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Author Topic: Anti-fractal?  (Read 5457 times)
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heneganj
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« on: February 23, 2007, 01:39:27 PM »

I stumbled across this interesting article.  What do you call something that is infinitely complex but not self-similar?  I like the name 'anti-fractal', but would 'chaotic' suffice?  Does chaos imply lack of order?  These patterns are completely mathematically ordered...

http://www.reuters.com/article/scienceNews/idUSN2245118920070222?src=022207_1643_ARTICLE_PROMO_also_on_reuters&pageNumber=1

Medieval Muslims made stunning math breakthrough



WASHINGTON (Reuters) - Magnificently sophisticated geometric patterns in medieval Islamic architecture indicate their designers achieved a mathematical breakthrough 500 years earlier than Western scholars, scientists said on Thursday.

By the 15th century, decorative tile patterns on these masterpieces of Islamic architecture reached such complexity that a small number boasted what seem to be "quasicrystalline" designs, Harvard University's Peter Lu and Princeton University's Paul Steinhardt wrote in the journal Science.

Only in the 1970s did British mathematician and cosmologist Roger Penrose become the first to describe these geometric designs in the West. Quasicrystalline patterns comprise a set of interlocking units whose pattern never repeats, even when extended infinitely in all directions, and possess a special form of symmetry.

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gandreas
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« Reply #1 on: February 23, 2007, 04:41:49 PM »

I stumbled across this interesting article.  What do you call something that is infinitely complex but not self-similar?  I like the name 'anti-fractal', but would 'chaotic' suffice?  Does chaos imply lack of order?  These patterns are completely mathematically ordered...

In complexity theory, "chaos" is that fine (or sometimes broad) line between "order" and "random".  (For a good introduction, check out Stuart Kauffman's "At Home in the Universe")

And the term "Fractal" derives from fact that it has fractional dimensionality - for example, that jagged line that represents the shoreline isn't straight (1 dimensional), yet it doesn't actually fill 2 dimension, but is like (for example) 1.3245 d.  It does not require self similarity (there are many fractals that are not self similar, but they aren't all that aesthetically interesting for the most part either)

And speaking of Islamic star patterns, Craig Kaplan wrote a Java based applet that lets you experiment with them (and has good paper on his techniques at http://www.mi.sanu.ac.yu/vismath/kaplan/index.html - his web site also has a bunch of other interesting material that isn't directly fractal related, but is definitely useful for those interested in the whole "procedural/generated art".
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heneganj
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« Reply #2 on: February 25, 2007, 10:10:15 AM »

Excellent thanks for the link.  Although it is repeating (tiling) the pattern.  Would be nice to see a non-repeating version of this program.
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bradorpoints
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« Reply #3 on: April 11, 2007, 04:40:58 PM »

Good article; it got me interested.  I've often wodnered about such patterns for the mid and far east.  It didn't have any detail (for good reason).  Here's a link to a paper which goes into some of the how's and why's. 

http://www.physics.harvard.edu/~plu/publications/Science_315_1106_2007.pdf
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