Fractals
chaoticutopia | November 17, 2006What's so special about a spiral? Why does it catch our eye, inspiring our art and architecture? Why is it even there? This week, I reviewed a program about the emergence of order, showing how organized patterns appear in nature and society, without the guidance of a leader. Spiral forms are an excellent example for this subject... did some creator or guide put these elegant forms in nature? Or do they appear almost by accident, a result of simple, repeated processes? Sounds familiar. Before comparing today's fractal with an image from nature, compare these two definitions:
A self-similar…
chaoticutopia | November 10, 2006 Hidden deep within the layers of the Mandelbrot set, subtle, yet familiar forms can almost leap out at you. This happened to me as I applied the coloring formula (based on Gaussian integers) to this week's fractal. I was merely looking for a pattern that was somewhat spotty. I ended up with this:
Originally, I was trying to mimic this...
...The spotted back of this Vietnamese moss frog, which I admired last weekend at the Denver Zoo:
A Vietnemese Moss Frog (Theloderma corticale)
The moss frog, with a gift for camouflage, can be found along streams high in the mountains of Vietnam,…
chaoticutopia | November 3, 2006Fractals, like so many sights in nature, can seem both static and dynamic at the same time. A cloud can change its shape right before your eyes, and so can a slice of the Mandelbrot set, with a slight nudge of the bailout values. Try to find the same spot later on, without the same exact values, and you may never see it again. This fractal, in shades of tangerine, is a typical Mandelbrot set, colored with an "alternative" fBm algorithm using 4n linear arrays. (The algorithm was written by David Makin, whose spectacular fractals can be seen here.) The combined effects of the chaotic,…
chaoticutopia | October 30, 2006 In order to make up for my recent shortage of Friday Fractals, I've assembled a few at once, with a Halloween-ish theme. I browsed over the Mandelbrot set, seeking the spookiest angles. What seems freakiest is the unending depths of the set... I could have been wandering through forever. It is sort of like a dream of falling, but never hitting the bottom.
I began using the same set of colors displayed in my Halloween banner, and soon found images reminiscent of monsters and insects:
Freaky Fractal I-Monsters
That was slightly unsettling, so I brightened things up with shades of orange.…
chaoticutopia | October 6, 2006To compare this evening's fractal with nature, take a walk. Look beyond the scaling patterns in the autumn leaves, the branching trees, the billowing clouds, and up into the darkness of space. If the time is right, you'll see a familiar shape, similar to the following image. This, of course, isn't a photograph, but a computer-generated set of patterns based on numbers. This Julia Set has been trapped within a spherical shape and slightly randomized, then highlighted with nature's palette:
In case you can't find the Harvest Moon in the sky tonight, try this picture, taken by Bev Brink in 2003…
chaoticutopia | September 22, 2006Deep within the pockets of a Mandelbrot set, delicate branches display endless variations. When highlighted with the colors of autumn, (since today is, after all, the Autumnal Equinox,) patterns of exquisite beauty emerge:
These patterns can remind us of many forms in nature, including a grove of quaking aspen:
Quaking Aspen (Populus tremuloides)
Aspen are members of the poplar family, along with their cousins, the cottonwoods (featured in last week's fractal.) While they are known for their brilliant foliage in the fall, the species has another claim to fame: the world's largest organism…
chaoticutopia | September 15, 2006The other day, I put up a small question about history. What better place could there be to put my answer, but in the form of a fractal? Patterns seem to almost repeat themselves. Sweeping changes result from a single, initial circumstance. Each point is connected to another, within the same set. Are these descriptions of events in history, or the rules defining a rippled Julia set?
Or perhaps the rings of a tree?
The trunk of a cottonwood tree, showing rings formed over many years. Cottonwood trees (below) line the bank of Walnut Creek, which appealed to Sarah H. Church when she arrived…
chaoticutopia | September 11, 2006
In memory of our loved ones; lost five years ago, on this day.
September 11, 2001
Their silent wounds have speech
More eloquent than men;
Their tones can deeper reach
Than human voice or pen.
-William Woodman
chaoticutopia | September 8, 2006Fractals are like landscapes. From a simple process, be it a formula or continental drift, one area can be strikingly different from another. This is true for my favorite type of fractal, a Julia set, "colored" with a bit of fractal Brownian motion:
Or the ripples on the Great Sand Dunes:
Stone, Steam and Sand: A Geologic Photo Tour of Southwest Colorado, Part III
In the first part of this series, I described the formation of the San Juan Mountains, and then a bit of the more recent history of the lands to the west of the range. The valley east of the San Juans had similar initial…
chaoticutopia | September 1, 2006Picture a Julia set, in shades of deep blue and green, with stretching, white spiral arms, decorated with fractal Brownian motion, which leaves hazy boundaries between shades of cloud, land, and open ocean:
It might resemble a hurricane. Of course, a relatively simple geometric structure like a fractal has nothing on Mother Nature. You can adjust variables all day, but never get the delicate coastline of Baja California, as a hurricane approaches:
Hurricane John nears Baja, California, moving north-northwest at 8 mph, with sustained winds near 110 mph.
I've been keeping up with the latest…
chaoticutopia | August 28, 2006Life, at times, seems rather abstract. So, with little comment, I present an abstract fractal. Unlike most of my fractals, which resemble some form in nature, this one represents a figment of a dream. Since I can't exactly take a photograph from a dream, I'll let the fractal stand alone:
A Mandelbrot Set, using a radial wave function to color the outside of the set. (The inside remains black.)
I'll leave it at that, for now, until I can return to explain the symbolism in greater detail. In the meantime, here is a hint, disguised in one of my favorite poems:
A Dream Within A Dream
Take…
chaoticutopia | August 18, 2006This was actually the first Friday Fractal, originally posted on ChaoticUtopia.com on April 28th of this year. I've found many images in fractals which are reminiscent of forms in nature. Both the precise mathematical displays and the delicate shapes of life have an essence of intricate beauty, provoking strong feelings of awe--especially when they are strikingly similar.
Here is a section taken from the Mandelbrot set:
And its corresponding form in nature:
An ammonite fossil, Discoscaphites conradi, from the upper Cretaceous age, found in the Pierre Shale Formation here in Colorado.
This…
chaoticutopia | August 11, 2006Emergence: Complex patterns arise from the simplest rules. From lighter elements emerge heavier compounds; from clouds of gasses and particles arise galaxies, stars, and planets. From basic atmospheric reactions between basic chemical compounds, the building blocks of life, amino acids emerge. From these acids, patterns of DNA emerge; from DNA, simple life. From simple life forms, complex forms arise, filling countless niches and functions, each form intricately involved from another. From these interactions, ecosystems arise. In these ecosystems, complex beings, dependent upon vast arrays of…
chaoticutopia | August 4, 2006Some 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…
chaoticutopia | July 28, 2006I thought I'd do today's fractal a little backwards. I usually begin with a fractal, then show a matching picture from nature, and explain a bit of the science behind the nature. This time, I'm going to start off with a nature photo, and follow with the fractals. Everyone should be familiar with the science behind rainbows, so I'll let the simple beauty of the image speak for itself:
And a similar fractal:
The Mandlebrot set might not be quite as simple, although the formula used to create it sounds simple enough: Z = Z2 + C
But the art found within the set can be deeply complex and…
chaoticutopia | July 21, 2006I never did match any of Lillybridge's photography work on my visit to the South Platte. The past seemed to be lost under layers of roadways and rails. Still, within that changed land, patterns of natural beauty still exist. Sometimes, looking through fractals is the same way... there are many complex layers, and sometimes I have to search long and hard to find the pattern I seek. Today's fractal was no different. The lower layers are familiar Julia sets (similar to these) while the upper is a unique formula written by Andras Szolek. His formula is a complex conglomeration based on other…
chaoticutopia | July 14, 2006When you peer into a fractal, you're seeing the edge of chaos. If you sift through enough Julia or Mandelbrot sets, you might catch a hint of fractal fever. When you find that point, where order is filtered out of randomness, and glimpse a familiar pattern, you might feel tempted to shout "Eureka!" That triumphant feeling is, of course, much older than the computers that generate fractals. We've been seeking precious patterns for centuries. Compare this fractal image, taken from a section of a Julia set colored with fractal Brownian motion...
...with a much older sorting method:
Panning for…
chaoticutopia | July 7, 2006Today's fractal is so bushy and branching, that you might almost miss the regular curving patterns of a Julia set. (Actually, this is two Julia sets, but who's counting?) That seems fitting, as nature can be the same way. You might not notice a particular spot, even though you pass by every day. Then, one day, your eye happens to catch a familiar pattern, and suddenly the ordinary is strikingly beautiful. I created this fractal to honor that awestruck feeling that comes along with discovering a world right under your nose:
And here's a matching spot, which I pass by every day, while walking…
chaoticutopia | July 4, 2006Freedom is a beautiful thing. In some sense, it can be found in every particle, every molecule, and every living cell. (To divide, or not to divide...) As sentient beings, we are mass conglomerations of freedom, individual harmonious machines, adapting by choice to the changing world around us. I've always appreciated living in a country based entirely on the idea of freedom. It reflects our nature, whether scientific or social. While we may squabble about what responsibilities come along with our freedom, we can all agree: liberty is precious.
In honor of this anniversary of liberty, I've…
chaoticutopia | June 28, 2006Why does chaos always strike in threes? I've spent the past three days* dealing with what I can only describe as personal chaos. Of course, I don't mean it in the classic sense; I haven't been floating in a void of disarray. (It just feels like it sometimes.) Rather, I refer to the variables in life... those which we knew were possible, but seem unbelievable when they occur. I'm still seeking order (riding the waves, with a touch of battle, here and there) so it will be a few days before I have time to write. In the meantime, I'd like to share a few bits from my old site--about chaos, of…