Fractals: irregularity to beauty
Fractal geometry is the geometry of nature. Everything around us that are not man-made, are fractals. Take yourself for an example. A human body has a symmetry to it. But, Euclidean geometry could never describe it. This was the void in the world of math and science until recently when a way was found to describe such figures. This way was Fractal Geometry.
The discovery of Fractal Geometry by Benoit Mandelbrot dates back to 1975. Well, maybe not discovered, but finally put into words.
There were many Fractals that were discovered and studied, one of them found by Georg Cantor and named after him as the Cantor set. He came across this Fractal (one of the first ones) while working on ‘continuity’. The cantor set was explained as follows:
Take a line, remove the middle third, leaving two equal lines. Likewise remove the middle thirds from each of these two lines. Repeat this process an infinite number of times, and you are left with the Cantor set. When the number of times this process is repeated tends to infinity, a line will be obtained that has zero measure and is nowhere dense. That is, it is a line that is filled with holes. But, it still contains as many points as the line it is carved from. You may not see the line but, it is present. Interesting, don’t you think?
There is another interesting Fractal called the “Space filling curve” by Giuseppe Peano. A line or a simple curve is one dimensional. Peano made a curve such that it was continuous and it filled a particular space completely without leaving any gaps. Now the curve was two dimensional as it filled in the space and it had an area. But what about the curve that is obtained in between these two situations? That curve neither has one nor two dimensions! This was an unpleasant situation for many mathematicians. They denied the existence and possibility of such a curve. This problem of dimensions was resolved by Felix Hausdorff.
Felix Hausdorff led a different take on dimension. Focusing on the manner in which shapes fill the space around them, Hausdorff derived a measure which extends our intuitive ideas of dimensionality. For more complicated shapes, other than normal Euclidean objects, Hausdorff’s approach gave a fractional dimension. This could make dimensions such as 1.5, 1.33, etc, possible.
Fractals have a property of self-similarity. Every fractal contains a smaller copy of its bigger self in the fractal itself. That smaller self again contains a much smaller version of the parent fractal. The ideal Euclidean shapes do not have this property. But, the shapes in nature do. Let’s take an example of a cauliflower to explain this better. When you cut a cauliflower, you obtain a floweret. This looks exactly like the original cauliflower, but, smaller. Then cut out another sub-floweret out of that floweret. This again looks like the original cauliflower! If you continue to do this infinitely, you’ll only get smaller cauliflowers. Thus, a cauliflower is an example of a fractal having the property of self-similarity. You can see this a lot in nature. The leaves, the branches in a tree, etc, all have the property of self-similarity.
In the same amount of given space, Fractals have infinite length. I found this out through a 1961 paper by Richardson that caught my attention. This paper was entitled “How long is the coastline of Britain?”. When you measure a coastline, you are only measuring an approximate outline of the coast, not the real coastline. If you take the real coastline and try to measure it accurately, you will notice that the more you zoom in, the more rough and irregular the boundary gets. Trying to measure an entire coast filled with such complexity finally results in the value of the length of the coastline tending to infinity. After an exhaustive analysis of all the cartographical data available to him, Richardson plotted a graph of his results against the logarithm of the size of the measuring stick (Graph given below). It was observed that the slope of Richardson’s graph was none other than the Hausdorff dimension of the coastline.
There are many more fascinating Fractals like Sierpinski triangle, The fig tree, Julia set, Mandelbrot set, etc, each having interesting specialities and peculiar features. I will be elaborating on these in my next article. All of these variety of fractals and its uses has changed the view of mathematics in the real world. This, thus, changed Fractal Geometry from being called a “Gallery of Monsters” with ‘pathological’ shapes to a “Museum of Science”.
