Saturday, February 11, 2012

The fractals of winter


Winter creates many unique patterns. I have to catch them now because they will disappear soon.
Some patterns are easy to explain, but others raise questions.

1) Mandelbrot set - salt
2) Mandelbrot set - ice - this is my attempt at an explanation
3) Koch curve - salt
4) Diffusion-limited aggregation (DLA) - salt - this is a well researched process
5) Fractal landscape - ice floes on the river

6) Fractal landscape - snow - I wonder - this looks so very much like a real archipelago landscape - maybe the erosion processes are similar in a real landscape and in snow on the sidewalk - is the water erosion process similar to the process caused by walking feet?

References:
en.wikipedia.org/wiki/Mandelbrot_set
en.wikipedia.org/wiki/Koch_curve
en.wikipedia.org/wiki/Diffusion-limited_aggregation
en.wikipedia.org/wiki/Fractal_landscape

6 comments:

  1. The "snow landscape" example probably is an interaction of:

    1) a homogenous layer of snow
    2) random perturbation by footsteps
    3) feedback powered self-organisation - places with steep angles will erode more quickly when people step on them - probably snow melts faster near the dark pavement

    I'm sure you can model that very nicely using 2-d differential equations.

    See also:
    http://sprott.physics.wisc.edu/pubs/paper270.pdf
    http://www.miqel.com/fractals_math_patterns/visual-math-natural-fractals.html

    ReplyDelete
  2. Also:
    http://www.cscs.umich.edu/~crshalizi/notabene/self-organization.html

    ReplyDelete
  3. Interesting pics!
    I am not capable of the Dif EQ math equation. The math for Brownian motion eludes me too. I just recognize them when I see em; I do love patterns and observing natural phenomena. The colour palettes of ice landscapes are strikingly lonely and peculiar. The colours (especially the background)often appear to be predominantly muted mauves,teals,grays an hazy yellows. Look at various depictions Monet's did of Ice floes. The colours are very similar to the ones you captured
    P.S.
    I don't know much about colour theory either.

    ReplyDelete
  4. Interesting pics!
    I am not capable of the Dif EQ math equation. I just recognize them when I see em; I do love patterns and observing natural phenomena. The colour palettes of ice landscapes are especially lonely and peculiar. The background colours mauve,teal,gray an hazy yellow predominate Look at various renderings of Monet's Ice floes and his color choices. The colours are very similar to the ones you captured.
    P.S.
    I don't know much about colour theory either.

    ReplyDelete
  5. Hi

    I love these pictures. I've enjoyed your blog for a long time too. So much so that I've featured it on my new blog, Psychogeographic Review, as my psychogeographic 'website of the month'!

    Why not have a look? It's at:

    http://psychogeographicreview.com/

    With best wishes,

    Bobby Seal.

    ReplyDelete
  6. http://psychogeographicreview.com/

    A very interesting website indeed. It's a mystery how I've managed not to find it yet. I've immediately subscribed myself.

    ReplyDelete