Fractals Dustin Tench Math 6120 What is a

Fractals Dustin Tench Math 6120

What is a Fractal? §Examples §Earth – video §Mandelbrot – video §Nature

Different Types §Base-Motif §Mandelbrot Sets §Paper folding

Base Motif

Base Motif

Base Motif

Mandelbrot Sets Take a point z such that z = 0 + 0 i and another point c on the complex plane and make z = z 2 + c. Repeat this many times. If the number does not go to infinity, it belongs to the set and we can mark it. If it goes to infinity, we can color the point depending on how fast it escaped to infinity.

Paper Folding Take a long piece of paper and fold it in half. Unfold it so that the angle between the two halves is 90°. Repeat for each half, then to each quarter.

Paper Folding The more iterations we do, the more complicated the figure gets. The result of doing this an infinite number of times would be called the Dragon Fractal.

Applications § Data Compression § Diffusion § Special Effects

Data Compression Consider this graphic. This graphic is 8 kilobytes. However, the formula that produces the graph, z = z 2 + c, only takes up 7 bytes. That’s a 99. 91% compression. Might be applicable to larger images, even if it requires hundreds of fractals.

Diffusion-Limited Aggregation Link

Special Effects § The biggest use of fractals in everyday live is in computer science. Many image compression schemes use fractal algorithms to compress computer graphics files to less than a quarter of their original size. Fractals were used to create the jagged unfinished edges of the new death star in Return of the Jedi A fractal landscape

Megner Sponge § Take a cube and cut it into 27 smaller cubes. Remove the center cube and the center cube of each side. Repeat this process for each remaining cube. Repeat an infinite number of times. What’s the sum of the lengths of all the edges? What’s the total surface area? What’s the volume?