Fractals
chaoticutopia | April 13, 2007What happens when you cross two immensely complex patterns, and then picture them at the perfect unique angle? Consider first, the merging of a skewed Mandelbrot set with a cloud-like plasma fractal. Zoom in on one, out on the other, and blend:
Now compare this with a photograph of a distant galaxy:
NGC1356
Like my fractal, the photo was taken at a particular and unique angle, capturing an eclipse of enormous scale. In this case, the Chandra X-Ray observatory captured the eclipse of a black hole with a galactic cloud. This gave astronomers the perfect opportunity to measure the halo of…
chaoticutopia | April 6, 2007While here in Colorado, freezing rain and snow is drizzling from the skies, spring is sweeping across the northern hemisphere. In celebration, I designed this petal-like Julia set and laid it on top a wavy Mandelbrot set, creating this arousing union. (What can I say? It's spring!)
In creating the fractal, I used the colors from this native Colorado wildflower, which will be blooming here, very shortly:
Bell's Twinpod (Physaria bellii)
These pretty little blooms are only found in Colorado, and even then only in certain areas. Specifically, the plant grows in sandstone or shale formations (…
chaoticutopia | March 30, 2007"It is what it is." This phrase has been bothering me lately, though I couldn't quite put my finger on the reason. It was enough that I was reluctant to post this fractal last week. Being an optical illusion, it contradicts the whole idea of "it is what it is". Of course, the more I thought about the phrase, the more I noticed people saying it. (Apparently, it was even voted the #1 cliche of 2004 by USA Today.) The more I heard it from people, the more I started thinking they might be wrong. "It is what it is" seems to deny the complex and dynamic aspects of nature. "Maybe it is something…
chaoticutopia | March 17, 2007You've probably noticed that people are running around drinking green beer, pinching and kissing one another. Yes, it is St. Patrick's Day. But, the other day was Pi Day. So, this week's Friday Fractal was due to be sandwiched between two contrasting holidays, one objective and arbitrary, one colorful and subjective. It seemed to me that a fractal would be the perfect opportunity to combine these two contrasting worlds, and explore the fuzzy edge between.
I began with the usual Julia set, and threw in a slice of pi. Usually, in this Julia set, "z" represents a pixel. We tell the computer that…
chaoticutopia | March 9, 2007Today's tentacle fractal (with a bonus haiku) is dedicated to PZ Myers of Pharyngula:
Happy 50th Birthday, PZ!
Nova fractal created by the author using ChaosPro.
chaoticutopia | March 2, 2007For today's fractal, here's something abstract:
The Lines Between*
*Left open to interpretation.
If that seems too intangible, how about no lines at all, just points? This fun little toy, called MandelSwarm, allows you to explore the Mandelbrot set with "magnetized particles." (That's the best description I could come up with, anyways.) It's one of those things you have to try for yourself. Drag the window around as you like, using + or - to zoom in and out, and enter to refresh the particles. MandelSwarm is one of the more unique forms of fractal exploration that I've run across, and…
chaoticutopia | February 23, 2007A fractal wave:
This wave-like x/y Julia set (enhanced with fBm noise) was inspired by one of my favorite works of art, a Japanese woodcut:
The Great Wave off Kanagawa by Hokusai
Incidentally, I borrowed this image from a Belgian workshop specializing in intricate designs on silver, Wolfers. Many of their pieces have been inspired by the same image. These works, sculpted from solid metal, seem to flow like waves of water. If you like that sort of thing, be sure to check out their other works.
Silver art and woodcut via Wolfers. All fractals made by the author using ChaosPro
chaoticutopia | February 16, 2007What do you get when you cross squid-mania with fractal art? "Mandelbrotopus! Octomandelpus! Cephalbrotopod! Fractalpod! Fractalpus! Iteratopod! Octomandelfractalpusopod!" to quote ScienceBlogs reader and wildlife photographer, Martin Richard. Of course, Martin was refering to Mandelbrot sets and octopi, and today we have a bioluminescent squid and a Nova Julia set. So, perhaps we should call it Squidlyglofractolopod! Or Cephanovataninga? Maybe Julioctonovadanae? Well, whatever you want to call it, this week's fractal is quite tentacle-like:
Unfortunately, my fractal tentacles don't quite…
chaoticutopia | February 9, 2007Fractal Cognition:
An abstract layered fractal, incorporating two types of Julia sets and an L-System fractal (pictured at right, below)
This fractal was inspired by some of the ideas I've posted lately. Here are a few tidbits for context:
We only see a fraction of an ever-changing cosmos. Each fragment that we observe, whether within ourselves or gleaned from the sensations registered in our minds, adds to the entropy of the cosmos, inducing more change. It is an imperfect pursuit, but we are active participants, whether we choose to look, or not. It is important for humankind, armed…
chaoticutopia | February 2, 2007Well, technically, it's still Friday somewhere...
It is bitterly cold in Colorado right now, somewhere around 10 degrees below 0, Fahrenheit. It was a bit warmer than that at sunset, when I spent some time admiring the fractal patterns in tree branches. (I'll confess, it's a favorite hobby of mine.) The windswept clouds, highlighted by the setting sun, made a lovely backdrop for the bare limbs. I realized, as snow drifted around my ankles, that I might be able to share the scene without the biting freeze. So, I was inspired to create a last minute layered fractal of trees:
The geometric…
chaoticutopia | January 26, 2007Trying to merge a flame fractal with a Julia set and a 3D sphere was almost enough to make my computer have a thermonuclear meltdown.... or at the very least, run out of memory. With a little coaxing, however, I was able to make it work. I'd say the results were pretty hot:
...but not quite as hot as another flaming sphere that we see every day:
Sol (image taken this morning by the Solar and Heliospheric Observatory)
This seemed like a good time to make a corona-like fractal. Coronal Mass Ejections have been spurting out the side of our grand star over the last week. (Check out these cool…
chaoticutopia | January 20, 2007I must apologize for being a bit slow with blogging lately, and now a day late with the Friday Fractal. A series of events and observations left me in an existential mood, pondering the fuzziness of the line between reality and fiction. Since my story on the subject, Illusions in Lavender was published this week, I have to admit, it is just that sort of philosophical pondering that can drive a person to madness. I've gotten used to these moods, however, and the most that suffered was my blogging. This week, however, as I emerged from that feverish hermit state, I caught another sort of fever…
chaoticutopia | January 12, 2007Many things in nature seem to catch our eye, simply by hovering between a turbulent, chaotic mess, and a vacant, serene sense of order. Art tends to reflect these dichotomies in nature, even when it is fairly abstract in nature. Some say it stems from a fascination from good and evil, or light and dark, or lightness and weight. But I suspect we just enjoy exploring the point in between. This is the draw of fractal art... the edge of the fractal itself, the border between the "inside" and the "outside" of the shape, can be followed endlessly. To our eye, the wonders and variation never seem to…
chaoticutopia | January 5, 2007I'd be hard pressed to choose a favorite from all of the fractals I've done as a Friday Feature. I started the feature in April of last year, after playing around with a fractal toy that a friend of mine built. Since then, I've created a piece of nature-inspired fractal art nearly every week. I've pasted them all into one block here, with a few abstract fractals included in the bottom row:
(Descriptions and links can be found beneath the fold)
In the first column, from top to bottom, are 1. Ammonite (which was also 15), 2. Red Spot Jr., 3. Rose, 4. Reflection of the Sun, 5. Grasshopper, 6.…
chaoticutopia | December 29, 2006Even in a snowy winter landscape, I feel as if I'm surrounded by fractals. In order to capture this feeling, I turned to the complex topography of the Mandelbrot set. I added an fBm coloring algorithm, to mimic the soft hues of a winter storm. When the fractal was complete, I decided to superimpose a sketchy map outline for clarity. (The NOAA does it, why can't I?) Here's what I came up with:
...which is rather similar to a recent image from the NOAA:
The National Oceanic and Atmospheric Administration's Image of the Day, showing snow accumulation over the Rockies from this week's…
chaoticutopia | December 23, 2006In order to create a festive holiday fractal, you don't even need a computer. All you need are a few standard Christmas decorations: glass balls and lights. (Actually, you don't even need Christmas lights, any light source will do.) It's quite simple: put the balls in a small pile under the light, pressing up against one another. Voila:
If you manage to get 4 balls together in a tetrahedron (each ball touching every other ball) the fractal image in the middle is called a Wada basin. In this type of fractal, there are three basins of attraction--areas where the patterns tend to converge, but…
chaoticutopia | December 17, 2006I'm a bit late on the Friday Fractal this week, mostly because I've been busy with my son's 6th birthday party. Since Roland adores these fractals almost as much as I do, I told him he could make this week's fractal. (That's another reason we're late... he has school on Fridays, and had the party yesterday.) He understands the basic premise behind a fractal--a pattern repeats, but with differences--enough to occasionally point them out to me in nature and art. When he sits down in front of the computer, however, he just wants to pick wild colors and weird shapes. I can't say as I blame him--…
chaoticutopia | December 8, 2006Be it winter or spring, I can find fractals growing all around me. Patterns which seem defiantly simple emerge with branching, crystalline beauty. This week's fractal is a Julia set, transformed with inverse circles, then painted with the colors of a winter's day:
...similar to a snowy scene in thicket of coyote willow:
Ice crystals growing on a branch of coyote willow (Salix exigua Nutt.) bush
...which somewhat resemble the flowers (catkins, in this case) of the plant in the spring:
All images by the author, fractals made by the author using ChaosPro.
chaoticutopia | December 2, 2006My apologies for getting the fractal out a day late, and for keeping it simple. Just as soon as I started to get caught up with things, I caught a nasty little cold. So, here are the basics-I'll get back to more detailed posts as soon as my head clears. Thanks for your patience!
For this fractal, I chose to forego the typical Mandelbrot or Julia sets, and stick with a pure fractal coloring algorithm. I use "fractal Brownian motion" to color the fractals rather often, as the stochastic patterns lend themselves well to fuzzy patterns in nature, from fog and clouds to ripples in the Great Sand…
chaoticutopia | November 24, 2006Occasionally, I run across a fractal in nature that is so perfectly formed, that the repeating, scaling patterns are actually 3-D. In these cases, it is difficult to do the image justice by creating a similar fractal on a 2-D surface. I encountered this in today's fractal. The Julia set, colored with a formula named Azar por Amplitud (literally, chance by amplitude) by Marcelo Anelli, mimics nature to some degree:
But doesn't reach nearly the fractal depth of field found in this vegetable:
Broccoli Romanesco (Brassica oleracea var. Botrytis) florets (photo by Alfredo Matacotta)
(Ok, I…