Fractals Everywhere? Our Fractal Universe Ep. 2

[gentle music] Fractals were formalized by mathematician Benoit Mandelbrot just in 1975, so they're not that old.

The word comes from the Latin fractus, meaning broken or fragmented, but the ideas do go back further.

In 1904, Helge von Koch created the Koch snowflake.

You start with a triangle, add smaller triangles to each side, repeat forever, and you get an infinite perimeter in a finite space, and the pattern looks the same at every zoom.

So that was in 1904.

In 1915, you have the Sierpinski triangle.

That was described by Waclaw Sierpinski.

You remove the middle triangle repeatedly, and you get a self-similar pattern at every scale.

And then we get to the most famous.

That's the Mandelbrot set, discovered in 1980, and it comes from one simple iterative equation, and that's Z, N squared plus C.

If the result stays bounded, i.e., it doesn't escape to infinity, then it's in the set.

And so the colors that you see in the Mandelbrot animation, they show the escape speed.

Zoom forever, and new structures keep emerging.

It's infinite complexity from simple rules.

And here's a key point of fractals is fractals have fractional dimensions, so more than a line, one-dimensional, but less than a plane, two-dimensional.

So a coastline might be 1.26-dimensional, so it means it's efficiently filling the space through repetition.

Nature uses this everywhere for optimization, and here's what blew my mind when I started looking into this was that so do we.