Fractals are repeated patterns given by some function that, when magnified, resemble the pattern as a whole, a defining property known as self-similarity. The iteration of this function makes the entire fractal extremely sensitive to the initial conditions ("the butterfly effect"), creating dramatic, unpredictable changes that strongly influence subsequent pattern formation. How could a chaotic mathematical monster of such disorder stay in accordance with the original form, and produce the beautiful configurations ubiquitous in nature found in plants, lightning bolts, snowflakes, and even our own DNA? Not only are fractals visual patterns, they can also be a process in time. For example, a theory known as fractal cosmology suggests that the way matter was distributed in the universe is the result of a fractal process, and that we are so deep within the chaos that we can't see the order beyond it. Studying the mathematics that governs these types of pattern formation can yield new advancements in our knowledge of the world around us.

The goal of this project is to convey the idea of fractals and communicate their emerging connectedness to important fields such as math, science, computer graphics, and even pathology. Fractals are exceptionally complex, exhibiting similar levels of detail to infinite degrees of magnification. The length of a fractal line segment can never be measured exactly due to this endless nature, and is therefore too complex to be 1-dimensional, yet too simple to fill an area and be considered 2-dimensional [2]. This analysis brings forth the idea of a non-integer fractal dimension. We will use an equation that relates the number of new segments to the resulting length of a segment to characterize a body as having 1, 2, or 3 dimensions, and show that the same calculations can be applied to relatively simple fractals [1]. Using reflected light, we will also create much more intricate fractals so as to give a visual interpretation that can be appreciated more aesthetically. Such complex fractals are generated using light rays contained in a controlled system [3]. A reflective sphere enclosed in a mirrored cube acts as a Sinai diffuser so as to circularly scatter the light rays in chaotic trajectories that, like fractals, are sensitive to the initial conditions. When these light rays are reflected from the sphere onto the walls of a well ordered system, the mirrored cube, they are continuously reflected, creating infinite geometrical magnification on smaller and smaller scales. Although the fractal dimension of this type of system is too complicated to easily calculate, this setup gives a striking visual example of the manifestation of a fractal dimension and demonstrates that a higher fractal dimension corresponds to a higher degree of complexity. In the future, we would like to use an advanced ray tracing program such as Cinema 4D to calculate the fractal dimension of these patterns.

References

[1] Glenn Elert, "About Dimension".

[2] Anthony Barcellos, "The Fractal Geometry of Mandelbrot".

[3] B C Scannell, B Van Dusen, and R P Taylor, "An Optical Demonstration of Fractal Geometry".