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.
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:
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method=multipass caption=”Background” opacity=100
maxiter=50000 filename=”Standard.ufm” entry=”Mandelbrot” p_start=0/0
density=0.42 transfer=log filename=”Standard.ucl” entry=”Smooth”
smooth=yes rotation=29 index=28 color=6555392 index=92 color=13331232
index=196 color=16777197 index=285 color=43775 index=371 color=3146289
smooth=no index=0 opacity=255