Nothing Finer than Living in North Caroliner

Nothing Finer than Living in North Caroliner
Blue Ridge Smoky Mountains

Tuesday, September 20, 2011

Day 248 - Fractals - Self-Similarity

What is a fractal?  A fractal is "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," a property called self-similarity. Roots of the idea of fractals go back to the 17th century, while mathematically rigorous treatment of fractals can be traced back to functions studied by Karl Weierstrass, Georg Cantor and Felix Hausdorff a century later in studying functions that were continuous but not differentiable; however, the term fractal was coined by BenoĆ®t 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. There are several examples of fractals, which are defined as portraying exact self-similarity, quasi self-similarity, or statistical self-similarity. While fractals are a mathematical construct, they are found in nature, which has led to their inclusion in artwork. 

Mondelbret's fractal
Click on the link at the beginning of the blog to understand the full mathematical world we live in.  It is truly fascinating.  If Mondelbret hadn't discovered fractals in nature and eventually mountains, our simulators and video games would not have the "real" look to them. 

Approximate fractals are easily found in nature. These objects display self-similar structure over an extended, but finite, scale range. Examples include clouds, river networks, fault lines, mountain ranges, craters, snow flakes, crystals, lightning, cauliflower or broccoli, and systems of blood vesselspulmonary vessels, and ocean waves. DNA and heartbeat[12] can be analyzed as fractals. Even coastlines may be loosely considered fractal in nature. and

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 frond from a fern is a miniature replica of the whole: not identical, but similar in nature. The connection between fractals and leaves is currently being used to determine how much carbon is contained in trees.
In 1999, certain self similar fractal shapes were shown to have a property of "frequency invariance"—the same electromagnetic properties no matter what the frequency—from Maxwell's equations (see fractal antenna).

I'm thinking now that fractals are also in genetic offspring.  Each of our kids has a portion of ourselves.  Depending on which part of ourselves each child gets, it can be a blessing or a vice.  If we look closer, and closer and closer, and closer, our children our a fractal of ourselves.  There are fractals in our eye movement to take in information.  They are in everything and we are just now starting to understand them.  I guess this fascination has to do with the fact that I have a little fractal of my father in myself....the love of math.  Glad I didn't get the "crazy" fractal from him too! Wouldn't want to keep up the tradition of throwing batteries at drivers on the highway.  

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