Proving Symmetry Properties of Mandelbrot Sets

Overview

Objectives/Goals To prove Symmetry rules found by the iterations of complex equations by utilizing fractal geometry. Methods/Materials Materials: Linux operating system, C code compiler, and knowledge of using Linux and C. Results From the data generated by the programs, I noticed that the symmetry of the image seems to be related to the power of the function. This is pretty interesting because it demands a reason why. The reason why lies in the mathematics. If someone examines the function x^2 or x^3 then that person will notice that for complex numbers, this is a rotation. In the data we saw what happens when we multiply the original function by 2d. The point behind 2d was its relation to the circumference of a circle 2d r. In this case r is assumed to be 1. When simplified, the function doesn#t change at all. It acts like the 2d was 1. The 2d created an effect of #circling# all the way back around to the same starting place. Since the power was 1, the circle wraps around a whole time producing a totally equal effect. It generates a screen full of black. But when the function is raised to the 2nd power, things change. Instead of wrapping all the way around on the 2nd power, it only wraps ½ way around. Then it wraps around again. This produces the two symmetric parts we see when we raise the function to the 2nd power. For the 3rd power, we see 3 symmetric parts. For the 4th power, we see 4 symmetric parts. Conclusions/Discussion I conclude that complex number equations have symmetry when raised to a certain power. This symmetry can be proved both thru observing firsthand through empirical observation and mathematical work. From this project I learned many things including how to create a fractal and the discovery of the more hidden parts behind fractal images. I learned why fractals have a certain pattern.

Competition history

• CSEF 2002 Mathematics & Software · Entry S1211

Resources

• CSEF project summary ↗

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