Fractals
chaoticutopia | December 4, 2007The winds are blowing off of the Rockies, hitting the Front Range with brute force. The winds make walking around campus either fun or near impossible, and shake my townhome with enough force to rattle the ornaments on the mantel. The odd thing about the winds is the warmth.... it isn’t the slightest bit chilly. Still, the leaves have fallen from the trees around the school buildings, left to now dance around in the breeze. That shaking mantel is covered in tinsel and lights.... nevermind the warmth; it’s nearly the holidays!
So, here’s an odd assortment of things to do on a windy Tuesday…
chaoticutopia | December 1, 2007Some time ago, I gave instructions for making your own paper fractal. I’ve decided to do another today. This time, instead of ending up with a crumpled paper ball, you’ll have a design worthy of becoming a holiday decoration.
Remember those old paper snowflakes that we would cut as children? We’re going to use the same concept today to create a fractal snowflake.
To make this fractal, you’ll need a piece of white paper, some scissors, a straight edge (a ruler, or even another folded piece of paper) and a compass. The latter two items are actually optional--you could trust your eye instead--…
chaoticutopia | November 18, 2007Sorry for the delay, folks... I’ve been bogged down with homework this past week, and now have a sick kid on my hands. So, the Friday Fractal was bumped to Saturday, and then Sunday. I’ve said it before; I ought to just call these the "Weekend Fractals".
When two people see the same thing, do they necessarily have the same experience? This is a problem which troubles philosophers, particularly philosophers of science. I just finished writing a paper on the subject, comparing the views of several scholars. I won’t post it until it has been graded, but in the meantime, I’m still stuck thinking…
chaoticutopia | November 10, 2007Speaking of unpredictable climate changes, there was always that surprising storm on Jupiter that started brewing last year (and still blows strongly.) I figured now would be as good a time as any to repost the fractal I made in tribute. (This works out especially well, as I didn’t have anything else prepared.)
Pictures released to the media [May 5, 2006] seemed absolutely perfect for the Friday Fractal. A breathtaking example of sensitive dependence on initial conditions, today’s image shows the enigmatic beauty of chaotic patterns. No scientist has yet been able to explain the famous deep…
chaoticutopia | November 3, 2007It’s time to set clocks back an hour again, if you are in an area that practices daylight savings time. I sometimes wish we didn’t use it here in Colorado; it always manages to confuse my schedule somehow. At any rate, I figured I’d honor the turning of the wheel of time, and the changing seasons with an abstract fractal:
Seasonal Cycles and Fractal Concentric Circles
It might not seem too recognizable, but I used the same sort of formula to create this fractal as I did these fractal trees earlier in the year. Since I’ve been messing around with animations lately, it may actually be easier…
chaoticutopia | October 27, 2007It’s getting to be that time of year again, when I pull the severed heads and oversized rats from the crawlspace... a time to deck the halls with ultraviolet, spider webs and witches brooms. My passion for the night of fright even bleeds into my blogging. For instance, there’s the fractal in my banner. It is the exact same Julia set that is normally there, just with a slightly different set of colors. If I were to change the fractal itself, it might come out looking something like this:
But... since I actually posted this fractal as my banner last year, I figured it wouldn’t hurt to come…
chaoticutopia | October 20, 2007After questioning how easily we might create useful models of our environment the other day, I started to wonder if I could even mimic our planet with a fractal. I’ve played around with spherical fractals in the past, for instance, my Paper Ball and my Harvest Moon. As with most of my fractals, I could only come so close. These patterns, as with most fractals, are based on a "seed" number; a number chosen at random. So, after choosing an earth-like palette and setting up the basics, I started trying different seeds. Some produced worlds covered in oceans, others bare and rocky. Sometimes they…
chaoticutopia | October 14, 2007I suppose, if I wanted to make things easier, I would just start calling these the "Weekend Fractal" but it just doesn’t have the same ring. Besides, this week, Carl Zimmer beat me to the Friday Fractal, on naked skin even. (The owner of the fleshy fractal shares some interesting insights on his Julia set; be sure to check it out.) Still, I had this section of a Mandelbrot set lying around, whose autumn hues would be ill fit if posted later in the season:
You can see where this slice fits in to the entire set in this short movie:
     
Here, if…
goodmath | September 29, 2007 As pointed out by a commenter, there are some really surprising places where fractal patterns can
appear. For example, there was a recent post on the Wolfram mathematica blog by the engineer who writes
the unlimited precision integer arithmetic code.
Unlimited precision integers are numbers represented in a variable-length form, so that you can
represent very large numbers by a sequence of smaller numbers. For an overly simplified example, suppose you wanted to represent the number "340123461297834610029346129834761298734612483". One way of doing that
would be as a list of four-digit…
goodmath | September 25, 2007 In the course of the series of posts I've been writing on fractals, several people have either emailed or commented, saying something along the lines of "Yeah, that fractal stuff is cool - but what is it good for? Does it do anything other than make pretty pictures?"
That's a very good question. So today, I'm going to show you an example of a real fractal that
has meaningful applications as a model of real phenomena. It's called the logistic map.
The logistic map is a way of describing the expectations about the size of a population which
is primarily bounded by limited resources. In a…
chaoticutopia | September 23, 2007This week’s fractal has been delayed slightly, to coincide with the Autumnal Equinox. It isn’t your usual Friday Fractal, either.
I was fiddling around this week, thinking about ancient symbols which may have represented some sort of dynamic changes. Were our ancestors fascinated by the relentless cycles of nature on which their lives were so dependant? Changing seasons meant changes in food supply to a hunter-gatherer culture, and thus changes in survival strategies. Rhythmic patterns were the way of life. (Is today really any different?) It stands to reason that some ancient artists would…
chaoticutopia | September 15, 2007Note from your fractalist: Sorry, folks, this one is a day late. I discovered early yesterday that my old website had been hacked. It has been fixed, now, although I plan to eventually remove everything from there, and repost it here somewhere. Just getting the bad scripts out has kept me plenty busy. Never fear, I did finish the Friday Fractal. Other (current) posts are forthcoming. -K
I’m not the only one around here who gets into fractals. I’ve noticed a few other science bloggers occasionally blog on the topic. Mark, over at Good Math, Bad Math, has been working on a series describing the…
chaoticutopia | September 7, 2007Life is complex. The last week has been particularly so for me, but I’d rather not go into details about it. So, I’m keeping this week’s fractal somewhat on the simple side. I suppose in fractals, just like life, simplicity and complexity are often found side by side. I’ve always liked to describe it in terms of waves; complexity rises and falls, almost rhythmically. It is always the edge between that seems most interesting. In a fractal, like today’s Mandelbrot set, simple circles bend into seemingly infinite forms, revealing complex edges. Is life any different? We never seem to notice our…
goodmath | September 6, 2007 When you mention fractals, one of the things that immediately comes to mind for most people
is fractal landscapes. We've all seen amazing images of mountain ranges, planets, lakes, and things
of that sort that were generated by fractals.
Seeing a fractal image of a mountain, like the one in this image (which I found
here via a google image search for "fractal mountain"), I expected to find that
it was based on an extremely complicated fractal. But the amazing thing about fractals is how
complexity emerges from simplicity. The basic process for generating a fractal mountain - and many other…
chaoticutopia | August 31, 2007You might recall a young girl named Alice. Alice liked to contemplate things, such as what would happen if sage advice were ignored. "If you drink much from a bottle marked "poison", it is almost certain to disagree with you, sooner or later," she wisely noted. You don’t have to look into children’s literature, or even for a corrupt pharmacist, in order to find a "bottle" marked "poison". In nature, markings which warn of danger, similar to the crossbones pictured on old-fashioned bottles of toxic substances, are not altogether rare. In animals, and sometimes plants, this trait is known as "…
chaoticutopia | August 24, 2007This fractal is rather basic; it is simply a coloring formula called "Chips are Us". I’m not sure exactly what that means--perhaps that we, like computer chips, can generate complex patterns--but I really liked what it did. After playing with the variables, I tried a number of color themes. For some reason, it looks best in black and white:
A Satin Sun
The pattern reminded me of both a sunflower and rippling fabric, hence the name. Other forms seem to lurk around the edges of this fractal blossom, besides petals and folds. What ever those may be is left to the reader’s interperetation.…
goodmath | August 22, 2007
Aside from the Mandelbrot set, the most famous fractals are the Julia sets. You've almost definitely seen images of the Julias (like the ones scattered through this post), but what you might not have realized is just how closely related the Julia sets are to the Mandelbrot set.
Remember what the mandelbrot set is? I'll run through a very brief refresher, but if you want more details, you can look at my earlier post about it.
Take a simple quadratic function in the complex plane: f(x)=x2+c, where c is a complex constant. If you iterate f, starting with f(0) - f(0), f2(0) = f(f(0)), f3(0)=…
chaoticutopia | August 10, 2007How do you mimic a classic rendition of mollusk, sea, naked flesh, and love with a computer? Stop... I know what you’re thinking. I suppose there are some sites out there that specialize in such a thing, but today, we’re just going to stick with fractals. Oh, don’t look so disappointed. What if I throw in a little 4D action? (We’ll get to the raging sea part in a bit.)
If you look at the fractal below, you’ll see one of my layered fractals. If you casually ignore the morphed Julia set in the background, you’ll notice an odd shape in the middle. This unique form is called a quaternion, a…
goodmath | August 8, 2007
One of the most fundamental properties of fractals that we've mostly avoided so far is the idea of dimension. I mentioned that one of the basic properties of fractals is that their Hausdorff dimension is
larger than their simple topological dimension. But so far, I haven't explained how to figure out the
Hausdorff dimension of a fractal.
When we're talking about fractals, notion of dimension is tricky. There are a variety of different
ways of defining the dimension of a fractal: there's the Hausdorff dimension; the box-counting dimension; the correlation dimension; and a variety of others…
goodmath | August 6, 2007
So, in my last post, I promised to explain how the chaos game is is an attractor for the Sierpinski triangle. It's actually pretty simple. First, though, we'll introduce the idea of an affine transformation. Affine transformations aren't strictly necessary for understanding the Chaos game, but by understanding the Chaos game in terms of affines, it makes it easier to understand
other attractors.
An affine transformation is a simple set of linear equations which has the effect of doing a simple
scaling of any geometric figure put through it.
So, for a two-dimensional affine…