Friends, I urge you to take this moment to ponder the vast beauty of nature.
The more closely you examine objects in nature, the more you will find examples of interesting structures called fractals. A fractal is “a complex geometric pattern exhibiting self-similarity in that small details of its structure viewed at any scale repeat elements of the overall pattern.”[1] Examples of fractals can be found in the structures of fern leaves, shells, and even lightening.
The first mathematics behind fractals didn’t begin to show up until the 17th century. It wasn’t until the work of Benoît Mandelbrot in 1975, however, that mathematical representations of fractals began to take on the now mainstream picture of fractals. The Mandelbrot Set is a special kind of fractal on the complex plane in which the complex polynomial x^2 + c remains bounded after iterations.[2]
Below are examples of the Mandelbrot set I created using python.
The coloring in this ’solar eclipse’ image is representative of the relative speed at which the complex polynomial escapes to infinity. The central black image is the range of points at which the polynomial always remains bounded. As you move away from the set, the colors start bright (slowly escapes to infinity) and change to dark (rapidly escapes to infinity).
This image is created using the same function as the image above except that the coloring pattern is just a cosine function of the red, green, and blue values of the color.
As you can see, mathematical representations of fractals are extremely interesting. In the fast-paced lives we live, we often forget to take the time to stop and look at the real world around us. For this week, I challenge you to stop and spend at least 5 minutes a day just to take a break and really take in the beauty of nature. You will be surprised at what you find!
-Kevin Renfrow
Notes:
[1] - “fractal.” The American Heritage® Science Dictionary. Houghton Mifflin Company. 14 Nov. 2007. <Dictionary.com http://dictionary.reference.com/browse/fractal>.
[2] - Mandelbrot Set Explorer: Mathematical Glossary.
