Fractals
goodmath | August 3, 2007 Most of the fractals that I've written about so far - including all of the L-system fractals - are
examples of something called iterated function systems. Speaking informally, an iterated function
system is one where you have a transformation function which you apply repeatedly. Most iterated
function systems work in a contracting mode, where the function is repeatedly applied to smaller
and smaller pieces of the original set.
There's another very interesting way of describing these fractals, and I find it very surprising
that it's equivalent. It's the idea of an attractor. An attractor is…
chaoticutopia | July 27, 2007Hi! My mom has been busy this week, getting ready for vacations and back-to-school, so she let me do the Friday Fractal this week. I like making my own fractals, because they are always different, sort of like taking a picture. I made this one, and then showed it to Mom. She said I made "a Mandelbrot set using Carlson Orbit traps, named for Paul Carlson, who developed the original textures and parameters for the coloring algorithm." That’s right.
Here it is:
I’d like to call it "My First Mandelbrot Set".
This one was neat, because it looks like it is made of circles, but it goes everywhere…
goodmath | July 27, 2007
While reading Mandelbrot's text on fractals, I found something that surprised me: a relationship
between Shannon's information theory and fractals. Thinking about it a bit, it's not really that suprising;
in fact, it's more surprising that I've managed to read so much about information theory without
encountering the fractal nature of noise in a more than cursory way. But noise in a communication channel
is fractal - and relates to one of the earliest pathological fractal sets: Cantor's set, which Mandelbrot
elegantly terms "Cantor's dust". Since I find that a wonderfully description,…
goodmath | July 25, 2007
One of the strangest things in fractals, at least to me, is the idea of space filling curves. A space filling curve is a curve constructed using a Koch-like replacement method, but instead of being
self-avoiding, it eventually contacts itself at every point.
What's so strange about these things is that they start out as a non-self-contacting curve. Through
further steps in the construction process, they get closer and closer to self-contacting, without touching. But in the limit, when the construction process is complete, you have a filled square.
Why is that so odd? Because you've…
chaoticutopia | July 21, 2007If we look at the natural world around us, fractals abound. Sometimes, not. This is the greatest puzzle to me... not that fractals appear in nature, but the fact that not everything is a fractal. Working on this week’s layered set (which took a while, mostly due to unrelated circumstances) I found myself questioning that inconsistency.
As I try to imitate some iterative, aperiodic pattern with my computer, I often find myself layering one fractal on top of another, to match the foreground and background (i.e., tree and sky, or clouds and land.) Sometimes, I’ll use more than a couple. This…
goodmath | July 17, 2007
I just finally got my copy of Mandelbrot's book on fractals. In his discussion of curve fractals (that is, fractals formed from an unbroken line, isomorphic to the interval (0,1)), he describes them in terms of shorelines rather than borders. I've got to admit
that his metaphor is better than mine, and I'll adopt it for this post.
In my last post, I discussed the idea of how a border (or, better, a shoreline) has
a kind of fractal structure. It's jagged, and the jags themselves have jagged edges, and *those* jags have jagged edges, and so on. Today, I'm going to show a bit of how to…
chaoticutopia | July 13, 2007I’ve never liked the taste of cream in my coffee, but I like to watch the way it swirls as you add it in. Such elegant forms appear when you add one liquid to another, especially when they are contrasting colors. It seems like the viscosity of the cream adds more complexity to the patterns than milk. Since the cream is poured in at a slightly different rate, or in a slightly different position each time, the patterns are always unique. Consistant in shape, but inevitably varied... you guessed it, I see fractals in my coffee.
Here’s a Julia set that has been modified with artificial symmetry…
goodmath | July 12, 2007Part of what makes fractals so fascinating is that in addition to being beautiful,
they also describe real things - they're genuinely useful and important for helping us to
describe and understand the world around us.
A great example of this is maps and measurement.
Suppose you want to measure the length of the border between Portugal and Spain. How long is it? You'd think that that's a straightforward question, wouldn't you?
It's not. Spain and Portugal have a natural border, defined by geography. And in Portugese books, the length of that border has been measured as more than 20% longer…
goodmath | July 11, 2007
The most well-known of the fractals is the infamous Mandelbrot set. It's one of the first
things that was really studied *as a fractal*. It was discovered by Benoit Mandelbrot during his early study of fractals in the context of the complex dynamics of quadratic polynomials the 1980s, and studied in greater detail by Douady and Hubbard in the early to mid-80s.
It's a beautiful
example of what makes fractals so attractive to us: it's got an extremely simple definition; an incredibly complex structure; and it's a rich source of amazing, beautiful images. It's also been glommed onto by an…
goodmath | July 9, 2007
I thought in addition to the graph theory (which I'm enjoying writing, but doesn't seem
to be all that popular), I'd also try doing some writing about fractals. I know pretty
much *nothing* about fractals, but I've wanted to learn about them for a while, and one
of the advantages of having this blog is that it gives me an excuse to learn about things that that interest me so that I can write about them.
Fractals are amazing things. They can be beautiful: everyone has seen beautiful fractal images - like the ones posted by my fellow SBer Karmen. And they're also useful: there are a lot of…
chaoticutopia | July 6, 2007After talking about eagles yesterday and mimicking parrot plumage last week, I decided to stick with a feathery theme for this week’s fractal. To color this layered fractal set, I pulled the hues from the tail of a red-tailed hawk (Buteo jamaicensis) and a few shades from a cloudy sky. Then, I wove them together to create this:
A Fractal Dreamcatcher
I must confess, I think I’m coming down with a cold, so I’m going to leave this fractal as it is, without any explanation. Instead, I’m about to curl up on the couch and try to catch a few dreams of my own.
All fractals made by the author using…
chaoticutopia | June 29, 2007Fractals must lend themselves well to feathery forms. Last time, I used a Mandelbrot set to mimic the soft leaves of the Yarrow plant. Here, I decided to use the same Mandelbrot set to create feathery forms of a different sort:
These varying shades of gray were borrowed not from a plant, but from one of our most intelligent feathered friends:
A close-up view of the feathers under the eye of an African Gray parrot (Psittacus erithacus). Photo by Martin Richard.
What a marvelous creature the African Gray is--not only can it posses an extensive vocabulary, but it displays beautiful…
chaoticutopia | June 15, 2007Here we are, at my 50th Friday Fractal. I have yet to tire of the beautiful spiraling and branching forms of the Mandelbrot set. I've found no shortage of matching forms in nature, either. Even in my own garden, I find lovely fractal shapes, some as delicate as a feather, but as hardy as a weed.
But don't get me wrong... this layered fractal, with a leafy Mandelbrot base, isn't meant to mimic a weed, but one of my favorite plants:
Yarrow (Achillea millefolium 'Moonshine')
I should note, the native yarrow in Colorado is white, not yellow. I picked this one up at the nursery a few years ago…
chaoticutopia | June 8, 2007It seems that cephalopods, from giant squids and octopi to camouflaging cuttlefish, are all the rage these days. As I've shown before, cephalopods can be quite fractalish (or fractals tend to be tentaclish, take your pick.) I'm not exactly sure why these creatures are so loved these days, but who am I to disagree with popular opinion?
So, for this week's fractal, I took two Carlo Julia sets, and colored them with different variations of epsilon crossing. I then layered these atop a Mandelbrot set, and some fBm "plasma". In other words, this fractal includes just about everything and the…
chaoticutopia | June 1, 2007Today, you can create your own fractal. (Don't worry, I'll still include one of my artistic fractals at the end of this post.) You don't need to download any programs, or learn any new techniques. In fact, the only thing that you need is probably already sitting on your desk: a single piece of paper. (It can be any size, fresh or scribbled on; it doesn't really matter.)
Before we begin, take a look at the surface of the paper. How many dimensions does it have? (For this exercise, we're just looking at the surface, disregarding the edges.) Like the image here on your computer screen, it is 2-…
chaoticutopia | May 25, 2007I've been running late again this week, but at least this time, I had a decent excuse. While most of my free time lately has been spent cleaning out closets and other fun springtime tasks, last night, I had a wicked time. That is, I had a chance to see the musical, Wicked, at the Temple-Buell Theater in the Denver Center for the Performing Arts. I adore theater, most especially, the lyrical forms (I was named after an opera, after all.) I thought the play was wonderful, although the musical score seemed a little a dry. (I think it might be based on the theme from M*A*S*H.) The sets and the…
chaoticutopia | May 18, 2007I think I'm going to take this Nova Julia set home, color it with fBm noise, and call it "Find Nemo":
Didn't find him? Try here:
A Percula Clownfish (Amphiprion percula) swimming around in an aquarium
Ok, so, this one is for the kids. (And the grownups, who, like me, couldn't resist watching Finding Nemo once or twice... or a dozen times.) As any child who has seen the movie knows, clown fish live with poisonous sea anemones, which protect them from predators. In the movie, Nemo's father says he "gets stung all the time" and so he's "used to it". However, real clownfish (often known as…
chaoticutopia | May 4, 2007Words cannot quite describe this fractal piece, which is abstract, yet seemingly (and sensually) familiar. As any of my regular readers will know, it isn't too hard for me to find fractals which subtly imitate natural forms. Still, it was quite a surprise to find these anthropomorphic forms lurking in the set. The flesh-like curves appeared as I was feeling around a vortex Julia set, using 3D fractal Brownian motion as a coloring formula. Originally, I was trying to find something that looked like a feather or a leaf. (When I first saw the body, I had it colored green. Talk about abstract.)…
chaoticutopia | April 27, 2007
Tracking wildlife in my neighborhood wetlands this week made me reflect on the complex network of organisms in a habitat. Everything in an ecosystem is so intimately tied together, that a single species can have drastic effects on the entire habitat. The ecosystem, like all systems containing that elusive chaotic aspect, has sensitive dependence on the initial conditions... like a fractal. In the fractal or in nature, one small change can ripple through the entire set. For an example, I took two copies of the classic Mandelbrot set, and laid one atop the other. Then, I used slightly…
chaoticutopia | April 20, 2007The Perfection of Wisdom: A Fractal:
A layered fractal, incorporating a Lissajous curve
(For Melinda)
The Perfection of Wisdom: An 11th Century Illuminated Manuscript:
A page from the illuminated manuscript, Astasähasrikä Prajñäpäramitä (The Perfection of Wisdom)Created on pattra pages in a Tibetan monestary during the late 11th century (Pala age)
And thus Sariputra says:
The perfection of wisdom gives light.
I pay homage to the perfection of wisdom!
She is worthy of homage.
She is a source of light,
She removes darkness, and leads
Away from the blinding darkness
Caused by a…