Greetings, humans! I'm from Ukraine. My English is very bad. So I will use a Google Translate.
In 2002, I came up with an interesting piece. I was only 14 years old. I was thinking about fractals and chaos theory, and did not want to learn. Did not want to learn, and were forced to walk to school. I came up with a way to kill time on the boring lessons in history, geography, and other humanities. I'll try to explain in detail. All we need - a piece of paper into a cell and an ordinary pencil. If a piece of paper is not available and also no pencil - an online version of JavaScript: http://xcont.com/pattern.html

The algorithm is simple to indecency. Actually these things look like this:

Select the rectangle and let in the corner of the "quantum beam" (as I called it in 2002) The beam reflected from the walls and is lost at the other corner.

If certain conditions were met - it turns fractal pattern.
If the conditions are not met (for example the obvious - the rectangle are equal) - pattern is not obtained. Of the less obvious example - the same pattern as it is impossible, if the size of the parties have a common divisor. In fact, the patterns are obtained only if the dimensions of both sides - are relatively prime (have no common divisor).

Сlickable:



In the picture all the numbers from 1 to 30.

And now a little about Fibonacci and fractals.

Patterns represent a fractal.

What determines the pattern?

So you need to make a difference, too, was a prime number (the largest), its difference with the numbers, too, was simple and small, and so on, then there will be something interesting.

That suggests to us - what if you try to Fibonacci numbers? "Пацан сказал - пацан сделал" (do not know how to translate it to English.


Painted the largest enclosed area.

233х144:

987х610 (pressed 5 times):

233х144 и 987х610 - identical

As you can see, fractal repeats part of the overall fractal increased

Fractals, as they are.

Site: New kind of fractals - Fractals in relatively prime integers (coprime integers)

P.S.

And a little bit schizophrenic 11-year-old:

Then, thinking about the relations of chaos and order where chaos is taken in order, and in order - chaos. So that was then thought that when everything exploded (big bang, which I firmly believe), was a ray of electromagnetic energy that is in the early running in a small space (which is further expanded.) Since electromagnetic waves can be in the form of photons - this beam is uninterrupted. Where we see the intersection of electromagnetic waves - there appears the "matter" (https://en.wikipedia.org/wiki/Pair_production) in the form of fractal patterns. Thus is born the order out of chaos.

(2) there is no space in the quantum distance - so there is no common divisor. Always work pattern (what we call matter).
(1) The universe is expanding continuously and smoothly. A fractal is continuous (and also smoothly) goes from one to the other - what we call the motion of matter.
(3). The universe is not two-dimensional and three-dimensional (and from the point of view of the General Theory of Relativity - four-dimensional).
So fractals are much more difficult.

Perhaps the chaos - it is also a fractal?

 

GIF:

Сlickable (enclosed area of fractals - "brain slugs"):


and this

 

 

Fractals are fun.