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Number Theory Fractals in relatively prime integers

Gerasimov

New member
Sep 17, 2013
2
Greetings, humans! (Tongueout) 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: New kind of fractals - Fractals in relatively prime integers (coprime integers).

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. (Crying)

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

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?
 

Opalg

MHB Oldtimer
Staff member
Feb 7, 2012
2,708
Beautiful idea, intriguing pictures. This looks like a fascinating topic (and much more educational for a 14-year-old than boring school lessons). Thank you for sharing it with us.

Everyone should click on the Javascript link in Gerasimov's post, to explore the patterns he creates there.
 
Last edited:

topsquark

Well-known member
MHB Math Helper
Aug 30, 2012
1,123
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.
The fractals are beautiful but what does this quote mean? For instance in (1) "the universe is expanding continuously and smoothly." This makes no sense to me. Can you explain it?

-Dan
 
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Bacterius

Well-known member
MHB Math Helper
Jan 26, 2012
644
Very nice! I wish I was thinking of fractals when I was 14 years old, haha.. much more interesting subject than stupid "school arithmetic" as I have come to call those pointless homework sheets that seemingly have infinitely many sides.

The patterns are very interesting, and indeed this is an application of chaos theory. I wonder if there is a fast local algorithm to compute large patterns efficiently without following a beam through the rectangle until it gets back to where it starts (kind of like how you are able to calculate individual digits of $\pi$ without having to calculate every previous one).

Have you investigated what the effect of changing the beam's starting point is?

This type of fractal could have applications in maze generation.. I am impressed. :)
 

Gerasimov

New member
Sep 17, 2013
2
Have you investigated what the effect of changing the beam's starting point is?
same fractals

What if, instead of the beam (black, white, black, white ...) to use the beam that smoothly changes intensity?