Like fractals?

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I also had the impression that the second one would be related to insects...the first one reminded me of spring, and the third one of autumn as well... I like the last one too...

I was pretty interested to learn where the word fractal actually comes from. For those that don’t know, it comes from the fact that fractals have fractional dimension. To see this, you need a new concept of dimension, one that derives from the concept of self-similarity.

If you take a D-dimensional figure, and divide it into N equal parts, the similarity ratio, r, between the entire figure and a single part, will be given by

r = D√N

so if you take a 2-dimensional rectangle, and divide it into nine parts, you get a similarity ratio of three

r = D√N = 2√9 = 3




This indicates that one side of the larger rectangle is three times longer than the corresponding side of one of the smaller rectangles.

So then you can try this formula with an actual fractal, e.g., with a Koch curve.
A Koch curve is constructed by starting with an equilateral triangle and proceeding as follows:



And so on and so on ad infinitum.

For the Koch curve, we know N and r, but not D. The replication procedure for one length of coastline is as follows: a single line is replaced by four lines (so N=4), each one-third the length of the original line (so r=3). This holds for the curve as a whole.

So we have 3 = D√4 which means that D = 1.2618. The Koch curve has fractional dimension.

Another surprising fact about the Koch curve: it has a finite area, but an infinite perimeter

I like it!

They sound like they could Pink Floyd or Tangerine Dream tracks.

These fractal pictures are great. I mainly know the Mandelbrot and Julia sets. It's one thing I like to do when I'm by a coast line - think that it's a fractal because you could always measure it's length more accurately and that's like a fractal, the more you zoom in the more detailed it is - forever.