Chaos Game in a Square

In this short, we show what happens when iterating the procedure of choosing a square vertex or edge midpoint at random and moving two thirds the distance from the current dot to the chosen point.

#chaos #chaosgame #square #sierpinski #sierpinskicarpet #mathvideo​ #math​ #mtbos​ #manim​ #animation​ #theorem​​​ #iteachmath #mathematics #dynamicalsystems #iteratedfunctionsystem #dynamics #fractals

If you want to know more about the Chaos game, see the following links:

To learn more about animating with manim, check out:

[Marco_Viva]

amazingly, the points start to become colored

[quay]

What if we do this in 3D? Will it produce Menger Sponge?

[phillyphakename]

My mind: "oh, it's a serpinski triangle, but the one with the square instead of the triangle!"

Video: "Serpinski's carpet"

Me: "I like my name better"

[jasmint]

I love this kind of free-form exploratory math.

[stephanelem]

That carpet was introduced to me with a big square without smaller square into, to be applied recursively. An uniform figure function.
I'm amazed to discover the way you build it by using a point.

[allanatal]

“Starting with a point at anywhere inside the square”. What if this point started at the very center of the square?

[briancase]

That's absolutely beautiful. The real world is incredible.

[robertpaton]

you can do this with any n sided figure. for 3 sides you get the Sierpinski triangle and the new points have to be 1/2 way along. I forget the formula but I worked it out manually in a program which creates these sierpinski fractals for any number of side - you just have to adjust how far along the new point has to be. They look great for 5 6 7 ...

[IAA]

When chaos isn't actually chaotic.

[bluematter]

oh come on, thats amazing,
i wonder what happens if you change the parameters
(instead of 2/3 use something else,
use some other shape, like a hexagon or a triangle, what about irregular shapes?)
thanks for showing this,
im off to do some coding

[hakanazak]

who would've thought chaos game would be so predictible

[mrbrown]

Those first 3 moves didn't look so 'random',
but that's just the randomness of random....

[verity]

Kudos to the guy who did this manually 30k times.

[IdealIdeas]

What if you start the dot where there would have been an empty space?

[robdom]

Look, if you want a new carpet just say so.

[colinjava]

What happens if you change the factor, I might have to code this to see what happens.

[Joerg]

Mh, cool idea! Engineering me had to reproduce for scientific reason on Excel, it works quite nicely :)

[Psylar]

Does the serpinski carpet match the serpinski drapes?

[egochka]

“Chaos, chaos? No, order, order!”
-Tasque manager

[amareyez]

the way I was expecting actual chaotic behavior