Question Corner and Discussion Area

Asked by Anuradha (last name unknown) on July 2, 1997:

Please assist me in learning about "fractals." What are fractals? How are they applied in mathematics at secondary school levels and later at higher levels of school or education?The name "fractal" arises from the concept of a fractional dimension. What this means exactly is a difficult to say in simple language and we will simply try to give a feel for what fractals are and what sorts of behaviors they exhibit.

Generally fractals are the result of some which is procedure repeated again and again. One of the most basic examples of a fractal is obtained in the following way. Start off with an equilateral triangle which has sides of length 1. Now on each edge of the triangle, add a new equilateral triangle with sides of length 1/3. Now in the middle of each side of this new shape, add a triangle with sides of length 1/9. Continue this process, each time adding new triangles to each side which are 1/3 the size of the triangles added in the last stage. When you are done (infinitely many steps later!) you have the desired fractal.

* * * * * * * * * * * * ******* ******* * * ----> * * * * * * * * * * * * * * * * * * ******************* ******* ******* * * * * * * * * * * * * * * *** *** *** *** ---> * * ---> . . . * * * * * * (better pictures in * * * * graphical version) * * *** *** *** *** * * * * * * * * *

Now let us take a moment to examine some of the properties of this strange new "shape." The area of this object can be calculated by adding up the area which we added at each stage. It is not difficult to see that this sum is actually a geometric series which converges to some finite area. One can also check that, after adding the new triangles at some stage, the perimeter of the shape is four-thirds what it used to be. Thus after repeating the process n times, the perimeter of our shape is (3) (4/3)^n. After repeating this process infinitely many times to get our fractal, the "perimeter" becomes infinite. Some of you may be uneasy about this last part since it is not really easy to tell how to define the perimeter of such an object. It turns out though that there is a way to define a notion of perimeter for such a shape and that we come to the same conclusion when things are handled with a bit more care.

Another interesting thing to note is that if we magnify our fractal along its edge, it looks the same, no matter how large the magnification is. This is another property typical of fractals.

A more well known fractal, the Mandelbrot set, is a little harder to describe mathematically. It's definition is based on the multiplication of the complex numbers. Start with a complex number z_0. From it define z_1 = (z_0)^2 + z_0. Assuming that we know what z_n is, define z_(n+1) to be (z_n)^2 + z_n. The points in the Mandelbrot set are all those points which stay relatively close to the point 0 + 0i (in the sense that they are always within some fixed distance of 0 + 0i) as we repeat this process. As it turns out, if z_n is ever outside of the circle of radius 2 about the origin for some n, it won't be in the Mandelbrot set. When you see colored pictures of the Mandelbrot set, what you are really seeing are the points outside the set. The color code corresponds to how many iterations it takes to place these points outside this circle of radius 2.

(See the graphical version for a picture)

It turns out that fractals occur frequently in nature. This is often due to how nature performs its own iterations. For instance, as a tree grows, new branches grow from old branches. As these branches mature, they sprout new branches of their own. It is possible to implement a relatively simple algorithm on a computer and have it produce a remarkably accurate rendition of a tree. The shore line of a lake or large body of water also tends to exhibit fractal behavior. One can tell by comparing aerial photos of shore lines to the contours of certain fractals whether or not the shore is natural.

Asked by Jill (last name unknown), student on December 16, 1996:

What can you tell me about the history of fractals?Fractals have their roots in 19th century mathematics. In Jean Perrin,

In 1918 both Fatou and Julia worked on what we would think of as the more standard types of fractals.

One of the most inspiring works however seems to have been
Poincare's
*Vorlesungen uber die Theorie der Automorphen Funktoren* published in 1897
which contained many influential illustrations.
His drawings of hyperbolic tesselations were embellished by M.C. Escher and
made into a form of art which itself could be argued to be closely related
to fractals.

Some of the modern interest in fractals among the general public comes from the computer-assisted work of the IBM fractal project and 20th century mathematicians like Benoit Mandelbrot after whom the Mandelbrot set is named. Pictures of this set are quite fascinating and show surprising features at every level of detail. Until the introduction of the computer, very little from this area of mathematics was in the form of graphics. IBM's fractal project added the pictures to what is inherently an incredibly visual field of mathematics. This added dimension of the field in turn gave rise to new discoveries and incarnations of fractals such as mountains and clouds.

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