A Definition of Fractal Geometry. The Mathematics of Self-Similarity and Infinite Shape. One of the interesting developments of modern mathematics is the exploration of new forms of geometry which do not adhere to “standard” definitions.
Fractal geometry is, like all other forms of geometry, the mathematics of size, shape and special relationships. Where it differs from standard (that is, Euclidean) geometry, is that it deals with shapes that are, by definition, infinitely irregular.
While mathematicians possess many complicated computational methods to define a true fractal, they are really not that simple to comprehend (as opposed to many other complex mathematical ideas). A fractal is a complex shape which, when viewed in finer and finer detail, shows itself to be constructed of ever smaller parts, similar to the original.
Sound complicated? A simple analogy should fix this.
The Shoreline Analogy
The most common such familiar analogy that is used in explanations of fractals, is that of the measurement of a shoreline.
Imagine being tasked to measure the length of a coastline, such as the western coast of the united states. Perhaps the simplest place to begin such a task would be to purchase an accurate, scale map of the coast, then use the given scale and a piece of string to get a good guess as to the overall length of the coast from the boarder of Mexico to the border of Canada.
Realizing that the size of the map itself may have been limiting the accuracy however, the next step would be to maybe purchase a larger map, or a series of large maps detailing individual regions of the coast. Using these, one would find details that were not evident on the larger map – bays, inlets, peninsulas and the like. Taking all of these into consideration, the length of the coastline nearly doubles in size!
Still, however, the measurement could be more accurate.
The next best thing might be to walk the entire length of the coast (for the truly committed measurer only). Doing this, the gentle curves of the coast, the individual rock formations, and many other small nuances can be taken into account, and the length doubles again.
Not good enough.
Take a microscope and begin to measure the coastline pebble by pebble as the water curves in and out of these spaces… and suddenly one realizes that this will never end. The closer one looks at this coastline, the longer it will get, to the point where individual atoms are being taken into account!
History of Fractals
The coastline analogy exemplifies an important type of fractal – though surely mathematicians don’t linger on such “story problems” for long. There are important mathematical considerations to fractals that can’t be explained using mere stories (unfortunate for the mathematically ill-equipped).
The first example of such mathematics may very well have been a function developed by Karl Weierstrass in 1872. This function, when displayed on a graph, showed a series of jagged lines at first, then as one refined the variables even further, zooming in on any area, that portion of the graph would also show this same pattern. When one zoomed in on that pattern, an even finer version of the same pattern was displayed. This could be repeated infinitely, to finer and finer levels – a feature which is known as self-similarity.
As other mathematicians began to work with similar problems and find other “continuous and non-differentiable” (a fancy way of saying “fractal”) functions in mathematical language, it also began to become clear that, as in the coastline analogy, fractals appeared very commonly in nature itself, and thus the mathematics which define them very well might allow one to better understand the world around them.
Clouds, snowflakes, mountains, ferns and many other naturally-occurring entities demonstrate fractals to mathematicians. At the same time, though, many fractal examples are simply fabrications of the clever minds of mathematicians.
Fractal geometry has aided in the continuing development of smarter computers and in the study of potential “cellular automata” (machines which can build copies of themselves – fodder for science fiction writers everywhere), among countless other technological advancements.
Seismologists are attempting to use fractals in better describing the constitution of the Earth and the distribution of earthquakes, digital artists and video game designers incorporate fractals into graphic codes in order to create random and more “organic” environments, and chaos theorists use fractals to help them understand the movement of chaotic fluids.
And this is only the beginning. There are great steps still to be taken in fractal geometry. It is a truly exciting field of mathematical study.