Fractal Geometry

Fractal geometry is a fascinating concept of dimension and shape. After being assigned this project I was recalled to the cookie jar that is on top of the fridge that I sought after as a child. The cookie jar features an image of a chef reaching into a cookie jar that featured the same repeating image of a chef. This particular ceramic piece of art was my first thought about the concept of infinity.

The pioneering genius of fractal geometry, Benoit Mandelbrot, was a highly visual thinker who earned good grades through his outstanding visual representations. He stated "Clouds are not spheres, mountains are not cones, coastlines are not circles, bark is not smooth, nor does lightning travel in straight line," by which he meant that some of the shapes found in nature were not adequately measured by traditional Euclidean geometry. He also believes that fractal geometry is "the geometry of deterministic chaos" and can be used to "describe the geometry of mountains, clouds, and galaxies". Mandelbrot even coined the term fractal in 1975 from the Latin verb fragere, which means 'to break.'

What I find particularly interesting about fractal geometry is that it is so modern, unlike Euclidean geometry which dates back to two thousand years ago. Euclidean geometry is defined by algebraic formulas, while fractals are the result of iterative constructive algorithm. While Euclidean geometry fits man made objects, fractal geometry accommodates objects in nature. A good example of fractal geometry in nature is the fern plant. Observing the fern from across a room, you see large individual fern leafs, but from a yard away it appears a branching limbs, each with it's own extending set of leaves. I had never thought that I could so closely tie such foreign concepts of geometry with something as simple as the structure of a plant in my own living room.

To qualify as a fractal, a shape must have two particular elements, self-symmetry and fractal dimension. If a pattern has self-symmetry, it looks the same at all magnifications. Mandelbrot used a coastline to explain self-symmetry, as there is no difference between a map as made from a satellite from the same area mapped by a person surveying a small portion by foot. The fractal dimension falls between the whole numbers that define standard shapes. If one were to crumple a sheet of notebook paper if would not be not two nor three dimensional, but a fractional dimension of approximately 2 ? . Mandelbrot thus created a new language fit to describe the ornate shapes only found in the complexity of nature.

Researchers have found the fractal dimension of the human lung to be 2.17. The branching bronchial tubes fill up more space than a flat surface, but less than a solid, three-dimensional object. Biological fractals can be found just about anywhere scientists care to look. This implies the idea that instead of separate construction plans, all the information needed to build a complex biological system could be contained in a few basic instructions that are continually repeated.

Fractal geometry is a fascinating concept of the measure of infinity. I can spend hours on end thinking about the delightfully ornate shape that can be produced by fractals or the elaboration of simple patterns to create intricate visual figures

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