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Visual Math *April 26, 2010*

*Posted by zach in Math, Science & Culture.*

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Look at the mountains, it may look as if it would be impossible to use a mathematical model to show the relationships between all the peaks and valleys. In the 1970’s the mathematician Benoit Mandelbrot discovered a formula that would allow a model to be made for natural objects such as mountains and branching structures such as limbs off a tree. Fractals use geometric shapes that can continually be split into smaller fragments. With an infinite number of points both real and imaginary art can be made using math. Lets look at all the cool aspects of visual math fractals opened up.

**Fractal showing the surface of the mountains.**

**Many many iterations**

This first image is of a fractal with only five iterations. In the second image is the number of iterations when it is increased to 75, giving the image a much greater level of complexity.

The image below explores the edges around a Mandelbrot set.

**Now lets explore how fractals can been shown in plants. **

The branching structures of the California Oak tree and the smaller sprouts keep coming off the Romanesco Broccoli. The fractal concept is displayed in the broccoli when one closely examines how the whole is made up of multiple, smaller, repeating parts.

**Patterns that resemble fractals are not limited to the biological world.**

Fractals can also be seen in architecture, this is most apparent in European Gothic Cathedrals. The image below has a central spire with multiple other spires, repeatedly surrounding the central one, in varying sizes.

**Snowflakes**

Kock snowflake starts with an equilateral triangle, each iteration replaces the middle thrid of the line segment, with a part of line segments that form an equilateral bump.

**Sierpinski triangle**

Cool!