<> home <> galleries <> essays <> links <> contact me <> update log <>

<> essay hub <> fractal basics <> fractal adventures <> getting started <> are they art? <> rosewatching <> poetry <>

f r a c t a l   a d v e n t u r e s
The art I showcase on my site is made from shapes called fractals. If you’ve been browsing the site for long, you’ve probably begun to wonder just exactly what a fractal is. The question is simple enough… it’s the answer that gets a little complicated. There’s an entire branch of mathematics locked in that answer. But don’t worry—this explanation won’t get mathematical. Read on.
Let’s start with another deceptively simple question. How long is the coastline of Britain? Well, you can get an approximate answer easily enough with a map, a ruler, and a bit of string. But the approximation is just that—imprecise. What if someone were to ask you for the exact length of the British coast? Maybe you’d go out to the beaches with a long tape measure and try to follow the coastline’s contours… however, you’d find that very difficult as well, for no matter how precise you are or how thin your tape measure is, there are always smaller and smaller details to elude you! 

Every peninsula and bay is covered in smaller inlets and promontories, which in turn are themselves thoroughly crinkly.
  

What’s funny about this process is, it continues ad infinitum! And since every repetition makes the measured length a little longer, it can be safely said that in mathematical terms, the coast of Britain is infinitely long!

In terms of conventional mathematics, this is of course absurd—but conventional mathematics has long had trouble describing natural shapes. As chaos theoretician Dr. Benoit Mandelbrot put it in his groundbreaking book The Fractal Geometry of Nature, “Clouds are not spheres, mountains are not cones… nor does lightning travel in a straight line… Nature exhibits not simply a higher degree but an altogether different level of complexity.” Mandelbrot’s book goes on to outline an entire branch of mathematics that does have the ability to describe such shapes. That mathematics is known as fractal geometry.

 
Fractal geometry may seem like an incredibly complicated thing, far above the head of the much-deprecated Everyman—but that’s not the case. Granted, the esoteric depths of the fractal field yawn deeper than a casual amateur can truly fathom, but around the edges there are jewels enough to make anyone gasp in wonder.
In fact, using the coastline example, I’ve already described the basic characteristics of a fractal. Take a look at the crinkly, dented bays, for example—look at how they share the same visual hallmarks on many scales. That is called self-similarity, and it’s the first of four characteristics shared by virtually all fractals. The image at right is an example of self-similarity—see how the big blobby shape repeats itself within its own borders? Absolute self-similarity, such as that of an idealized coastline, results naturally in an infinite length contained in a finite area.
Most fractals also resemble or even describe objects in nature. The fractal coastline is an obvious example. Other objects, such as ferns, rock formations, clouds, and trees, look very fractal; the image at left was generated using only a computer and a fractal algorithm. Still others, such as turbulence and population fluctuation, are not as obviously fractal until you try to describe them using math.
Third, most fractals can be produced through the process of iteration—that is, performing the same operation again and again, recursively. If you want to make a fractal out of a triangle, for example, you might connect the midpoints of its sides and remove the smaller triangle thus formed. You’re left with three small triangles and an empty hole. Do the same thing as before to each of the new triangles. Then do it again, to each of the triangles thus produced. Repeat it ad infinitum and you’re left with a famous fractal called the Sierpinski triangle. A limited iteration of the Sierpinski triangle is shown at right; it only shows four iterations, whereas the "real" Sierpinski triangle would have an infinite number. No computer can actually deal in infinity; thankfully, close approximations are quite enough for all practical purposes. 
The fourth and most important characteristic, however, is even odder than the first three. Instead of being one-dimensional like a line, two-dimensional like a square, or three-dimensional like a cube, every fractal object has a fractional number of dimensions. Not only that, but every fractal has a different number of them! The dimension of a particular fractal can be calculated by looking at the degree of its self-similarity. Take the Sierpinski triangle, for example. The drawing at left makes it apparent that it is made up of three small copies of itself, each one-half the original size. 

From that information we can calculate what’s known as the fractal’s similarity dimension. Dr. Mandelbrot describes it as equal to the logarithm of the number of small copies divided by the logarithm of the size reduction. The Sierpinski triangle is made up of three copies of itself, each one-half the original size; therefore its similarity dimension is the logarithm of 3 over the logarithm of 2, or about 1.584.

Not all fractals actually have such wonky numbers of dimensions as those you’ve looked at so far. One famous fractal, known as the Mandelbrot set, has a border that wiggles so wildly that it is actually two-dimensional! That same border is full of astoundingly beautiful moments: as you zoom in on the edge of the Mandelbrot set, awe-inspiring vista after awe-inspiring vista climbs into view. In fact, it is with scenes from the Mandelbrot set and its cousins, the Julia sets, that much of the art on this site begins.
  
It is amazing, then, to learn that the endless complexity of the Mandelbrot set arises from a very simple rule—the equation znext = z2 + c. When this equation uses zero for the first z and a complex constant for c, the result is not nearly as simple as it would seem. Square z, add c to it, and then feed it into znext; iterate again and again, then graph your results. The wriggling border denotes a Julia set, such as the one seen at left.
If the numerical designation of each point on the graph serves as that point's c-value, the resulting graph is the Mandelbrot set shown at left. Using a different equation produces a different Mandelbrot set; the Mandelbrot set, however, remains the child of the simple z = z2 + c.
   
   
I'm not particularly good at explaining this, but I can easily point you in the direction of people who are. Juan Luis Martinez, of third.apex.to.fractovia, has written a good introduction. There is also a basic explanation on the Infinite Fractal Loop's homepage. Many other sites will explain the concept to you in varying levels of detail, from the very basic to the practical, the highly complex and the intricately mathematical.
   
Image credits:
Fractal fern (Culcita sp.?) © CN and DS of http://www.home.aone.net.au/byzantium/ferns/fractal.html
Coastline photo © Jan Tucker, http://www.aberporth.com/around.htm  
   

<> home <> galleries <> essays <> links <> contact me <> update log <>

Website, web design, and fractal art copyright 2004 admin [at] fractalhaven.com.