Superb Mandelbrot Set On Wikipedia

Same building different angle on Fuxing North Rd.(復興北路)

These are superb images I found of a Mandelbrot Set on wikipedia. They were so gorgeous and reminded me of the time when I was studying Complex Mathematical Analysis that I just had to post them.

[ A small digression:

At the time, I was obsessed with Chaos Theory and complex mathematics.

Strange attractors, fluid dynamics and partial differential equations.

One of the things that really got me out of mathematics was that it was all too abstract; all the concepts were not grounded in palpable reality.

Head in the clouds.

And I wanted to be on Earth…

Digression ends here.] 

A Mandelbrot Set is a fractal.

The Mandelbrot set is a fractal that has become popular outside of mathematics both for its aesthetic appeal and a complicated structure arising from a simple definition. Benoît Mandelbrot and others worked hard to communicate this area of mathematics to the public.

A fractal is:

In colloquial usage, a fractal is “a rough or fragmented geometric shape that can be subdivided in parts, each of which is (at least approximately) a reduced-size copy of the whole”. The term was coined by Benoît Mandelbrot in 1975 and was derived from the Latin fractus meaning “broken” or “fractured”.

A fractal as a geometric object generally has the following features:

  • It has a fine structure at arbitrarily small scales.
  • It is too irregular to be easily described in traditional Euclidean geometric language.
  • It is self-similar (at least approximately or stochastically).
  • It has a Hausdorff dimension that is greater than its topological dimension (although this requirement is not met by space-filling curves such as the Hilbert curve).
  • It has a simple and recursive definition.

Because they appear similar at all levels of magnification, fractals are often considered to be infinitely complex (in informal terms). Natural objects that approximate fractals to a degree include clouds, mountain ranges, and lightning bolts. However, not all self-similar objects are fractals—for example, the real line (a straight Euclidean line) is formally self-similar but fails to have other fractal characteristics.


The images were produced by Wolfgang Beyer with the Ultra Fractal 3 program:

mandelZoom00MandelbrotSet {
fractal:
title=”mandel zoom 00 mandelbrot set” width=2560 height=1920 layers=1
credits=”WolfgangBeyer;8/21/2005″
layer:
method=multipass caption=”Background” opacity=100
mapping:
center=-0.7/0 magn=1.3
formula:
maxiter=50000 filename=”Standard.ufm” entry=”Mandelbrot” p_start=0/0
p_power=2/0 p_bailout=10000
inside:
transfer=none
outside:
density=0.42 transfer=log filename=”Standard.ucl” entry=”Smooth”
p_power=2/0 p_bailout=128.0
gradient:
smooth=yes rotation=29 index=28 color=6555392 index=92 color=13331232
index=196 color=16777197 index=285 color=43775 index=371 color=3146289
opacity:
smooth=no index=0 opacity=255
}

Author: range

I'm mathematician/IT strategist/blogger from Canada living in Taipei.