Manipulating Reflection to Generate Fractal Patterns
Natalie Mezzadonna and John Noé
Laser Teaching Center Department of Physics and Astronomy
Stony Brook University
Fractals are repeated patterns given by some function that, when
magnified, resemble the pattern as a whole, a defining property known as
selfsimilarity. 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 1dimensional, yet
too simple to fill an area and be considered 2dimensional [2]. This
analysis
brings forth the idea of a noninteger 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".
