Chaotic Utopia

Friday Fractal XIII

i-5fd7033b559099bb30d564d1291fdf7f-dfwhole.jpgSome people have been asking how I make the fractals for the Friday feature. A few just assumed that I took a photograph and plunked it into a computer program, which automatically spit out a fractal. I wish it were that easy… or, I suppose, if it were, everyone could do it, and I could sleep in on Fridays. (There’s that wishful thinking again, eh?) I decided to keep this week’s fractal rather simple, so I could explain a little about how I create them. While I don’t just drop in a picture, I don’t exactly sit and work out the math to match it, either. The real process is somewhere in between.

First, I find a formula. Some of the most famous were written (relatively) long ago, such as Newton’s methods. Others have made exquisite variations on these classics. As I said before, I chose a simple one today:

z= z/ func1( 1 / z) + c

Here, z defines a pixel within the set. For instance, if you drew a circle, and point z was inside, you’d color in point z. Do this for every point on the page, and you’ll have a filled in circle. The fractal is similar, but the coordinates lie in the complex plane. So, point z is defined by two numbers; one number defines a place on the “real” axis, the other on an “imaginary” axis. So, instead of being a “flat” circle on a regular 2-D page, the shape has a “bumpy” edge, with hills and crevices. Func1 is a trigonometric function–just like you see on a calculator. Here, it is neg–just like it sounds, it makes the variable negative. (Those grouchy functions….) The variable c, the “seed” for the fractal, is also a complex number. These numbers can be adjusted… and make a huge difference as to what you see:


At this point, I feel around for the right look. Sometimes, the right shape just seems to jump out of the screen. This is where I find fractals utterly fascinating. There seem to be endless shapes, usually so alien and abstract. Yet, among these are shapes which are reassuringly familiar, as if certain forms work with such a natural efficiency that they persist in nature. It is as if the forms are waiting there, to be discovered and perfected. I feel the same way when I write a story, or rearrange my knick-nacks. Sometimes, when writing a fractal, I use other tools to transform the fractal even further. These can adjust the way the program colors “z”, or add a twist or mask to the whole thing. Since we’re keeping this simple, I skipped that today. The next part I leave up to nature.

To color the fractal, I use an art program to note the RGB coordinates of various pixels in the photograph. I add these colors to the fractal palette, as many or as few as I wish. This is where the fractal seems to come to life–all that is left is to find the best place to crop:


And compare it with the photograph:


A Dandelion “Clock” (Taraxacum officinale)

Warning! This plant reproduces by taking advantage of small children!


Aw, c’mon, you know you used to blow them too!

“To some the dandelion is a weed; but not to me, unless it takes more than its share of space, for I always miss these little earth stars when they are absent. They intensify the sunshine shimmering on the lawn, making one smile involuntarily when seeing them. Moreover, they awaken pleasant memories, for a childhood in which dandelions had no part is a defective experience.”
(E. P. Roe in The Home Acre via Dandelion Pictures and More)

All photos and fractals were created by the author, fractals using ChaosPro.


  1. #1 Katherine Sharpe
    August 6, 2006

    Is that your son? He is BEYOND CUTE!

  2. #2 Abel Pharmboy
    August 6, 2006

    Complete concurrence with Katherine…We are SO out there for a playdate with PharmPreSchooler! She is also an expert in the diaspora of Taraxacum seeds!

  3. #3 Karmen
    August 7, 2006

    Thanks, guys. I love that picture. And yep, that’s Roland. Don’t let the sly grin fool you, though. That’s the look of genuine mischief.

  4. #4 Kim
    August 9, 2006


    I hope he grows up to find his own Tower somewhere…
    And I hope it won’t be as gruesome and obsessive a quest as that of that other Roland.
    Anyhow, he’s definitely way cuter than “Double Ugly”, and I’m sure Eddie would agree. (The mischievous grin only ads to that)

  5. #5 Karmen
    August 10, 2006

    Oh, sometimes I wonder… he seems to like to take the tough, stubborn paths. He takes after his mommy too much. That’s my lil’ gunslinger.

  6. #6 Kim
    August 10, 2006

    Well, my greetings to that lil’ gunslinger, and (since it’s been a while) greetings to Otto too.

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