Fractus

In mathematics, a self-similar object is exactly or approximately similar to a part of itself (i.e. the whole has the same shape as one or more of the parts). Many objects in the real world, such as coastlines, are statistically self-similar: parts of them show the same statistical properties at many scales. Self-similarity is a typical property of fractals.

Fractals have always intrigued me since I learned of their existence. I feel this would be hard to explain as I have always known about them. So even as a child I would have experienced the beauty of fractal self-similarity. I feel that this is perhaps how we identify individuality in objects as humans. I perceive fractal symmetry as being one of the most important elements that can be observed in our world.

Self-similarity has important consequences for the design of computer networks, as typical network traffic has self-similar properties. For example, in teletraffic engineering, packet switched data traffic patterns seem to be statistically self-similar. Fractals are an important part of our physical world as well as mathematical and abstract forms. Many parts of our own bodies are fractal in nature, just think of the form of the DNA double helix structure within the nucleus of our own cells.

A fractal is generally "a rough or fragmented geometric-shape that can be split into parts, each of which is (at least approximately) a reduced-size copy of the whole. The roots of fractals within mathematics be traced back to the late 19th Century, the term however was coined by Benoit Mandelbrot in 1975 and was derived from the Latin fractus meaning "broken" or "fractured." A mathematical fractal is based on an equation that undergoes iteration, a form of feedback based on recursion.

Fractals appear similar at all levels of magnification, fractals are often considered to be infinitely complex (in informal terms). Natural objects that approximate fractals to a degree include clouds, mountain ranges, lightning bolts, coastlines, snow flakes, even various vegetables (cauliflower and broccoli). However, not all self-similar objects are fractals—for example, the real line (a straight Euclidean line) is formally self-similar but fails to have other fractal characteristics; for instance, it is regular enough to be described in Euclidean terms.

Trees and ferns are fractal in nature and can be modeled on a computer by using a recursive algorithm This recursive nature is obvious in these examples—a branch from a tree or a fond from a fern is a miniature replica of the whole: not identical, but similar in nature. The connection between fractals and leaves are currently being used to determine how much carbon is contained in trees. This connection is hoped to help determine and solve the environmental issue of carbon emission and control.

Fractals have the ability to capture your imagination and keep you transfixed more so than other forms of art. Fractals have allowed us to view mathematical algorithms and complex occurrences in nature from a different perspective. I chose to review the subject because fractals have a great influence in the natural world. They are one of my favourite art forms, the fact that fractals can be seen in nature, mathematics, and art makes them very unique.

A short video that displays graphic fractals and audio that relies on self-similarity