Fractals in coprime integers

Discussion in 'Math' started by Gerasimov, Sep 18, 2013.

  1. Gerasimov

    Thread Starter New Member

    Sep 18, 2013
    3
    0
    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:

    [​IMG]

    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:

    [​IMG]

    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?
    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:
    [​IMG]

    [​IMG]

    [​IMG]

    987х610 (pressed 5 times):
    [​IMG]

    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:
    Perhaps the chaos - it is also a fractal?
     
  2. Gerasimov

    Thread Starter New Member

    Sep 18, 2013
    3
    0
    GIF:
    [​IMG]

    [​IMG]

    Сlickable (enclosed area of fractals - "brain slugs"):
    [​IMG]

    and this :)
    [​IMG]
     
  3. Gerasimov

    Thread Starter New Member

    Sep 18, 2013
    3
    0
  4. studiot

    AAC Fanatic!

    Nov 9, 2007
    5,005
    513
    Fractals are fun.
     
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