5
$\begingroup$

I am a programmer by trade, and am very interested in fractals.

To be very basic about the concept, one might say a 'circle of circles' is a fractal. Where each circle is made up of circles, and those circles are made of smaller circles and so on.

This idea is similar to a Sierpinski triangle, which I interpret basically a triangle of triangles.

To test my understanding, I decided to draw out this notion of a 'circle of circles'. Realize, I did this by hand with MS paint, copy and paste. I really should have just written out a simple program using some derivative of pi to get a perfect, mathematical display.

In any case, this image is just to illustrate the concept.

enter image description here

It's obviously not perfect, my question is: Is the notion of a 'circle of circles', 'triangle of triangles', or 'function of functions' truly the definition of a fractal? And, consequently, is the application of this notion, in this image, a fractal? (I realize it isn't because the alignment is incorrect.)

Note: The coloring was just so I could see the pattern easier, lets assume there is no color. Though if the color had been also repeating, would this still prove true?

I included another, expanding on the use of color:

enter image description here

$\endgroup$
8
  • $\begingroup$ I quite like the picture you've generated :P $\endgroup$ Commented Apr 8, 2016 at 5:33
  • $\begingroup$ The inside of the entire circle should technically be purple ;) $\endgroup$ Commented Apr 8, 2016 at 6:02
  • $\begingroup$ Or one of the colors actually, depending on which part of the sequence it is zoomed to. $\endgroup$ Commented Apr 8, 2016 at 6:05
  • $\begingroup$ Haha agreed, if you're happy with my answer, could you please consider accepting it as the correct answer :) $\endgroup$ Commented Apr 8, 2016 at 6:40
  • $\begingroup$ Your set is the attractor of an iterated function system, close to one that generates a 9 fold polygasket. Have a look at this poly-gasket visualization. Set the number of sides to 9 and the scaling ratio to 0.25. You will generate an image very close to yours. $\endgroup$ Commented Apr 8, 2016 at 12:02

1 Answer 1

1
$\begingroup$

A fractal is just a simple rule that you repeat over and over again. You'd be right to say that it is recursive sometimes, but it isn't always generated recursively, indeed many fractals would be impossible to generate with a recursive "seed" so to speak.

Mandelbrot Set

For example, here is the mandelbrot set:

Mandelbrot Set

You can generate this picture by making a grid on your computer to represent the complex plane, and assigning a single complex number to each grid cell (for example, the origin would be 0+0i and the right-most point would be 1+0i).

After this, you can just run this simple mathematical function say, 50 times (arbitrary choice):

$$z_{n+1} = z_n^2 + c $$

Where c is a constant of your choosing. If after these 50 iterations, the number you've generated is greater than 2; you've "diverged", so you color that pixel white. Else, you've converged, and you color that pixel white. *you'd want $|c| < 1$ for that reason.

You can generated a colored mandelbrot set by coloring the pixel based on the magnitude after 50 iterations.

Your Fractal

Your fractal is certainly a fractal if you continue to iterate it. In fact, to make this fractal, you'd just have a function that you will repeat over and over again, however; the manner in which you generate the fractal may be different. I'd therefore say that a good definition for a fractal is just an object that displays self similarity at different scales, or an object that is produced by repeated applications of a simple rule (which is exactly what you've done).

This extends fractals from the realm of graphics and the physical world into other areas where they may someday find an application :)

Hope that helps.

$\endgroup$
11
  • $\begingroup$ I thought to be a fractal, the pattern had to the same at every scale. Does the mandelbrot do that? If not, what is the difference? $\endgroup$ Commented Apr 8, 2016 at 5:09
  • $\begingroup$ Yes the mandelbrot set definitely does do that :P check this out $\endgroup$ Commented Apr 8, 2016 at 5:12
  • $\begingroup$ Oh man, that's really trippy haha. $\endgroup$ Commented Apr 8, 2016 at 5:16
  • $\begingroup$ So what makes the 'circle of circles' different from the mandelbrot set? $\endgroup$ Commented Apr 8, 2016 at 5:18
  • 1
    $\begingroup$ The Mandelbrot set does not display self-similarity. On the contrary, you can actually tell where you are in the Mandelbrot set by observing the local structure. This behavior is described nicely in this paper by Bob Devaney. On page 3, he calls the Mandelbrot set "the antithesis of a fractal" because of this. $\endgroup$ Commented Apr 8, 2016 at 12:07

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .