Fractals
goodmath | October 20, 2009
So I'm trying to ease back into the chaos theory posts. I thought that one good
way of doing that was to take a look at one of the class chaos examples, which
demonstrates just how simple a chaotic system can be. It really doesn't take much
at all to push a system from being nice and smoothly predictable to being completely
crazy.
This example comes from mathematical biology, and it generates a
graph commonly known as the logistical map. The question behind
the graph is, how can I predict what the stable population of a particular species will be over time?
If there was an unlimited…
goodmath | July 13, 2009
Sorry for the slowness of the blog; I fell behind in writing my book, which is on a rather strict schedule, and until I got close to catching up, I didn't have time to do the research
necessary to write the next chaos article. (And no one has sent me any particularly
interesting bad math, so I haven't had anything to use for a quick rip.)
Anyway... Where we left off last was talking about attractors. The natural question
is, why do we really care about attractors when we're talking about chaos? That's a question
which has two different answers.
First, attractors provide an interesting way…
chaoticutopia | January 21, 2009Here is one final thanks to all of my readers, especially for all of your wonderful comments over the last week. I know I'll miss blogging here. In fact, I doubt I'll be able to stay gone for too long. I promise, (barring any truly chaotic circumstances) that I will return. I may start a new blog then, since I was ready to change themes anyways. I'll find a way to let you know when I return to blogging, even if it isn't under the same "Chaotic Utopia" banner.
In the meantime, you can find me in any of the following places:
Facebook (though I warn you, I'm just barely starting it--it'll be a…
chaoticutopia | October 11, 2008Barbed wire stretched across the prairie, the slaughter of animals (and people)... the founding of the old wild west wasn’t so beautiful. Despite that, of the classes I’m taking this fall, titled "Don’t Fence Me In", has gotten me to take a closer look at the Western genre in literature and film, and the commentaries they have made on our history. So, I’ve been turning back to the history of Church Ranch (my own pet wild west story,) watching Clint Eastwood films, and reading books by Cormac McCarthy. The latter really caught my interest... about halfway through Blood Meridian, (which is…
chaoticutopia | September 26, 2008Ok, I’ll confess--school has been blowing my f*$#!^& mind this past week--but in a good way, really. I’m studying a variety of interesting subjects which are all tying together in surprising ways... which has lead me to do quite a bit of extra reading and writing. Not that I mind. I just wish I had time to wrap my head around it all, enough to make coherent blogs posts out of these ideas. I did end up making a rather intense fractal, abstract yet full of intense symbolic imagery. But I looked at it, and thought, ew, that’s a little too dark. It will, at least, take some explaining... it…
chaoticutopia | September 13, 2008After blogging about my experiences earlier this week, I had crows on the brain. So, as I went drifting through fractal sets, trying to come up with something for this week’s Friday Fractal (now a Saturday Set), I kept running across crow-like shapes. One in particular stuck out, perhaps because I found it while adjusting the parameters of a set labeled "flight". It seemed like a crow caught in a superposition of states... one of Schroedinger’s crows, perhaps. Is it a perched crow, glaring down at its foe? Or is it a crow in flight, wings beating furiously against the wind? Before you decide…
chaoticutopia | August 15, 2008Summer around here must be coming to a close. The temperature has dropped and the kids are going back to school. Strangely, these circumstances have gotten me thinking more about the sun, rather than less. I blame my son... he came home after one of his first days at school talking about life on other planets. (Apparently, this week, he is aspiring to be an exobiologist. Previously, he wanted to be a pirate.) He asked what planets might have life and what they looked like, so I pulled out a book titled "Empire of the Sun: Planets and Moons of the Solar System" and we began to leaf through our…
chaoticutopia | July 11, 2008Finally, after months of painting, planning, organizing, and flinging, my home is starting to look like a home. I’m down to the little finishing touches: trim and accessories, windows and doors. Some of these just need to be purchased and installed, so, thankfully, this means I can start getting back to normal... cooking, filing, writing, e-mailing, monitoring comments and even (gasp) blogging!
To avoid burnout, I’m returning to blogging on a gradual basis, starting with some interesting but pointless light blogging. Some of it (like this post) might be related to my remodeling, but I…
chaoticutopia | June 13, 2008I mentioned in my last post (a few weeks ago) that I was taking a hiatus from blogging to work on my overgrown home remodel. I’d like to say I’m finished: "woohoo! All done, come and see!" but I’m not. I am, however, getting close. Close enough to start thinking about what type of art I’m going to hang on my newly painted walls. Why not fractals?
A few years ago, I would never have considered hanging a fractal on the wall. Sure, maybe if you have a college dorm room with tie-dyed sheets and Jimi Hendrix posters... but in a family home? Then again, a few years ago, my family strongly disagreed…
chaoticutopia | May 9, 2008Maybe it’s because it’s nearly summer. Maybe it’s because I finally finished my finals. Maybe it’s all a result of coincidence and random web browsing. Whatever the reason, I feel like filling my blog with warm, happy colors this weekend. What better place to begin, than with a fractal?
A Fractal Sunset(look below to see it "set")
This is a combination of sets, each transformed to simulate ripples on a lake. One (the mountain range) is a Julia set, while the other (the sun) is the modified interior of a Mandelbrot set. Usually, when you see a Mandelbrot set or Julia set you see busy spirals…
chaoticutopia | April 20, 2008For some of my friends, today is a holiday. So, to celebrate, here’s a fractal:
Fractal 420(Click the image for a desktop sized version, or here for an alternate version.)
Note: If you’re wondering where I’ve been, or why this post is so brief, don’t worry. I’ve just been having some issues with my keyboard (it keeps typing the wrong things or jumping a "page up") which makes any writing a serious pain in the wrist. I’m going to try to clean my keyboard out with canned air later today, or, if all else fails, I’ll just buy a new one.
Fractal created by the author using ChaosPro.
chaoticutopia | April 11, 2008What if you could escape this busy world, rise above the clouds, and see everything from a new perspective? From that astronauts-eye-view, you could see the greenhouse effect in action: Sunlight pouring in, some reflected off of clouds in the upper layers of the atmosphere, some filtering down below. The light that does manage to reach the earth is absorbed or reflected by the surface below. That reflected earth shine bounces off the clouds as well, in colors imperceptible to you or I. Would it be an alien sight? Colors we can’t see, our homes obstructed by that foggy greenhouse roof, with…
chaoticutopia | February 29, 2008
Leap through Labor to Leap through Labor to Leap
I created a bit of ambiguous poetry recently, and I simply couldn’t resist trying another. And well, it is Leap Day, after all... which is also my birthday, so it seems like a good time for a "free choice" fractal. That said, I couldn’t bring myself to simply post the art and line. I’d fidget all weekend wondering if anyone understood it. Even if I do explain, it still may not make any sense.
When I initially started to create this fractal (from a formula ambiguously named Andrextrandom) I called it "Flight", for several reasons. Most…
chaoticutopia | February 10, 2008This weekend’s fractal isn’t very late, when you consider its roots are found 50 million years in the past. The earth was a bit different then... think of it as a post-global-warming world, where greenhouse gasses have already run amok, driving global temperatures. The poles are virtually frost-free, and rainforests reach far into the northern latitudes. Some of our familiar coastline is entirely underwater, including Florida, Louisiana, and a large swath of Texas. On the other side, most of California and Oregon isn’t quite done yet... large chunks of it are still sliding up from Mexico. The…
chaoticutopia | February 1, 2008I figured I'd post this fractal set while it is still Friday somewhere (here in Colorado, for instance.) My thoughts on it follow below.
Flying in Vapor: A poem and a fractal for riding the waves
I know what it's like
To be down in the water
And tossed by the waves
Each day, crashing into the next
Pulling me, seething, frothing
Falling
Exhilarating ride through time
Through the tumbling surf
Dance on the shimmering crests
Only then to be pushed beneath
Into the surrounding, suffocating
Cold
Plunging and diving
Grows wearisome after a while
And so I'll ride above
Flying, so to speak…
chaoticutopia | January 18, 2008 The following fractal is a tribute to our new overlords, Dendroctonus ponderosae.
Lindenmayer Trees and Fractal Brownian Motion
I wrote about the role of pine beetles in Colorado’s future last week. The conclusion of that piece: a slight rise in temperatures means a strong advantage for pine beetles, which will be able to decimate Colorado’s vast lodgepole pine forests, thus increasing fire danger and erosion, not to mention damaging our tourism. Well, our annual aerial survey of the forests was completed recently, and the news isn’t good:
The growth of the beetle epidemic affecting…
chaoticutopia | January 13, 2008Something about climate change makes people want to argue. Take Greenland, for instance. A few weeks ago, I posted a photo essay about the recent acceleration of melting in the Greenland Ice Sheet. Not only is the entry is still getting comments, but it also spurred a discussion on a political message forum that went on for six pages. Watching all these opinions fly, there were a number of times that I wanted to dive in and start defending science... but I’m a hermit, and it is more fun to lurk and watch. Besides, every point that I wanted to bring up can be found in a paper I’ve been waiting…
chaoticutopia | December 23, 2007Here’s wishing you the very best this holiday season, with a fractal Christmas tree:
Now, wait.... shouldn’t the lights on the tree be blinking?
Merry Christmas!
Update: A few people have expressed interest in decorating their own pages with this tree. (It certainly is easy to put up!) For ease of decoration, here is an animated gif. Thanks to Alan for putting it together for me!
Regular posting (whatever that may be) will resume Wendsday. All fractals created by the author using ChaosPro.
chaoticutopia | December 12, 2007
Are you looking to tell someone to "adapt" this Christmas? Or perhaps you’ve been seeking the perfect gift for the nerd in your life, one that merges science and art. In either case, you’re in luck.
The Chaotic Utopia Curio Shop has just opened, after much anticipation. Just in time for the holidays, the shop (via cafe press) has an odd variety of gifts in addition to the Adapt Fish stickers. There are fractal t-shirts, jewelry boxes, clocks, and even an "I see fractals in my coffee" mug.
So, wether you’re looking for last-minute stocking-stuffers or just like looking at fractal art, come…
chaoticutopia | December 7, 2007Is it just me, or are the holidays getting more and more high tech and environmentally friendly? Maybe it is me. I haven’t used real pine to decorate for the holidays in years. There may be some bits among the potpourri in the centerpiece, but just about everything else has been replaced with paper or plastic versions of the real thing: the tree, the garlands, and the wreath. This year, I’ve been seeking better ways to do the holiday thing. For instance, my tree and garlands now are strung with LED lights. (The odd "lunar white" glow was a bit disconcerting at first--they seem to flicker with…