Next game number is a 2;
Address of previous game point
is 3211111111…
![Page 158: Fractals - SFU.carpyke/mathcamp/Fractals presentation.pdfAll fractals have an address system you can use to label parts of the fractal Let‟s begin the Sierpinski Chaos game at the](https://reader034.fdocuments.in/reader034/viewer/2022042601/5f964e5c05aed62d6f1f2a11/html5/thumbnails/158.jpg)
Next game number is a 2;
Address of previous game point
is 3211111111…
Address of this game point is
![Page 159: Fractals - SFU.carpyke/mathcamp/Fractals presentation.pdfAll fractals have an address system you can use to label parts of the fractal Let‟s begin the Sierpinski Chaos game at the](https://reader034.fdocuments.in/reader034/viewer/2022042601/5f964e5c05aed62d6f1f2a11/html5/thumbnails/159.jpg)
Next game number is a 2;
Address of previous game point
is 3211111111…
Address of this game point is
23211111111…….
![Page 160: Fractals - SFU.carpyke/mathcamp/Fractals presentation.pdfAll fractals have an address system you can use to label parts of the fractal Let‟s begin the Sierpinski Chaos game at the](https://reader034.fdocuments.in/reader034/viewer/2022042601/5f964e5c05aed62d6f1f2a11/html5/thumbnails/160.jpg)
Next game number is a 2;
Address of previous game point
is 3211111111…
Address of this game point is
23211111111…….
![Page 161: Fractals - SFU.carpyke/mathcamp/Fractals presentation.pdfAll fractals have an address system you can use to label parts of the fractal Let‟s begin the Sierpinski Chaos game at the](https://reader034.fdocuments.in/reader034/viewer/2022042601/5f964e5c05aed62d6f1f2a11/html5/thumbnails/161.jpg)
Next game number is a 1;
![Page 162: Fractals - SFU.carpyke/mathcamp/Fractals presentation.pdfAll fractals have an address system you can use to label parts of the fractal Let‟s begin the Sierpinski Chaos game at the](https://reader034.fdocuments.in/reader034/viewer/2022042601/5f964e5c05aed62d6f1f2a11/html5/thumbnails/162.jpg)
Next game number is a 1;
Address of previous game point
is 23211111111…
Address of this game point is
![Page 163: Fractals - SFU.carpyke/mathcamp/Fractals presentation.pdfAll fractals have an address system you can use to label parts of the fractal Let‟s begin the Sierpinski Chaos game at the](https://reader034.fdocuments.in/reader034/viewer/2022042601/5f964e5c05aed62d6f1f2a11/html5/thumbnails/163.jpg)
Next game number is a 1;
Address of previous game point
is 23211111111…
Address of this game point is
123211111111…….
![Page 164: Fractals - SFU.carpyke/mathcamp/Fractals presentation.pdfAll fractals have an address system you can use to label parts of the fractal Let‟s begin the Sierpinski Chaos game at the](https://reader034.fdocuments.in/reader034/viewer/2022042601/5f964e5c05aed62d6f1f2a11/html5/thumbnails/164.jpg)
Next game number is a 1;
Address of previous game point
is 23211111111…
Address of this game point is
123211111111…….
![Page 165: Fractals - SFU.carpyke/mathcamp/Fractals presentation.pdfAll fractals have an address system you can use to label parts of the fractal Let‟s begin the Sierpinski Chaos game at the](https://reader034.fdocuments.in/reader034/viewer/2022042601/5f964e5c05aed62d6f1f2a11/html5/thumbnails/165.jpg)
So, if the game numbers are s1, s2, s3, …, sk,…,
the addresses of the game points are
game point 1: s1……..
game point 2: s2s1…….
game point 3: s3s2s1…..
⁞ ⁞ ⁞ ⁞ ⁞ ⁞ ⁞
game point k: sk…….s3s2s1…….
![Page 166: Fractals - SFU.carpyke/mathcamp/Fractals presentation.pdfAll fractals have an address system you can use to label parts of the fractal Let‟s begin the Sierpinski Chaos game at the](https://reader034.fdocuments.in/reader034/viewer/2022042601/5f964e5c05aed62d6f1f2a11/html5/thumbnails/166.jpg)
So, if the game numbers are s1, s2, s3, …, sk,…,
the addresses of the game points are
game point 1: s1……..
game point 2: s2s1…….
game point 3: s3s2s1…..
⁞ ⁞ ⁞ ⁞ ⁞ ⁞ ⁞
game point k: sk…….s3s2s1…….
Which means we can put a game point in every
address region of the Sierpinski triangle if the
game numbers produce every pattern of 1‟s, 2‟s,
and 3‟s.
![Page 167: Fractals - SFU.carpyke/mathcamp/Fractals presentation.pdfAll fractals have an address system you can use to label parts of the fractal Let‟s begin the Sierpinski Chaos game at the](https://reader034.fdocuments.in/reader034/viewer/2022042601/5f964e5c05aed62d6f1f2a11/html5/thumbnails/167.jpg)
Here‟s one sequence of game numbers that
produces every pattern of 1‟s, 2‟s, and 3‟s;
![Page 168: Fractals - SFU.carpyke/mathcamp/Fractals presentation.pdfAll fractals have an address system you can use to label parts of the fractal Let‟s begin the Sierpinski Chaos game at the](https://reader034.fdocuments.in/reader034/viewer/2022042601/5f964e5c05aed62d6f1f2a11/html5/thumbnails/168.jpg)
Here‟s one sequence of game numbers that
produces every pattern of 1‟s, 2‟s, and 3‟s;
1 2 3 - first, all patterns of length 1
![Page 169: Fractals - SFU.carpyke/mathcamp/Fractals presentation.pdfAll fractals have an address system you can use to label parts of the fractal Let‟s begin the Sierpinski Chaos game at the](https://reader034.fdocuments.in/reader034/viewer/2022042601/5f964e5c05aed62d6f1f2a11/html5/thumbnails/169.jpg)
Here‟s one sequence of game numbers that
produces every pattern of 1‟s, 2‟s, and 3‟s;
123
11 12 13 21 22 23 31 32 33
- then all patterns of length 2
![Page 170: Fractals - SFU.carpyke/mathcamp/Fractals presentation.pdfAll fractals have an address system you can use to label parts of the fractal Let‟s begin the Sierpinski Chaos game at the](https://reader034.fdocuments.in/reader034/viewer/2022042601/5f964e5c05aed62d6f1f2a11/html5/thumbnails/170.jpg)
Here‟s one sequence of game numbers that
produces every pattern of 1‟s, 2‟s, and 3‟s;
123111213212223313233
111 112 113 now all patterns of length 3
121 122 123
131 132 133
211 212 213
221 222 223
231 232 233
311 312 313
321 322 323 …..
![Page 171: Fractals - SFU.carpyke/mathcamp/Fractals presentation.pdfAll fractals have an address system you can use to label parts of the fractal Let‟s begin the Sierpinski Chaos game at the](https://reader034.fdocuments.in/reader034/viewer/2022042601/5f964e5c05aed62d6f1f2a11/html5/thumbnails/171.jpg)
12311121321222331323311111211312112
212313113213321121221322122222323…
This sequence will contain every pattern of
1‟s, 2‟, and 3‟s. So if we play the Sierpinski
chaos game with this game sequence, the
game points will cover all regions of the
fractal.
![Page 172: Fractals - SFU.carpyke/mathcamp/Fractals presentation.pdfAll fractals have an address system you can use to label parts of the fractal Let‟s begin the Sierpinski Chaos game at the](https://reader034.fdocuments.in/reader034/viewer/2022042601/5f964e5c05aed62d6f1f2a11/html5/thumbnails/172.jpg)
Another sequence that contains all patterns
is a random sequence
![Page 173: Fractals - SFU.carpyke/mathcamp/Fractals presentation.pdfAll fractals have an address system you can use to label parts of the fractal Let‟s begin the Sierpinski Chaos game at the](https://reader034.fdocuments.in/reader034/viewer/2022042601/5f964e5c05aed62d6f1f2a11/html5/thumbnails/173.jpg)
Another sequence that contains all patterns
is a random sequence
Choose each game number randomly, eg.,
roll a die;
![Page 174: Fractals - SFU.carpyke/mathcamp/Fractals presentation.pdfAll fractals have an address system you can use to label parts of the fractal Let‟s begin the Sierpinski Chaos game at the](https://reader034.fdocuments.in/reader034/viewer/2022042601/5f964e5c05aed62d6f1f2a11/html5/thumbnails/174.jpg)
Another sequence that contains all patterns
is a random sequence
Choose each game number randomly, eg.,
roll a die;
if a 1 or 2 comes up, game number is 1
![Page 175: Fractals - SFU.carpyke/mathcamp/Fractals presentation.pdfAll fractals have an address system you can use to label parts of the fractal Let‟s begin the Sierpinski Chaos game at the](https://reader034.fdocuments.in/reader034/viewer/2022042601/5f964e5c05aed62d6f1f2a11/html5/thumbnails/175.jpg)
Another sequence that contains all patterns
is a random sequence
Choose each game number randomly, eg.,
roll a die;
if a 1 or 2 comes up, game number is 1
if a 3 or 4 comes up, game number is 2
![Page 176: Fractals - SFU.carpyke/mathcamp/Fractals presentation.pdfAll fractals have an address system you can use to label parts of the fractal Let‟s begin the Sierpinski Chaos game at the](https://reader034.fdocuments.in/reader034/viewer/2022042601/5f964e5c05aed62d6f1f2a11/html5/thumbnails/176.jpg)
Another sequence that contains all patterns
is a random sequence
Choose each game number randomly, eg.,
roll a die;
if a 1 or 2 comes up, game number is 1
if a 3 or 4 comes up, game number is 2
if a 5 or 6 comes up, game number is 3
![Page 177: Fractals - SFU.carpyke/mathcamp/Fractals presentation.pdfAll fractals have an address system you can use to label parts of the fractal Let‟s begin the Sierpinski Chaos game at the](https://reader034.fdocuments.in/reader034/viewer/2022042601/5f964e5c05aed62d6f1f2a11/html5/thumbnails/177.jpg)
Another sequence that contains all patterns
is a random sequence
Choose each game number randomly, eg.,
roll a die;
if a 1 or 2 comes up, game number is 1
if a 3 or 4 comes up, game number is 2
if a 5 or 6 comes up, game number is 3
These game numbers will also draw the
fractal.
![Page 178: Fractals - SFU.carpyke/mathcamp/Fractals presentation.pdfAll fractals have an address system you can use to label parts of the fractal Let‟s begin the Sierpinski Chaos game at the](https://reader034.fdocuments.in/reader034/viewer/2022042601/5f964e5c05aed62d6f1f2a11/html5/thumbnails/178.jpg)
Adjusting probabilities
Suppose we randomly choose 1, 2, 3, but
we choose 3 2/3 of the time and 1 and 2
1/6 of the time each (roll a die: if 1 comes
up choose 1, if 2 comes up choose 2, if
3,4,5 or 6 come up choose 3)…..
![Page 179: Fractals - SFU.carpyke/mathcamp/Fractals presentation.pdfAll fractals have an address system you can use to label parts of the fractal Let‟s begin the Sierpinski Chaos game at the](https://reader034.fdocuments.in/reader034/viewer/2022042601/5f964e5c05aed62d6f1f2a11/html5/thumbnails/179.jpg)
![Page 180: Fractals - SFU.carpyke/mathcamp/Fractals presentation.pdfAll fractals have an address system you can use to label parts of the fractal Let‟s begin the Sierpinski Chaos game at the](https://reader034.fdocuments.in/reader034/viewer/2022042601/5f964e5c05aed62d6f1f2a11/html5/thumbnails/180.jpg)
We can calculate how many game points
land in a particular address region by
calculating how often that address (in
reverse) occurs in the game sequence.
(Homework!)
![Page 181: Fractals - SFU.carpyke/mathcamp/Fractals presentation.pdfAll fractals have an address system you can use to label parts of the fractal Let‟s begin the Sierpinski Chaos game at the](https://reader034.fdocuments.in/reader034/viewer/2022042601/5f964e5c05aed62d6f1f2a11/html5/thumbnails/181.jpg)
Are equal probabilities always the best?
![Page 182: Fractals - SFU.carpyke/mathcamp/Fractals presentation.pdfAll fractals have an address system you can use to label parts of the fractal Let‟s begin the Sierpinski Chaos game at the](https://reader034.fdocuments.in/reader034/viewer/2022042601/5f964e5c05aed62d6f1f2a11/html5/thumbnails/182.jpg)
Fern Fractal
Game rules; place 4 pins, choose 1,2,3,4
randomly. Actions; …….
If the numbers 1,2,3,4 are chosen equally
often;
![Page 183: Fractals - SFU.carpyke/mathcamp/Fractals presentation.pdfAll fractals have an address system you can use to label parts of the fractal Let‟s begin the Sierpinski Chaos game at the](https://reader034.fdocuments.in/reader034/viewer/2022042601/5f964e5c05aed62d6f1f2a11/html5/thumbnails/183.jpg)
![Page 184: Fractals - SFU.carpyke/mathcamp/Fractals presentation.pdfAll fractals have an address system you can use to label parts of the fractal Let‟s begin the Sierpinski Chaos game at the](https://reader034.fdocuments.in/reader034/viewer/2022042601/5f964e5c05aed62d6f1f2a11/html5/thumbnails/184.jpg)
Fern Fractal
Game rules; place pins, choose 1,2,3,4
randomly. Actions; …….
If the numbers 1,2,3,4 are chosen with
frequencies;
1; 2%
2,3; 14%
4; 70%
![Page 185: Fractals - SFU.carpyke/mathcamp/Fractals presentation.pdfAll fractals have an address system you can use to label parts of the fractal Let‟s begin the Sierpinski Chaos game at the](https://reader034.fdocuments.in/reader034/viewer/2022042601/5f964e5c05aed62d6f1f2a11/html5/thumbnails/185.jpg)
![Page 186: Fractals - SFU.carpyke/mathcamp/Fractals presentation.pdfAll fractals have an address system you can use to label parts of the fractal Let‟s begin the Sierpinski Chaos game at the](https://reader034.fdocuments.in/reader034/viewer/2022042601/5f964e5c05aed62d6f1f2a11/html5/thumbnails/186.jpg)
What probabilities to use?
Sometimes it is very difficult to determine the best probabilities to draw the fractal.
![Page 187: Fractals - SFU.carpyke/mathcamp/Fractals presentation.pdfAll fractals have an address system you can use to label parts of the fractal Let‟s begin the Sierpinski Chaos game at the](https://reader034.fdocuments.in/reader034/viewer/2022042601/5f964e5c05aed62d6f1f2a11/html5/thumbnails/187.jpg)
Another chaos game
![Page 188: Fractals - SFU.carpyke/mathcamp/Fractals presentation.pdfAll fractals have an address system you can use to label parts of the fractal Let‟s begin the Sierpinski Chaos game at the](https://reader034.fdocuments.in/reader034/viewer/2022042601/5f964e5c05aed62d6f1f2a11/html5/thumbnails/188.jpg)
![Page 189: Fractals - SFU.carpyke/mathcamp/Fractals presentation.pdfAll fractals have an address system you can use to label parts of the fractal Let‟s begin the Sierpinski Chaos game at the](https://reader034.fdocuments.in/reader034/viewer/2022042601/5f964e5c05aed62d6f1f2a11/html5/thumbnails/189.jpg)
What if action 4 was; “move ½ distance
towards pin 4” only (i.e., no rotation)……
![Page 190: Fractals - SFU.carpyke/mathcamp/Fractals presentation.pdfAll fractals have an address system you can use to label parts of the fractal Let‟s begin the Sierpinski Chaos game at the](https://reader034.fdocuments.in/reader034/viewer/2022042601/5f964e5c05aed62d6f1f2a11/html5/thumbnails/190.jpg)
![Page 191: Fractals - SFU.carpyke/mathcamp/Fractals presentation.pdfAll fractals have an address system you can use to label parts of the fractal Let‟s begin the Sierpinski Chaos game at the](https://reader034.fdocuments.in/reader034/viewer/2022042601/5f964e5c05aed62d6f1f2a11/html5/thumbnails/191.jpg)
Is this a fractal? (self-similar?)
![Page 192: Fractals - SFU.carpyke/mathcamp/Fractals presentation.pdfAll fractals have an address system you can use to label parts of the fractal Let‟s begin the Sierpinski Chaos game at the](https://reader034.fdocuments.in/reader034/viewer/2022042601/5f964e5c05aed62d6f1f2a11/html5/thumbnails/192.jpg)
Is this a fractal? (self-similar?)
Yes – 4 pieces, but
there is overlap
![Page 193: Fractals - SFU.carpyke/mathcamp/Fractals presentation.pdfAll fractals have an address system you can use to label parts of the fractal Let‟s begin the Sierpinski Chaos game at the](https://reader034.fdocuments.in/reader034/viewer/2022042601/5f964e5c05aed62d6f1f2a11/html5/thumbnails/193.jpg)
Is this a fractal? (self-similar?)
Yes – 4 pieces, but
there is overlap
![Page 194: Fractals - SFU.carpyke/mathcamp/Fractals presentation.pdfAll fractals have an address system you can use to label parts of the fractal Let‟s begin the Sierpinski Chaos game at the](https://reader034.fdocuments.in/reader034/viewer/2022042601/5f964e5c05aed62d6f1f2a11/html5/thumbnails/194.jpg)
Is this a fractal? (self-similar?)
Yes – 4 pieces, but
there is overlap
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Is this a fractal? (self-similar?)
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Pseudo Fractals
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Remove all 12‟s from game numbers…..
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Remove all 12‟s from game numbers…..
Remember, addresses of game points are the reverse of the game numbers.
So here no game points land in areas whose address contains a 21
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Remove all 12‟s from game numbers…..
Remember, addresses of game points are the reverse of the game numbers.
So here no game points land in areas whose address contains a 21
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Remove all 12‟s from game numbers…..
Remember, addresses of game points are the reverse of the game numbers.
So here no game points land in areas whose address contains a 21
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Note; this is not a fractal (is not self-similar)
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Back to adjusting probabilities
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Back to adjusting probabilities
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Back to adjusting probabilities
Equal probabilities
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Unequal probabilities;
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Unequal probabilities (again);
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Playing the chaos game with
non-random sequences
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Playing the chaos game with
non-random sequences
Sierpinski game. Initial
game point at bottom
left corner.
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Playing the chaos game with
non-random sequences
Sierpinski game. Initial
game point at bottom
left corner.
Game numbers;
123123123123123....
(i.e., 123 repeating).
![Page 217: Fractals - SFU.carpyke/mathcamp/Fractals presentation.pdfAll fractals have an address system you can use to label parts of the fractal Let‟s begin the Sierpinski Chaos game at the](https://reader034.fdocuments.in/reader034/viewer/2022042601/5f964e5c05aed62d6f1f2a11/html5/thumbnails/217.jpg)
Playing the chaos game with
non-random sequences
Sierpinski game. Initial
game point at bottom
left corner.
Game numbers;
123123123123123....
(i.e., 123 repeating).
Outcome?
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Looks like the game points are converging
to 3 distinct points;
![Page 250: Fractals - SFU.carpyke/mathcamp/Fractals presentation.pdfAll fractals have an address system you can use to label parts of the fractal Let‟s begin the Sierpinski Chaos game at the](https://reader034.fdocuments.in/reader034/viewer/2022042601/5f964e5c05aed62d6f1f2a11/html5/thumbnails/250.jpg)
Looks like the game points are converging
to 3 distinct points;
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Looks like the game points are converging
to 3 distinct points;
Why ?
![Page 252: Fractals - SFU.carpyke/mathcamp/Fractals presentation.pdfAll fractals have an address system you can use to label parts of the fractal Let‟s begin the Sierpinski Chaos game at the](https://reader034.fdocuments.in/reader034/viewer/2022042601/5f964e5c05aed62d6f1f2a11/html5/thumbnails/252.jpg)
Consider the 3 points with the following
addresses;
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Playing with game
sequence 123123....,
we see that
C → A → B → C etc.
(add game number to left
end of address of
game point)
![Page 254: Fractals - SFU.carpyke/mathcamp/Fractals presentation.pdfAll fractals have an address system you can use to label parts of the fractal Let‟s begin the Sierpinski Chaos game at the](https://reader034.fdocuments.in/reader034/viewer/2022042601/5f964e5c05aed62d6f1f2a11/html5/thumbnails/254.jpg)
Playing with game
sequence 123123....,
we see that
C → A → B → C etc.
(add game number to left
end of address of
game point)
That is, the game points
cycle through these 3
points in this order.
![Page 255: Fractals - SFU.carpyke/mathcamp/Fractals presentation.pdfAll fractals have an address system you can use to label parts of the fractal Let‟s begin the Sierpinski Chaos game at the](https://reader034.fdocuments.in/reader034/viewer/2022042601/5f964e5c05aed62d6f1f2a11/html5/thumbnails/255.jpg)
Playing with game
sequence 123123....,
we see that
C → A → B → C etc.
(add game number to left
end of address of
game point)
That is, the game points
cycle through these 3
points in this order.
No matter where the first
game point is, the
game points will end
up at A, B, and C.
![Page 256: Fractals - SFU.carpyke/mathcamp/Fractals presentation.pdfAll fractals have an address system you can use to label parts of the fractal Let‟s begin the Sierpinski Chaos game at the](https://reader034.fdocuments.in/reader034/viewer/2022042601/5f964e5c05aed62d6f1f2a11/html5/thumbnails/256.jpg)
Random Fractals
Random midpoints to
define triangle decomposition
Remove random triangle
at each iteration
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Random fractal curves
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Using fractals to create real life
images
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Fractal Clouds
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Fractal Clouds
Decide the self-similar pieces
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Fractal Clouds
Decide the self-similar pieces Generate the fractal
![Page 262: Fractals - SFU.carpyke/mathcamp/Fractals presentation.pdfAll fractals have an address system you can use to label parts of the fractal Let‟s begin the Sierpinski Chaos game at the](https://reader034.fdocuments.in/reader034/viewer/2022042601/5f964e5c05aed62d6f1f2a11/html5/thumbnails/262.jpg)
now blur a bit
Create a realistic cloud!
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Real Mountains
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Fractal Mountains
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Fractals in Nature
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Fractal Art
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Julia setsG. Julia, P. Fatou ca 1920
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For more information:
• http://www.sfu.ca/~rpyke/ “Fractals”
• Email: [email protected]
This presentation: www.sfu.ca/~rpyke/mathcamp