Fractals, demonstrating infinite complexity from infinite simplicity, showcase nature itself

Fractals, demonstrating infinite complexity from infinite simplicity, showcase nature itself

Seek and ye shall find… but there are fortuitous exceptions. BenoÃ®t Mandelbrot sought nothing, but he found. We aren't being historically accurate here, but essentially, Mandelbrot wrote some code and there appeared on his screen something very interesting. (This was way back in 1976.) He thought there was something wrong with the code, so he double-checked. But there it was, staring him in the face-something like the images on the next two pages! (In black and white, though.) He had discovered fractals-mathematical patterns of unmatched beauty, complex, delirious delights to the eye, stuff that makes you wonder what you've been smoking.

We hate history, so all we'll say is that we're talking about structures arising from extremely simple mathematical formulas. And what good are they? Mostly, they're trippy, fun, and addictive. That's why we're writing this-to encourage you to create your own, then preferably print them out on your T-Shirts. But in addition, they're mathematically significant and philosophically profound: fractals are self-repeating. They carry themselves within them. Some say fractals describe the structure of the universe: where one thing pervades all, where the smallest reflects the large. As Blake put it:

…Infinity in a grain of sand,

And eternity in a flower.

Now you're probably wondering what this writer's been smoking, so without much ado, look on this month's CD to find fractal programs such as Fractal Forge (FF) and Fractint. Neither of these require installation. FF, as well as the DOS version of Fractint, feature colour cycling-very, very trippy-while the Windows version of Fractint doesn't support it in most cases. Fractint is the older, most well-known fractal-generating program; FF is newer, and the UI is more user-friendly with more obvious options.

Creativity 101

Fractals are, generally speaking, so intricately beautiful that it's easy to think you're being creative when you generate a sufficiently complex one!

The fact is, it's nature doing it for you-it's numbers working their magic. Now what's hard is coming across something unique-like a fractal island or river in uncharted territory-which comes only with patience, hard work, and all the other virtues.

Right. So when you fire up, say, Fractal Forge, you get an interesting-looking structure. This is called the Mandelbrot Set. It's a fractal.

Zoom in, and the plot thickens. The pattern gets increasingly complex. Zoom in even more, and you'll notice that "deep inside" is the same shape you saw in the beginning-in fact, lots and lots of them! Tiny little buggers, representative of the whole. Swirling round them are rings, rivers, beaches, islands-all swathed in passionate, psychedelic colour.

Figure 1: An incredibly elaborate spiral (top left), when zoomed into (the white circle), gives more of the same... there's no getting to the centre!

Taking Control

We're not going into the mathematics here, but there are a few things you need to know. First read box How It Happens, then come back. Good. Now, the basic Mandelbrot set is generated without any special functions; it's just squaring and adding. Twists happen when you add functions such as the sine and the cosine to the equation.

However, if you follow the rule of continually squaring and adding, you're still in the Mandelbrot "family" of fractals. There are many more-explore the other sets in, for example, Fractint, which are generated using different formulas. Remember that the key attribute of a fractal is that it is self-repeating.

Figure 2: Hills, islands, rivers, and oceans-the Phoenix set

Happening Places

Now that you're hooked, you need to go out and look for where the interesting patterns are. Let's use the standard Mandelbrot set.

Generally, you need to look at two kinds of things, which are usually at the intersection of two major dark areas:

However, if you follow the rule of continually squaring and adding, you're still in the Mandelbrot "family" of fractals. There are many more-explore the other sets in, for example, Fractint, which are generated using different formulas. Remember that the key attribute of a fractal is that it is self-repeating.

Figure 2: Hills, islands, rivers, and oceans-the Phoenix set

Happening Places

Now that you're hooked, you need to go out and look for where the interesting patterns are. Let's use the standard Mandelbrot set.

Generally, you need to look at two kinds of things, which are usually at the intersection of two major dark areas:

- Places where the energy seems to shoot forth, and
- At where the force field seems to dissipate.

If you've been observing your fractal patterns carefully, then (and only then!) will the above make sense!

When we talk about energy shooting forth, it's in areas like the top left in Figure 3, and when we talk of the field dissipating, it's the circled part in Figure 4. Figure out for yourself where more such areas lie.

The thing is infinitely zoomable, only limited by computing horsepower! So when we say "find out such areas," don't limit yourself to the main fractal-zoom in to totally random areas, zoom in even more, find more and more balls and spirals and such, and try to find the "energy" and the "dissipation"!

Note, though, that you'll have to set high values for the number of iterations as you zoom in further, or you won't get the real picture.

An aside: talking about computing horsepower, back in 1995, on our 386 machines, the screens that now get generated for you in a few seconds would take hours, if not days!

Figure 3: The typical Mandelbrot Shooting Star (upper left), which can be seen in many forms in the Mandelbrot set, shoots into baby Mandelbrots like this one. Note that this is typical of an "explosive" landscape

When we talk about energy shooting forth, it's in areas like the top left in Figure 3, and when we talk of the field dissipating, it's the circled part in Figure 4. Figure out for yourself where more such areas lie.

The thing is infinitely zoomable, only limited by computing horsepower! So when we say "find out such areas," don't limit yourself to the main fractal-zoom in to totally random areas, zoom in even more, find more and more balls and spirals and such, and try to find the "energy" and the "dissipation"!

Note, though, that you'll have to set high values for the number of iterations as you zoom in further, or you won't get the real picture.

An aside: talking about computing horsepower, back in 1995, on our 386 machines, the screens that now get generated for you in a few seconds would take hours, if not days!

Figure 3: The typical Mandelbrot Shooting Star (upper left), which can be seen in many forms in the Mandelbrot set, shoots into baby Mandelbrots like this one. Note that this is typical of an "explosive" landscape

Natural Fractals

We did say that fractals showcase nature. How…? The fact is, nature abounds in fractal formations. Take the Java applet at http://snipurl.com/fract1

(http://math.rice.edu/~lanius/frac/koch/koch.html). There, the second iteration onwards, you'll see what looks like a snowflake. As you keep iterating, you'll find it getting more and more complex… and it becomes self-similar. That is, everywhere along the edges, it looks like the main shape itself. There are only so many iterations you can do with that particular applet, but you'll get the idea.

Figure 4: If you zoom into the areas of what looks like an unmentionable body part, you'll find the landscape is vastly different. Softer, gentler curves. No shooting stars

We did say that fractals showcase nature. How…? The fact is, nature abounds in fractal formations. Take the Java applet at http://snipurl.com/fract1

(http://math.rice.edu/~lanius/frac/koch/koch.html). There, the second iteration onwards, you'll see what looks like a snowflake. As you keep iterating, you'll find it getting more and more complex… and it becomes self-similar. That is, everywhere along the edges, it looks like the main shape itself. There are only so many iterations you can do with that particular applet, but you'll get the idea.

Figure 4: If you zoom into the areas of what looks like an unmentionable body part, you'll find the landscape is vastly different. Softer, gentler curves. No shooting stars

Coastlines, too, are something like this. If you were to measure a coastline with a mile-long ruler, you'd get a certain length.

Make it a foot-long ruler, and you'll see that it's longer than you thought. And so on. Ultimately, you'll find you just can't measure a coastline-it's infinitely long!

That's where the fractal part comes in-coastlines are infinitely detailed.

Mountains, snowflakes, leaves… are all fractals. Look at a leaf: it's got a main line with "branches." These branches have similar branches, and so on-and ultimately, if you examined it under a microscope, you'd probably find the same pattern repeated infinitely.

The above is just a rough introduction, and fractals in nature is an immensely interesting subject.

Visit http://snipurl.com/fract2 (http:// kluge.in-chemnitz.de/documents/fractal/ node2.html), and http://snipurl.com/fract3 (www.miqel.com/fractals_math_patterns/visual-math-natural-fractals.html).

Make it a foot-long ruler, and you'll see that it's longer than you thought. And so on. Ultimately, you'll find you just can't measure a coastline-it's infinitely long!

That's where the fractal part comes in-coastlines are infinitely detailed.

Mountains, snowflakes, leaves… are all fractals. Look at a leaf: it's got a main line with "branches." These branches have similar branches, and so on-and ultimately, if you examined it under a microscope, you'd probably find the same pattern repeated infinitely.

The above is just a rough introduction, and fractals in nature is an immensely interesting subject.

Visit http://snipurl.com/fract2 (http:// kluge.in-chemnitz.de/documents/fractal/ node2.html), and http://snipurl.com/fract3 (www.miqel.com/fractals_math_patterns/visual-math-natural-fractals.html).

How It Happens |

Someone once said the sales of a book would drop by a factor of two for each equation inserted. In that vein, we'll explain how fractals work without using a single equation! However, you'll need to know about imaginary numbers, otherwise you might as well skip this. OK, to ease up a bit, think of the square root of -1. It cannot have a real square root, of course, because no real number, when squared, will give a negative number! The square root of -1 is therefore called "i", standing for "imaginary"; it's a number that's not on the line of real numbers. Now, think of a number plane. "2 3i" represents the point which is 2 units from the origin on the X axis, and 3 units from the origin on the Y axis. See how the concept of an imaginary number leads to a number plane instead of just a number line? And now, take that number and square it. You get (-5 12i), because i2 is -1. Square it again. You get (-119 - 120i)... and so on. If the number runs away to infinity, it's not "part of the fractal set," and is painted white. If the number does not go to infinity, and stays within limits, it's part of the set, and is painted black. Take (0,0): it's obviously part of the set, because no matter how much squaring you do, it'll remain 0. So how do you get colours? A point is marked with a certain colour depending on how long it takes to move to infinity. Light colours are for those that become huge fairly quickly; dark colours are for the ones that take longer to become large. But how many times does one keep squaring? Obviously, one sets a stop point: "we'll iterate 200 times," for example. After the 200 iterations are done, a colour is assigned, based on how far it has moved to infinity (or how close it has stayed to zero). You'll often see the word "Julia" in fractals. This is the "complement" of the main set: there is a Julia set for each point in the plane. A point within the main set will generate a Julia set with only one shape; one outside the set will result in two shapes close to each other. |

Getting Down To It

Now, it's rather pointless telling you what parameters to change and why and how. FF features a comprehensive tutorial. We'll tell you this, though-that seemingly unrelated parameters, when changed at the same time, can lead to a drastically different picture.

Also, just try changing the real and imaginary parts of the "perturbations" in FF a wee bit and be amazed! In other words, just keep playing around...

It's a matter of discovery, and we can't teach you fractal art. Just explore as many regions of the initial fractal as you can.

If you come across something truly unique-such as interesting symmetries-do remember to save them and send them in to us.

Welcome to psychedelia-the natural way!

Now, it's rather pointless telling you what parameters to change and why and how. FF features a comprehensive tutorial. We'll tell you this, though-that seemingly unrelated parameters, when changed at the same time, can lead to a drastically different picture.

Also, just try changing the real and imaginary parts of the "perturbations" in FF a wee bit and be amazed! In other words, just keep playing around...

It's a matter of discovery, and we can't teach you fractal art. Just explore as many regions of the initial fractal as you can.

If you come across something truly unique-such as interesting symmetries-do remember to save them and send them in to us.

Welcome to psychedelia-the natural way!

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