Archive for October, 2010

The Beauty of Math: Fractals


Fractals are pictures that are constructed using mathematics that are also quite beautiful.  There are many many fractals in math but I will only describe a few.  Amazingly, it seems that fractals are all around us in the physical world as well, coming up in nature.

Take an equilateral triangle.  Break it into 4 equilateral triangles and shade in the middle one.  Repeat this process with the 3 equilateral triangles left.  Keep going forever.  What do you get?

The Sierpinski Triangle

(Image taken from http://math.bu.edu/DYSYS/chaos-game/node2.html)

This strange shape, called the Sierpinski Triangle, is one of many different fractals.  A fractal is a shape created through this process of doing something over and over again.  Another way to state this is to say that fractals have self-similarity; in other words that you can zoom in on part of the fractal and it is identical to the whole.  For example, one fourth of the Sierpinski triangle is identical to the entire thing!

Another fractal called the Koch snowflake is created just out of simple equilateral triangles as well, this time starting with an equilateral triangle and adding triangles with 1/3 the side length of the original to each side and repeating forever.  The resulting fractal looks like a snowflake!

For an awesome animation of the construction of the Koch snowflake, see this page on Wikipedia.

Fractals are absolutely gorgeous mathematical objects.  The prettiest fractal of all in my opinion is the Mandelbrot Set, which is constructed using advanced math:

These fractals are not just made up for the sake of looking pretty; they actually arise from different places in mathematics.  This is visual evidence for why math is beautiful.

Fractals come up in the study of nature as well.  The structure of leaves, DNA, and even the stars in the universe have been observed to be fractals with the self-similar property!  Do you see it in the fern below?  (Generating using a computer but it certainly resembles an everyday fern):

Fractals are all around us.

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