Day 28
I first came across the "chaos game" in a tiny book on fractals I read many years ago. I had coded it up at that time, but forgot about it subsequently. Later, in one of my Numberphile video watching binges, I came across Brady Haran talking to Ben Sparks about it.
So, what is it, and how does it work?
Well, in this version of the "chaos game", you start with an equilateral triangle (although Sparks says any old triangle will do). Label the vertices 1, 2, and 3. Pick any point within this triangle as the initial point. Get a three sided coin from somewhere (with options 1, 2, and 3, corresponding to the three vertices of the triangle). Toss the coin to choose a vertex. Plot the next point halfway between the current point and the chosen vertex. Repeat many many many times. And... drum roll... you get the Sierpinski triangle!
As Ben explains in the Numberphile video, this structure (i.e. the Sierpinski triangle) is the attractor of the given system, which is defined by the evolution I just described above.
By the way, do check out Ben Sparks' rendition of the chaos game here.
I have a question for future me or anyone else who is reading this: is this system really chaotic? This is clearly not a deterministic system, because of the coin toss step. Even Ben says that this is random behaviour producing something structured, So, is the term "chaos game" a misnomer?
Anyway, here is my notebook.
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Uploads remaining: 72
Days remaining: 42
Ideal rate of uploads: 1 upload per 0.58 days. Or, 1.71 uploads/day.
I first came across the "chaos game" in a tiny book on fractals I read many years ago. I had coded it up at that time, but forgot about it subsequently. Later, in one of my Numberphile video watching binges, I came across Brady Haran talking to Ben Sparks about it.
So, what is it, and how does it work?
Well, in this version of the "chaos game", you start with an equilateral triangle (although Sparks says any old triangle will do). Label the vertices 1, 2, and 3. Pick any point within this triangle as the initial point. Get a three sided coin from somewhere (with options 1, 2, and 3, corresponding to the three vertices of the triangle). Toss the coin to choose a vertex. Plot the next point halfway between the current point and the chosen vertex. Repeat many many many times. And... drum roll... you get the Sierpinski triangle!
As Ben explains in the Numberphile video, this structure (i.e. the Sierpinski triangle) is the attractor of the given system, which is defined by the evolution I just described above.
By the way, do check out Ben Sparks' rendition of the chaos game here.
I have a question for future me or anyone else who is reading this: is this system really chaotic? This is clearly not a deterministic system, because of the coin toss step. Even Ben says that this is random behaviour producing something structured, So, is the term "chaos game" a misnomer?
Anyway, here is my notebook.
---
Uploads remaining: 72
Days remaining: 42
Ideal rate of uploads: 1 upload per 0.58 days. Or, 1.71 uploads/day.
This changes eveeyrhjng. Is the universe working on predetermined equations and we all just for a design? Coz this fractal pattern, Fibonacci sequence, pingala sequence... All have patterns in nature. I am questioning my free will now. Or am I programmed to be kinda crazy in life?
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