Re: wait! stonehenge is a fractal?

sgt pinback (rob...@project.darkstar) wrote:
: but mandelbrot(stein)* himself called the gasket type thingy a fractal. 
: he talks about random and non-random fractals, so somethign doesn't have 
: to be random to be a fractal (last i heard there was no definitive 
: definition of a fractal tho. but what ol' benoit m. says is one should be 

Well, his definition of fractals as sets[1] which have a Hausdorff
dimension[4] not equal to their topological dimension[2] is reasonably
good IMHO, though it'll do you little good if you don't understand the
terms used. :(

Self-similarity could also be used as a definition, but I'd say it's a
different but related quality to fractality[3]. After all a straight
line is self-similar while the Mandelbrot set is strictly speaking not.

[1] For those not so comfortable with analytical geometry and the math
jargon involved, think "shape" or "thing".
[2] The ordinary one - points are 0d, lines 1d, planes 2d, etc.. I saw
an exact definition somewhere but I forget it.
[3] I just coined that. I wonder if it's in use or if there's a better
word for it.
[4] One of several not-quite-so-ordinary "fractal dimensions". Not quite
so simple to determine exactly for more complex shapes, but for the
simpler linear fractals like these in this thread it can be calculated
by the number and size of the smaller images in the whole.

The Sierpinski triangle that started this all contains three parts, each
half the size of the whole, so its fractal dimension is (log 3)/(log 2)
or about 1.585. The Koch snowflake (or rather one of the sides) has four
parts, each one third of the whole, giving (log 4)/(log 3) ~= 1.262.

: one. including (sneaking the ob ascii art in) koch curve interations:

Those were excellent. Let me contribute this fifth iteration Hilbert
curve. (Which has four segments each half the size of the original, so
a fractal dimension of (log 4)/(log 2) = 2, nice and integral but not
quite appropriate for a curve.. ;-) That's exactly why they call it a
plane-filling curve.)

   _   ___   ___   ___   ___   ___   ___   ___   ___   ___   ___
   _| |_  |_|  _| |_  |_|  _| |_  |_|  _| |_  |_|  _| |_  |_|  _|
  |  _  |  _  |_   _|  _  |  _  |  _  |_   _|  _  |  _  |  _  |_
  |_| |_| | |___| |___| | |_| |_| | |___| |___| | |_| |_| | |___|
   _   _  |  ___   ___  |  _   _  |_   _____   _|  _   _  |  ___
  | |_| | |_|  _| |_  |_| | |_| |  _| |_   _| |_  | |_| | |_|  _|
  |_   _|  _  |_   _|  _  |_   _| |  _  | |  _  | |_   _|  _  |_
   _| |___| |___| |___| |___| |_  |_| |_| |_| |_|  _| |___| |___|
  |  ___   ___   _   ___   ___  |  _   _   _   _  |_   ___   ___
  |_|  _| |_  |_| |_|  _| |_  |_| | |_| | | |_| |  _| |_  |_|  _|
   _  |_   _|  _   _  |_   _|  _  |_   _| |_   _| |  _  |  _  |_
  | |___| |___| | | |___| |___| |  _| |_____| |_  |_| |_| | |___|
  |_   _____   _| |_   _____   _| |  ___   ___  |  _   _  |  ___
   _| |_   _| |_   _| |_   _| |_  |_|  _| |_  |_| | |_| | |_|  _|
  |  _  | |  _  | |  _  | |  _  |  _  |_   _|  _  |_   _|  _  |_
  |_| |_| |_| |_| |_| |_| |_| |_| | |___| |___| |___| |___| |___|
   _   _   _   _   _   _   _   _  |  ___   ___   ___   ___   ___
  | |_| | | |_| | | |_| | | |_| | |_|  _| |_  |_|  _| |_  |_|  _|
  |_   _| |_   _| |_   _| |_   _|  _  |_   _|  _  |  _  |  _  |_
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  |  ___   ___  | |  ___   ___  | |_   _____   _|  _   _  |  ___
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   _  |_   _|  _   _  |_   _|  _  |  _  | |  _  | |_   _|  _  |_
  | |___| |___| |_| |___| |___| | |_| |_| |_| |_|  _| |___| |___|
  |_   ___   ___   ___   ___   _|  _   _   _   _  |_   ___   ___
   _| |_  |_|  _| |_  |_|  _| |_  | |_| | | |_| |  _| |_  |_|  _|
  |  _  |  _  |_   _|  _  |  _  | |_   _| |_   _| |  _  |  _  |_
  |_| |_| | |___| |___| | |_| |_|  _| |_____| |_  |_| |_| | |___|
   _   _  |  ___   ___  |  _   _  |  ___   ___  |  _   _  |  ___
  | |_| | |_|  _| |_  |_| | |_| | |_|  _| |_  |_| | |_| | |_|  _|
  |_   _|  _  |_   _|  _  |_   _|  _  |_   _|  _  |_   _|  _  |_
   _| |___| |___| |___| |___| |___| |___| |___| |___| |___| |___|

-- 
ii 3D .sigXIlmari KaronenAscii 3D .sigIlmari Karonen_Ascii 3D .sigIlmari Kar
fi....@graphic.iltzu@sci.fi.stereo@gra...@sci.fin.stereo@graphicilt
ltzu/ba/http://www.sci.fi/~iltzu/bahttp://www.sci.fi/~iltzu/1bahttp://www.sc
 is a stereogram!><This .sig is a stereogram!><This .sig is a stereogram!><T

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