Some affine maps have certain points that when plugged in, give themselves. Causing an infinite loop. Take FCF for example:
F1F( -1/2x+1/2, 1/2y).......F1F( 1/3, 0) ---> ( 1/3, 0)
F2C( -1/2x+1, -1/2+1/2).....F2C(2/3, 1/3)--->(2/3, 1/3)
F3F( -1/2x+1/2, 1/2y+1/2)...F3F(1/3, 1) --->(1/3, 1)
It is interesting to note that the repeating points in F1 and F3 are the end points of the straight line between the two sections. It is also interesting to note that if you took the point (1/3, 1/4) and put it through F1F it would approach the point (1/3 , 0) since 1/3 would stay 1/3 and 1/4 would become 1/8 which would become 1/16 getting infinitely smaller. These point seem something like the attractors I read about earlier. They help when trying to prove if two sections are connected.
Wednesday, June 17, 2009
Sierpinski Relatives: Straight Lines
Using the relative FCF as an example I was shown a simple way to prove there was a vertical line through S1 (bottom right) and S3 (top right). In order for this to work the x-components had to match, and the y values had to go from [0,1/2] for F1 and [1/2,1] for F3.
F1F(-1/2x+1/2, 1/2y) F3(-1/2x+1/2, 1/2y +1/2)
As you can see these two maps satisfy both properties. Since F2 was not used in this proof at all, an relative with F as the map for F1 and F3 will have a straight line through S1 and S3.
We can use almost the same technique to prove there is a horizontal line through S1 and S2 (bottom left). Instead the Y components must match and the X values must got from [0,1/2] for F1 and [1/2,1] for F2. The relative with the form EE_ will have a horizontal line through S1 and S2.
F1E(1/2x, -1/2y+1/2) F2E(1/2x+1/2, -1/2y+1/2)
-1/2y+1/2 = -1/2y+1/2 The y values match
1/2x=x ... x=0
1/2x+1/2=x ... x=1, x values go from [0,1]
F1F(-1/2x+1/2, 1/2y) F3(-1/2x+1/2, 1/2y +1/2)
As you can see these two maps satisfy both properties. Since F2 was not used in this proof at all, an relative with F as the map for F1 and F3 will have a straight line through S1 and S3.
We can use almost the same technique to prove there is a horizontal line through S1 and S2 (bottom left). Instead the Y components must match and the X values must got from [0,1/2] for F1 and [1/2,1] for F2. The relative with the form EE_ will have a horizontal line through S1 and S2.
F1E(1/2x, -1/2y+1/2) F2E(1/2x+1/2, -1/2y+1/2)
-1/2y+1/2 = -1/2y+1/2 The y values match
1/2x=x ... x=0
1/2x+1/2=x ... x=1, x values go from [0,1]
Tuesday, May 12, 2009
The Chaos Game
The Chaos Game can be played here. http://math.bu.edu/DYSYS/applets/chaos-game.html
I was asked to find an algorithm to solve the game. This post shows that with one initial choice of your original 3 moves, solving the game can be very easy be repeating a simple process until solved. I don't understand why the first 3 moves are what they are I just found the pattern based on observation.
How to beat the Chaos Game on Master
• First find your target (it can be hard on master)
• Focus looking at it on it’s first level
•Moves 1,2 and 3: The target is on the right triangle, so we move Blue, Red, Green(Note: If the target was on the top triangle we would have moved Blue, Red, Red.
If the target was on the left triangle we would have moved Red, Blue, Blue.)
This is the only unique step, all the other steps now follow a pattern.
• Now we zoom out our focus and look at the next level
• Move 4: As you can see the target is now in the top triangle in our new focus area. Making our next move to be Red. The next steps are just a repetition of this zooming out and seeing what triangle our target is in.
• Move 5: In this new focus our target is in the right triangle so our next move is Green.
• Move 6: Again in this focus we the target in the right triangle so we move Green.
• Move 7: Now in our final step we focus on the entire object. Our target is in the top triangle so our final step is to move Red.
This method can be used to solve The Chaos game on any level!!!
I was asked to find an algorithm to solve the game. This post shows that with one initial choice of your original 3 moves, solving the game can be very easy be repeating a simple process until solved. I don't understand why the first 3 moves are what they are I just found the pattern based on observation.
How to beat the Chaos Game on Master
• First find your target (it can be hard on master)
• Focus looking at it on it’s first level
•Moves 1,2 and 3: The target is on the right triangle, so we move Blue, Red, Green(Note: If the target was on the top triangle we would have moved Blue, Red, Red.
If the target was on the left triangle we would have moved Red, Blue, Blue.)
This is the only unique step, all the other steps now follow a pattern.
• Now we zoom out our focus and look at the next level
• Move 4: As you can see the target is now in the top triangle in our new focus area. Making our next move to be Red. The next steps are just a repetition of this zooming out and seeing what triangle our target is in.
• Move 5: In this new focus our target is in the right triangle so our next move is Green.
• Move 6: Again in this focus we the target in the right triangle so we move Green.
• Move 7: Now in our final step we focus on the entire object. Our target is in the top triangle so our final step is to move Red.
This method can be used to solve The Chaos game on any level!!!
Friday, May 8, 2009
Chaos: Making A New Science cont'd....
Chapter 7: The Experimenter
D'Arcy Thompson's thoughts on physical cause in shaping nature is interesting. Neat to see Libchaber discover a real life example proving Feigenbaum's universality constant. Are there other more simple experiments that prove this now?
Chapter 8: Images of Chaos
Why The Mandlebrot set, why does Z-> Z^2 + C produce the most complex object we've ever seen? What other effects can iterating a function have? What has the Mandlebrot set done for science so far (besides confuse us)? The chaos game and collage theorem seem very simple but quite interesting, I should try it out some time.
Chapter 9: The Dynamical Systems Collective
Lyapunov exponents and more bending/folding phase space confused me some. Dripping faucet example clearly shows how chaos is everywhere and can be simple to understand.
Chapter 10: Inner Rhythms
Huberman's vision test on schizophrenia is interesting because in The Colour of Infinity they mention how they believe our vision is somehow fractal related. Biological clocks are also quite an interesting topic.
Chapter 11: Chaos and Beyond
Interesting to talk about as something as simple as a snow flake to be so very complex. This chapter only this with the entire book shows how there is a new science out there and it applies to all feilds.
D'Arcy Thompson's thoughts on physical cause in shaping nature is interesting. Neat to see Libchaber discover a real life example proving Feigenbaum's universality constant. Are there other more simple experiments that prove this now?
Chapter 8: Images of Chaos
Why The Mandlebrot set, why does Z-> Z^2 + C produce the most complex object we've ever seen? What other effects can iterating a function have? What has the Mandlebrot set done for science so far (besides confuse us)? The chaos game and collage theorem seem very simple but quite interesting, I should try it out some time.
Chapter 9: The Dynamical Systems Collective
Lyapunov exponents and more bending/folding phase space confused me some. Dripping faucet example clearly shows how chaos is everywhere and can be simple to understand.
Chapter 10: Inner Rhythms
Huberman's vision test on schizophrenia is interesting because in The Colour of Infinity they mention how they believe our vision is somehow fractal related. Biological clocks are also quite an interesting topic.
Chapter 11: Chaos and Beyond
Interesting to talk about as something as simple as a snow flake to be so very complex. This chapter only this with the entire book shows how there is a new science out there and it applies to all feilds.
Thursday, May 7, 2009
Chaos: Making A New Science by James Gleick
After seeing Ashton Kutcher's film 'The Butterfly Effect' all scientific credibility Chaos Theory had in my mind went out the window. After reading the first chapter of Gleick's book also titled 'The Butterfly Effect' my mind was completely changed.
Chapter 1: The Butterfly Effect
How does such a small error of 0.000127 change the weather forecast completely? What 12 mathematical equations can obscure results so much based on so little change? Since our initial conditions are measurements with error themselves is that way long rang forecasting is almost always wrong. What if our initial conditions had no error in them? What are the bounds of the Lorenz Attractor and why are they the bounds?
Chapter 2:Revolution
Hard to believe we are still learning new things from something as simple as a pendulum. I don't really understand what Smale is trying to show with his stretching/folding of shapes, i.e. Smale's Horseshoe. Quite interesting that one of the first big questions Chaos solved was over 400 years old and about another planet (Jupiter's Giant Red Spot).
Chapter 3: Life's Ups and Downs
I agree with Yorke's notion that we learned not to see chaos. Why does a growth rate of 3 send the population into a chaotic behavior. What happens when the growth rate is a special number such as Pi?
Chapter 4: A Geometry of Nature
I am very interested in learning more about fractals and their prediction of the economy. Fractals are so simple is very hard to believe humanity has overstepped them for this long. This chapter and the previously linked videos have left me with too many questions. The Mandelbrot set is unbelievable, literally. It seems like something out of a movie. I'm glad my area of research is focused around fractals for it's by far the most interesting topic in this book.
Chapter 5: Strange Attractors
This chapter was a little abstract so I tried to get a more solid definition first, http://en.wikipedia.org/wiki/Strange_attractor#Strange_attractor, I understand the idea of a strange attractor (the Lorenz example helps) but it is still foggy and I should read more about phase space. I feel the generally idea of the chapter was to show that dynamic systems follow some sort of "strange attractor" and when the phase space of this system is plotted it turns out to be some geometrical pattern (ex butterfly-ish wings for the lorenz attractor).
Chapter 6: Universality
Goethe thoughts on colour are very interesting. When Feigenbaum talks about the picnickers becoming incomprehensible as distance grew between them, does this have anything to do with why the Sierpinski Gasket can start with any image and produce the same result? That is the further you iterate the less comprehensible the original image. How does Feigenbaum's universality constant work? Why is the doubling period always 4.6692016090 for any bifurcating function? What are the implications of this?
Chapter 1: The Butterfly Effect
How does such a small error of 0.000127 change the weather forecast completely? What 12 mathematical equations can obscure results so much based on so little change? Since our initial conditions are measurements with error themselves is that way long rang forecasting is almost always wrong. What if our initial conditions had no error in them? What are the bounds of the Lorenz Attractor and why are they the bounds?
Chapter 2:Revolution
Hard to believe we are still learning new things from something as simple as a pendulum. I don't really understand what Smale is trying to show with his stretching/folding of shapes, i.e. Smale's Horseshoe. Quite interesting that one of the first big questions Chaos solved was over 400 years old and about another planet (Jupiter's Giant Red Spot).
Chapter 3: Life's Ups and Downs
I agree with Yorke's notion that we learned not to see chaos. Why does a growth rate of 3 send the population into a chaotic behavior. What happens when the growth rate is a special number such as Pi?
Chapter 4: A Geometry of Nature
I am very interested in learning more about fractals and their prediction of the economy. Fractals are so simple is very hard to believe humanity has overstepped them for this long. This chapter and the previously linked videos have left me with too many questions. The Mandelbrot set is unbelievable, literally. It seems like something out of a movie. I'm glad my area of research is focused around fractals for it's by far the most interesting topic in this book.
Chapter 5: Strange Attractors
This chapter was a little abstract so I tried to get a more solid definition first, http://en.wikipedia.org/wiki/Strange_attractor#Strange_attractor, I understand the idea of a strange attractor (the Lorenz example helps) but it is still foggy and I should read more about phase space. I feel the generally idea of the chapter was to show that dynamic systems follow some sort of "strange attractor" and when the phase space of this system is plotted it turns out to be some geometrical pattern (ex butterfly-ish wings for the lorenz attractor).
Chapter 6: Universality
Goethe thoughts on colour are very interesting. When Feigenbaum talks about the picnickers becoming incomprehensible as distance grew between them, does this have anything to do with why the Sierpinski Gasket can start with any image and produce the same result? That is the further you iterate the less comprehensible the original image. How does Feigenbaum's universality constant work? Why is the doubling period always 4.6692016090 for any bifurcating function? What are the implications of this?
Tuesday, May 5, 2009
Intro Videos
Came across some videos on youtube that are a great introduction to fractals
Author Clarke's - The Colors of Infinity
http://www.youtube.com/watch?v=qB8m85p7GsU&feature=PlayList&p=9F915F8683A18D16&index=0&playnext=1
Nova's - Hunting the Hidden Dimension (will probably be taken down by youtube soon)
http://www.youtube.com/watch?v=E812h_xehM0
Author Clarke's - The Colors of Infinity
http://www.youtube.com/watch?v=qB8m85p7GsU&feature=PlayList&p=9F915F8683A18D16&index=0&playnext=1
Nova's - Hunting the Hidden Dimension (will probably be taken down by youtube soon)
http://www.youtube.com/watch?v=E812h_xehM0
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