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. The UL fractal is reached after an infinite number of steps 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. 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. Dr Eugene Gath is a Lecturer in Applied Mathematics. He has research interests in the theory and applications of non-linear dynamics.