Weekly #7- Fractals: The Nature of Mathematics

What is a fractal?
Before taking this course I was never formally introduced to the word fractal. It is very interesting that fractals occur frequently within in our daily lives. A fractal is "a never ending pattern." At each step of a fractal there is a repeating pattern. Shown below is an example of a fractal (found on wolfram alpha).

 As you can see each shape follows a pattern within itself and the pattern continues.

Brief History of Fractals
Benoît Mandelbrot came up with the term "fractal" as something "that when divided into parts, each part would be a smaller replica of the whole shape."
There are important fractals found including the Mandelbrot Fractals, Julia Fractals, and Newton Fractals. Below are pictorial examples of each fractal.

Mandelbrot Fractals 

Julia Fractals
 













 Newton Fractals

 

Fractals Found in Nature

Shown below is a typical fractal found in nature; a leaf is pictured below. From this picture I can see that the leaf pattern continues at each stage of the leaf. This is very interesting to see that such simple things in nature can create such an interesting bridge to mathematics.

Below is a photo of a Roman Cauliflower. As you can see the cauliflower is a fractal that is grown in nature. This pattern is has a very unique look; there are multiple branches that form on this Cauliflower that follow the same repeated pattern. Looking at this I am wondering how this vegetable might taste. Whether it tastes good or not, everyone should know how good mathematics is and the amazing connections that mathematics makes up in our world.

Below is a photo of the Grand Canyon. From the photo you can see that the Grand Canyon is also a fractal found in nature.  This is yet another example of a fractal found with our world.

Here is an example of a lightning fractal. As soon as I found this photo I was very curious whether all lightning strikes were considered a fractal. Looking more into it, I was not able to confirm or deny this idea.

In conclusion, fractals take up many forms throughout our daily lives. It is important to remember that this discovery has led to many other ideas relating back to the fractal. Studying fractals can be fun and can also lead to new discoveries.

 

Sources:
http://en.wikipedia.org/wiki/Fractal
http://motiv-designs.com/fractal-history.html
http://genuineaid.com/2009/07/21/romanesco-nutrients-and-benefits/
http://webecoist.momtastic.com/2008/09/07/17-amazing-examples-of-fractals-in-nature/
http://fractalfoundation.org/OFC/OFC-1-5.html