A fractal is not a simple object with a simple definition. Many people have different ideas as to how to express what a fractal is. The definitions can be divided into two basic types of definitions. There are technical definitions, and more understandable, practical definitions.

In Ian Stewart's book The Problems of Mathematics (p 239), he says "Such objects, possessing detailed structure on many scales of magnification abound in nature; … something worthy of study in its own right. The result is a new breed of geometric figure, called a fractal." John Allen Paulos agrees that a fractal is "…a curve or surface (or a solid or higher-dimensional object) that contains more but similar complexity the closer one looks," in his book Beyond Numeracy (p 84).

The technical definition that Benoit Mandelbrot gives of a fractal is "…a set for which the Hausdorff Besicovich dimension [see below] strictly exceeds the topological dimension." Hausdorff Besicovich, or fractal, dimension is really what separates fractals from other curves or surfaces. I will explain this later in the paper.

Some fractals are self-similar. There is really no disagreement about what a self-similar fractal is. My definition of a self-similar fractal is: a fractal such that if you enlarge a section, this enlarged section is identical to the whole.

The general consensus seems to be that a fractal is an object that is not smooth or simple no matter how closely you look at it. However, Mandelbrot's definition is actually the true official definition. This makes Mandelbrot's the one I prefer to use.

For most objects, you can easily tell how many dimensions it is. A piece of string is one dimensional, a square of paper has two dimensions. With a piece of string, you need two to double it. With a square of paper, you need 4 to double its size in all directions. With a cubic paperweight, you need 8. If you had a four dimensional hypercube, you would need 16. This eventually works out to a formula: C = 2^D This formula is how many copies (C) of a D-dimensional object it takes to double it. It can be further enhanced to C = A^D where A is the extent to which you are enlarging the D-dimensional object. By "extent to which you are enlarging," I mean how much you are multiplying it by in each dimension. To use an A of 3, we need to multiply each dimension of the fractal by 3. We can solve for D, and we get the following. log C = (log A) * D. So, D = log C / log A. With one side of the Koch Snowflake (the Koch Curve), we can easily see that 4 stuck together make something 3 times as large, so A = 3 and C = 4, and apparently, the Koch Curve is 1.26185… dimensional. What Mandelbrot means by topological dimension is that a curve is one dimensional, planar figures are 2-dimensional, and space figures are 3-dimensional. However, the surface of a sphere is considered 2-dimensional, and the surface of a hypersphere would be 3-dimensional. The Koch Curve's 1.26285 dimensions exceed 1, so it is officially a fractal. Apparently, this method of finding fractal dimension only works on self-similar fractals. I will elaborate more on the fractal dimension of the other self-similar fractals when I get to them.

The Koch snowflake is a very interesting fractal. The Koch snowflake is really 3 Koch curves stuck together. The curve can easily be generated by a simple process. You start with a line, and replace the middle third with an equilateral triangle, and then repeat that for each of the 4 new segments, and so on. The snowflake can easily be generated, without a computer, by the following process. It starts as the outline of an equilateral triangle. Each side is then divided into thirds, and the middle section is changed into an outward pointing equilateral triangle. With each order of the fractal, every line undergoes this process. Thusly, with each order, the length of this curve increases to 4/3 of its previous length. However, the area never exceeds a certain number. For example, if you drew a circle around order 1, the snowflake would never grow to be larger than that.

Benoit Mandelbrot created a "square snowflake" fractal, which can be generated by a similar process. It starts as a square, but the second fourth of each side is turned into an inwards-pointing square, and the third fourth of each side is changed into an outward pointing square. With each order, the length of the border doubles, but the area stays the same. This is an interesting example of how a fractal can have infinite length but it obviously has a finite area, as it doesn't change with each order.

The fractal dimension of this is also easy to determine. As with the Koch Snowflake, we are really only interested in one side of it (shown at left). It is important to stress that this is not really a fractal, merely my best pictorial representation. It is impossible to ever really draw a true fractal on a computer, as it can only be detailed down to the size of a pixel.

We can see from the coloring that 8 copies of this fit together to make something 4 times as big. If we plug this into our formula from before, D = log C / log A, with C = 8, and A = 4, we get that this curve is 1.5-dimensional. This exceeds the topological dimension of 1, so it is a fractal.

Another interesting fractal is the Sierpinski Gasket. It is also known as the Sierpinski triangle curve, or Sierpinski triangle. It was created by Waclaw Sierpinski.

The fractal dimension of this fractal can easily be determined. We can see from the coloring that 3 fit together to make something twice as big in every direction. So, using the aforementioned formula, D = log C / log A, D = log 3 / log 2. This comes out to a dimension of 1.585…

However, one might object that this is strictly less than the topological dimension of 2, and as such, it is not a fractal. However, the true Serpinski Gasket is really a curve, as illustrated in the accompanying diagram. It starts as an equilateral triangle, and in the middle of the bottom segment, draw another, upside down, equilateral triangle with sidelengths of half of the base line. Repeat this ad infinitum on every horizontal line, including the half lines left after constructing a triangle in the middle.

Perhaps the most well known fractal is the Mandelbrot set. The Mandelbrot set is on the complex plane, which requires some explaining in and of itself. The real number line, with 0 in the middle, the positive numbers on the right, and negative numbers on the left is important to the complex plane. This serves as the horizontal axis of the complex plane.

The vertical axis is based on the imaginary number i. i is the square root of negative one. The spot on the vertical axis twice as far away as i is 2i. Below the origin is the negative i's. The spot 3 units up and 6 units to the right is 3i + 6. Similarly, the spot 3 down and 2 left is -3i - 2.

The Mandelbrot set is defined as "the set of all complex c such that iterating z -> z^2+c does not go to infinity (starting with z=0)."

In order to draw the Mandelbrot set, one would have to iterate z -> z^2+c infinitely many times. In reality, this is hard to do. However, the more times you iterate this, the sharper and more accurate the final picture is. The formula also says "goes to infinity." The operative definition of goes to infinity is that it is eventually more than any given distance [such as 2,000,000] from 0. It has been determined that if a point is more than 2 units away from the origin, it will "go to infinity". In the colorful pictures of the Mandelbrot set, the artist or programmer chooses a maximum number of iterations. Each point is then iterated up to that many times. The point is colored depending on how many iterations it takes to get out of the circle with radius 2. That's why there is a circle on the graph. All the points outside that circle take 0 iterations to exit the circle (as they are already outside it), and all the points inside it of the same color take 1 iteration to "go to infinity". Sometimes 2 or more consecutive areas will have the same color. The black area in the center is the space that never gets a color, as it never escapes. This is the Mandelbrot set, the area that "does not go to infinity."

The Julia Set

There are many other fractals. Some of them exhibit the interesting properties that fractals can have, and some are very interesting.

This Menger Sponge, if combined with 19 others, makes a larger one three times it's size. This gives it a fractal dimension of log 20 / log 3. This comes out to 2.726…, but, like the Sierpinski gasket, it's topological dimension is actually 2, as it could possibly be made out of lots and lots of paper cutouts.

The Fractal Gasket pictured at right is another interesting 3-dimensional looking fractal. It starts as a tetrahedron, then similar tetrahedrons are removed from each of its faces, and this repeats infinitely. It takes 5 of these to make something twice as big, so its fractal dimension is 2.321…, but it's topological dimension is also 2. Many fractals that require a 3-space to construct have a topological dimension of 2, and many of the curves that are considered to have a fractal dimension of 1 need to be drawn on a 2-dimensional paper.

This Sierpinski Carpet on a black background is another good example of a fractal with a topological dimension of 1 that requires a 2-space to draw. 8 of these make another 3 times as large, its fractal dimension is 1.892.

Fractals exist in this world other than in computers. Some fractals happen naturally, and do not need to be created artificially. Trees, coastlines, and even human biology provide interesting fractals.

Looking at a tree, one can see a large main trunk, with many branches sticking out of it at angles. If we focus in on just one of those branches, we see it looks much the same, with smaller branches sticking out. These have similar twigs, and on the twigs are leaves, which are fractals in and of themselves.

Going along a coastline, we could measure how long it was. If we had to stay within a mile of the edge of the coast, we would get a certain value for the length. If we had to stay within 10 feet of the coast at all times, we would have to follow all the small bumps and dents in the coastline, and we would get a much longer coastline. If a mouse had to stay within 6 inches of the shoreline, the length of this coast would be much longer. An ant that had to alter it's path for each grain of sand could get something many times longer than the original person did who could be a mile away. This fractal coastline is an example of detail at all levels.

The human circulatory system is another natural fractal. The large arteries branch into smaller ones, and those into even smaller ones. At the end, there are capillaries, which are intricate and detailed and complex by themselves.

Fractals really are many places in nature, even where you wouldn't think they would be. So look around you, and maybe you'll see a fractal.

Stewart, Ian. The Problems of Mathematics. New York: Oxford UP, 1987.

Mandelbrot, Benoit B. The Fractal Geometry of Nature. New York: W. H. Freeman and Company, 1977

Paulos, John Allen. Beyond Numeracy. New York: Random House Inc, 1991

Gardner, Martin. Penrose Tiles to Trapdoor Ciphers. New York: W. H. Freeman and Company, 1989

"Free Cloud's Fractal Land" 05/25/00 <http://www.geocities.com/Paris/6976/what.htm> (May 25, 2000).

"Fractal Frequently Asked Questions and Answers" 2/13/95 <http://www.faqs.org/faqs/fractal-faq> (May 25, 2000).

"Iterated Function Systems" 5/20/99 <http://www.agnesscott.edu/aca/depts_prog/info/math/riddle/ifs/ifs.html> (May 25, 2000).

Matthew Warfel. "Fractals" 11/10/98 <http://www.cee.cornell.edu/~mdw/Default.htm> (May 25, 2000).

"Sample Pictures" <http://www.kcsd.k12.pa.us/~projects/fractal/pics.html> (May 25, 2000).

"The Fractory" <http://library.thinkquest.org/3288/fractals.html> (May 25, 2000)