Wouldn't this be a fractal without chaos? If you had 2 different starting points and plotted them moving 2/3s to the same random vertex (that is the starting points are different but the sequence of vertexes are the same) wouldn't you end up with a similar pattern?
The chaos is that you flip a random coin each time and move to the vertex. So running this two different times will result in completely different final locations for the dot. However, it is part of the chaos game that if you track all of the points visited, the final image will look exactly the same as this. Pretty cool. You are right that initial condition doesn't matter and the same imagery will show up. So the chaos is that there is no way to predict where the final point will be.
Yes! There are lots to play with. My channel has a variety of related constructions. You can also search chaos game and find examples of Sierpinski carpet and gasket. I’ll do others from time to time.
This is called the chaos game. Yes there is an algorithm but the algorithm depends on a completely random process. It is amazing that starting anywhere in the hexagon creates the same limiting set. That is truly amazing. So mathematical chaos studies these types of dynamical systems (in this case an iterated function system) that are sensitive to initial conditions. If you change the initial condition (or even if you don’t) the process of completing 30000 iterations creates a completely different collection of dots, but somehow that collection looks almost the same (and in fact the process will always produce the same set after enough iterations). So I don’t know what to tell you - I do have a decent understanding of mathematical chaos and what is studied in that field.
@@MathVisualProofs why this pattern emerges can be explained by the given conditions or limitations set by the rule of following half way to either of 5 corners. At each new step the last line can be explain by describing which corner must have been chosen.
These are really cool patterns that seem to appear out of random chaos.
You can play with other ratios too. :)
@@MathVisualProofs I bet!
Technically the universe, the planets, and by extension all of us are cool patterns that seem to appeared out of random chaos. Pretty neat, huh?
"Rule(s) per say is not Random"
Chaos disappears the moment you apply Rules. Order emerges from the basic instruction of the game.
But the selection of vertex to move to is random.
It should be “per se” not “per say”
Could you do an MVP of Morley's trisector theorem? That seems like it was made for your format.
Is a good idea! I will see what I can do. :)
i did this in excel, after watching a guy on tik tok talk about it. Very satisfying
Cool!
Wow what are the chances, I'm showing the Chaos Game to my Linear Algebra class tomorrow. I'm going to show them this video! :)
Awesome! I have some other ones I've been working that are even more linear-algebra-ey (creating k-reptiles) but they aren't ready to go yet.
Pretty cool
Thanks!
great video
Thank you :) Still have to work on my audio editing/mixing.
Wouldn't this be a fractal without chaos? If you had 2 different starting points and plotted them moving 2/3s to the same random vertex (that is the starting points are different but the sequence of vertexes are the same) wouldn't you end up with a similar pattern?
The chaos is that you flip a random coin each time and move to the vertex. So running this two different times will result in completely different final locations for the dot. However, it is part of the chaos game that if you track all of the points visited, the final image will look exactly the same as this. Pretty cool. You are right that initial condition doesn't matter and the same imagery will show up. So the chaos is that there is no way to predict where the final point will be.
Can we do this for other shapes? I'd love to see it
Yes! There are lots to play with. My channel has a variety of related constructions. You can also search chaos game and find examples of Sierpinski carpet and gasket. I’ll do others from time to time.
Trying this with a triangle results in a Sierpinski triangle
@@MrCubFan415 yes! I’ve done it for that one once. Got other shapes coming too. :)
I don’t got time for all those dots
i dub thee, the koch hexagon
:)
We’ve fractalized the Koch snowflake!
1:21 Amazing! How could it be?
Always interesting when structure appears from randomness right?
@@MathVisualProofs Yap. Randomness is awesome. That's why I has been attracted by chaos theory.
What? How? I must know!!!!!
200th like
:)
Making snowflakes has never been harder
Hah! True :)
Using an algorithm or set of rules is not fing chaos 🤦 it's very pre ordained by the very limited boundaries set
But who gives a rat's ass for bring correct. You say stuff that makes you sound clever to idiots?
But not to critical thinkers
You do not grasp chaos
This is called the chaos game. Yes there is an algorithm but the algorithm depends on a completely random process. It is amazing that starting anywhere in the hexagon creates the same limiting set. That is truly amazing. So mathematical chaos studies these types of dynamical systems (in this case an iterated function system) that are sensitive to initial conditions. If you change the initial condition (or even if you don’t) the process of completing 30000 iterations creates a completely different collection of dots, but somehow that collection looks almost the same (and in fact the process will always produce the same set after enough iterations). So I don’t know what to tell you -
I do have a decent understanding of mathematical chaos and what is studied in that field.
@@MathVisualProofs it is amazing. And it may be an "example" of chaos. But it's not chaos
@@MathVisualProofs why this pattern emerges can be explained by the given conditions or limitations set by the rule of following half way to either of 5 corners. At each new step the last line can be explain by describing which corner must have been chosen.