Self-similarity at all scales.

http://flea.sourceforge.net/fleaParasol.png - see FractalLifeEngine

Fractals are generated by iterative algorithms.  See PatternsGenerateArchitectures.  Fractals are patterns.  A pattern can be thought of as a rule that commands changes to be made in an environment.  All patterns are recursive in some way.  Many people think that there's a first time for all patterns and an original which is invoked again and again.  This is especially obvious of fractal generating patterns.  The archetype, design, exemplar, model, et cetera, is satisfactorily invoked if certain conditions obtain in the copy.  These are rule generated conditions.  The copy may be different from the original in various ways including low fidelity and incompleteness, and sometimes so different that the copy is not a copy at all.  A fractal is "a figure that shows self-similarity, possibly twisted, at all scales, such that the figure's boundary has fractional dimensions."

You can have a 2-D Cantor dust with fractal dimension 1, and what's more you can have a self-similar curve with fractal dimension 1, formed in a manner similar to the snowflake.  The first is definitely a fractal and the second would probably be called such by most mathematicians.  Going the other way, self-similarity is a really slippery concept, since we allow statistical rather than actual self-similarity, and I wouldn't be surprised if there are sets commonly called fractals that are severely lacking in this deparment.

http://pnorthov.future.easyspace.com/g107.jpg

From: http://pnorthov.future.easyspace.com/

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To correct some misconceptions above:

	 :	'Fractals are generated by computers according to rules.'

''Some'' fractals are generated by computers, but fractals are found all over. BenoitMandelbrot's TheFractalGeometryOfNature gives examples from nature (clouds, coastlines), biology (the circulatory system, lung surface), economics, and many more fields.

	 :	'A fractal is "a figure that shows self-similarity, possibly twisted, at all scales, such that the figure's boundary has fractional dimensions."'

A fractal can exhibit statistical self-similarity, while not being structurally self-similar.

See http://mathworld.wolfram.com/FractalDimension.html

	 :	'The copy may be different from the original [...] sometimes so different that the copy is not a copy at all.'

Makes one wonder where the boundary between copy and non-copy is, doesn't it?

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Wikis can also be fractal! What does this mean? 
	* Pages can have subpages and subsubpages without end. 
	* Any page can become a wiki of its own and contain further subwikis

This means that any page as a subwiki:
	* can have its separate properties, e.g. UI language or layout 
	* its own focus represented in RC, index and searching

See this at work at the FractalWiki development spike:
	* http://www.wikiservice.at/fractal/wikidev.cgi

-- HelmutLeitner

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Take a look at William Garnett's aerial photographs (in, for instance, ISBN:0520083482). He shot landscapes from a light aircraft. Many of them show huge topographic features, river deltas, alluvial plains and such, often without the context provided by human structures. After looking at the pictures for a while you realize that they have absolutely no sense of scale. The huge rock formations in Death Valley could almost equally well be a microgram of a crystal cleavage plain. The river delta a colony of bacteria. And so on. Magical stuff.

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I sure wish Escher were alive to see and perhaps use computer-generated fractals. He would have a ball with them.

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See PartToWholeAsWholeIsToPart, FractalComprehension, MetaPattern, PatternPattern, AlternateHardAndSoftLayers