Figure 1: The first step in making the UL fractal.
In the natural world we live in, we see a variety of shapes which are not easily drawn with smooth lines, curves and surfaces. Look around you and observe the branches of a tree, the rippled flow of a turbulent stream, the wispy clouds in a clear blue sky. Each one of these has the same characteristic that, when you look at it up close, you see much the same structure recurring but on a smaller scale. A single bough of a tree, for example, could be envisaged as a smaller copy of the full tree. This property is called self-similarity. For about a century, mathematicians had known about objects called fractals with this property, but it was only in the last decade or so that they attempted to use them to model images in the real world.
There are many popular texts dealing with some well known fractals, but how does one go about designing a fractal which looks close to some given image? To illustrate some of the ideas one uses in doing this, I will show you how to design a fractal based on any set of initials. First of all take out some old-fashioned squared "sum paper" . Represent your initials by shading in squares fully on the sum paper. You should try to use as few squares as possible in doing this. For some letters of the alphabet you may want to use parallelograms, but for simplicity let us stick to squares. Let me use as an example, the initials of this university, namely UL. I can represent the letters UL nicely on my sum paper by shading in the squares as in Figure 1.
Now we find the smallest rectangular box which encloses the shaded initials. For UL, the shape is bounded by a rectangle of width 7 units and in height 3 units on the sum paper. This rectangle is an area of 21 square units of which 12 have been shaded. The next iteration is the first of an infinite number of steps which will lead us to the UL fractal. We take this rectangular box and shrink it down so it exactly matches one of the squares, say the one at the bottom left corner of the U. If the bottom left corner is taken to be the point (0,0), then we can implement this by the scale transformation 
.
This maps all of Figure 1 into the square at the bottom left of Figure 2. We now repeat this 11 more times, mapping the original picture in Figure 1 into each square in turn by means of an affine transformation. For example, to map to the square on top of L, first note that its bottom corner is the point (5,2), so this map is achieved by the affine transformation 
.
In short first we shrink and then we translate.
We now have 12 little ULs making up the big one. The next step is to map Figure 3 onto each of the squares in Figure 1. This will give a UL made up of 12 x 12 even smaller ULs. We can repeat this procedure as often as we like. After n steps, the UL will be composed of 
little ULs. In the limit as 
, we end up with an infinite number of infinitely small UL's. That's the UL fractal!
Fractals have many unusual properties. Look at Figure 1 again. The shaded squares fill 12 square units i.e. 
of the total area of the bounding rectangle. In Figure 3, we have mapped this area onto one square, shrinking it by a factor of 1/21, so it becomes 1/21* units of area. However there are 12 copies of this little UL, so we get () = ()² units of shaded area in Figure 3. At the next step (not drawn) we would find ()³ square units. Now what happens to 
as n gets bigger and bigger? It get smaller and smaller! So in the imagined limit of , we get 
. The UL fractal has no area at all!
This fractal also demonstrates the property of self-similarity, as each little UL is a precise (albeit reduced) copy of the full fractal. However, this is really a mathematical idealisation, as no physical object will continue to have layer after layer of finer structures as our fractal. Eventually, e.g. by the time we reach the level of atoms, the self-similarity will long since have broken down. The image you see in Figure 4 was drawn with a computer and laser printer, with resolution of the order of a few tenths of a millimeter.
At that level of magnification, the image in Figure 4 just appears as a collection of square dots, whereas the actual UL fractal would look like 
perfect copies of itself. Indeed it is this self-similarity which allows us to compute the most unusual feature of fractals, namely the fact that their dimensions are typically not 1, 2 or 3 but something in between. The name fractal was coined from fractional dimension. One can show that the UL fractal has fractal dimension 
. The fact that this is less than 2 explains why its area is zero.
Fractals are just a part of a much wider area of research known as non-linear dynamics. In the University of Limerick, there are several people, including myself, who work on the theory and applications of non-linear dynamics. This is indeed a very fruitful research area as it relates mathematical ideas to the natural world around us, usually with the aid of computers.
