Wednesday, August 30, 2006

Standards


I had an unfortunate laundry experience the other day (turns out that “permanent press” is more wishful thinking than truth in advertising). I decided to entertain myself while ironing by listening to Eric Clapton tear through some blues standards (if you have to iron, this definitely helps). This got me wondering: Are there any “standards” in fractal art? That is, if fractal art is likened to blues or jazz or popular music, are there any images that might be the equivalent of “I’m Your Hoochie Coochie Man,” or “Take Five,” or “Night and Day”?

It seems to me that a fractal art standard would be an image intimately familiar to anyone intimately familiar with fractal art and vaguely familiar to those with less connection to the art. It would include a relatively simple formula, to increase its popularity and ease of calculation with various coloring schemes. Coloring formulas might be like genres in the musical analogy. (Here, I’m thinking in the Ultra Fractal schema, but such an image must not be tied to a particular platform.) The zoom and parameters should be such that the image is generally recognizable, but they could be varied as part of the artist’s rendition of that image.

Immediately, the overall Mandelbrot set leaps to mind. Other candidates might be zooms into the West Midget and the Seahorse and Elephant Valleys, the Koch curve, the Hilbert curve, and Newton's Method for z4 = 1. Any others?

5 Comments:

Blogger John S. Meade said...

Oh I can nominate a few...

Lupanov -- which I equate to beatnik jazz for some reason or other. And all those wonderful covers on the pulp scifi short story collection paperbacks of the early '60's.

Sierpinski* -- best contemplated while dining on appetizers and aperitif (as opposed to a pair of tiffs) in some sidewalk cafe on the outskirts of some city whose population is less than 229,423 give or take 4.669211660910299067185320382047...
... which brings us to:

the Bifurcation graph -- The sight of a touch, or the scent of a sound,
Or the strength of an arquebus deep in the ground. The wonder of flowers, to be covered, and then to burst up, thru tarmack, to the sun again -- Graham Edge (the Moodies)

Now on to the more specialized...

Man-o-War -- If I remember rightly reading through the comments in the fractint code, this fractal was the result of a "happy accident" (as was a lot of us).

I leave it to others to mention kam-torus hopalong, popcorn, and plasma.

And last but not last
anything in FE with "stalks".
-----------------------
* Never ask a triangle for directions. While they can tell you all about the signs to watch for, they can be very obtuse. In fact they can be downright unhelpful in other matters as well.

I once asked a triangle to cosign a loan for me and she very pointed replied, "It's not that I wouldn't or that I don't care it's just impossible." I handed her the pen anyway and she kept dropping it over and over and over again. And then said, "See? Can't."

8/30/2006 8:24 AM

 
Blogger Tim said...

I thought about this quite a bit when I was making up banners for the blog.

I think there's really only two or three. The plain mandelbrot you mentioned and also just about any spiral shape.

Maybe one or two julia variations on the plain mandelbrot, like the big "S" one and the sideway s taken from the points on the "neck" of the mandelbrot.

Koch curve, maybe. Beyond that I think the number of images multiplies so fast that I couldn't see any of them being a quickly identified or easily recognizable standard outside fractal circles.

I think outside of fractal circles the iconic fractal shapes would be anything that appears in common resources like an encyclopedia or math textbook. That's where ordinary folks probably get their first introduction to fractals.

8/30/2006 2:36 PM

 
Blogger ktazaco said...

It almost seems that fractals can be standardized as two or three types.

The first two types could be the boundary escape types such as the Mandelbrot and Newton. The second would be the orbital trajectory types such as the IFS and strange attractors.

Then we can get into the 2nd, 3rd and forth dimensional types. I suppost there can be a one dimensional type also.

Here is my list.

1.) The Mandelbrot method uses an algorithm that searches for a result that is "greater than" a value.

2.) The Newton method uses an algorithm that searches for a result that is "less than" a value.

3.) The Strange Attractor types
3.1) Gumowski-Mira fractal
3.2) Barry Martin fractal
3.3) The IFS fractal
3.4) Lorenz fractal
4.) Sierpinski and Koch fractal...
5.) Lyapunov fractal
6.) Lsystems

regards,
Steve

8/30/2006 8:48 PM

 
Blogger John S. Meade said...

Ok. I apologize. I misspelled Lyapunov (no disasemble number 5!). Maybe I got this subconscious thing going for autumnal wolves.

Anyway...It's a wonder I can get in here at all with all those extra visual only passwords like ftessi and exagga and cornholio and the like. Not that i'm complaining any, I like a good puzzle everynow and again

As far as fractals only showing up in math books... whose fault is that? I mean paisley is back in fashion this fall why not the Julia set? Can't be that much more difficult to stitch (but then I remember being part of the "jet set" (8 miles high 2-4-2 Foxtrot She packed my bags last night pre-flight)) At least I think I do.
If I can't remember it -- I must've really been there.

8/31/2006 6:01 PM

 
Blogger cruelanimal said...

I know using the sometimes cranky word verification passwords for posting comments can be annoying, but these visual checks do prevent spam-bots from swamping the blog with invasive, unwanted junk.

8/31/2006 6:51 PM

 

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