Fractals are repeated patterns given by some function that, when

magnified, resemble the pattern as a whole. The iteration of this

function makes the entire fractal extremely sensitive to initial

conditions and slight disturbances, a theory known as “the butterfly

effect”. This results in chaos, 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 such beautiful

configurations? Along with self-similarity, fractals have another

unique feature. While the structure has a finite boundary, the segments

of which it is made are infinitely long, and therefore can never be

measured. A sphere viewed from far away may look like a two-

dimensional circle, and even further away, a zero-dimensional point.

These same problems of perspective from different vantage points

account for the other intrinsic property of fractals, called fractal

dimensionality.

In the future, I hope to use a ray

tracing program so as to be able to calculate the fractal dimensions of these setups

myself, and compare the results to the ones given above.

Generates

non-chaotic

trajectories.

Generates

chaotic

trajectories

Geometric optics describes reflection and refraction

of light rays, as well as the resulting image formed

when they come into contact with mirrors or lenses.

By creating certain systems according to these

principles, fractal geometry can be formed in the

reflected patterns. To effectively generate fractal

trajectories, a mirrored sphere, referred to as a Sinai

diffuser, must be introduced to the system 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.

Manipulating Reflection to Generate Fractal Patterns

Natalie Mezzadonna and John Noé

Laser Teaching Center, Department of Physics and Astronomy

http://www.phys.unsw.edu.au/

STAFF/RESEARCH/taylor.ht

ml

Trajectories with and

without a Sinai diffuser:

The report that inspired this project had predicted fractal

dimensions for method 1 and 2 of 1.6 and 1.8

respectively [2]. Although method 2 didn’t produce as

good of a result as it would have using mirrors, the

patterns generated were still visually more complex than

the patterns shown in the first, as theorized.

1. Bernt Wahl, “Exploring Fractals” http://www.wahl.org/fe/HTML_version/link/FE4W/c4.htm

2. B C Scannell, B Van Dusen, and R P Taylor, "An Optical Demonstration of Fractal Geometry“

http://materialscience.uoregon.edu/taylor/fractalfiles/Optical(bridges).pdf

3. Anthony Barcellos, "The Fractal Geometry of Mandelbrot“

http://www.maa.org/sites/default/files/pdf/upload_library/22/Polya/07468342.di020711.02p000

26.pdf

4. Kerri Welch, “A Fractal Topology of Time: Implications for Consciousness and Cosmology”

http://www.academia.edu/245209/A_Fractal_Topology_of_Time_Implications_for_Consciousn

ess_and_Cosmology

Fractals are detailed patterns that are scale invariant, or self-similar on every

scale. When magnified, smaller features reveal higher levels of intricacy and

hidden patterns that resemble the structure as a whole. While the common

conception of a fractal is a computer generated pattern or work of art, fractals

actually take on several different forms. They are ubiquitous in nature, found

in plants, seashells, snowflakes, lightening blots, river networks, coastlines,

and even the structure of the universe.

Even more intriguing, fractals patterns are found not just in the matter

surrounding us, but in the matter that creates us. Some examples are blood

vessels, cells, DNA and the brain. Fractals can also be processes in time such

as weather, epidemics, biological evolution, physiology, fluctuations in

economic systems, turbulence, and inflation, some of which form fractal

structures as well.

These examples show that the above equation is true, since they accurately calculate the

topological dimensions for iterated lines and squares. The same equation can be used to

determine fractal dimensions, which will always exceed the object’s topological

dimension by a fraction of the topology.

Koch Curve:

Two different methods were used to create fractal reflection patterns which

used reflective ornaments to obtain circularly scattered light. Method 1 used

four ornaments instead of a mirrored cube, and method 2 used the setup

initially described. It was predicted that the walls of the cube had a greater

influence on the fractal scaling patterns created than the Sinai diffuser. This

would result in more chaos in the system, producing reflected fractal patterns

with visibly higher complexity, and therefore a larger fractal dimension.

Fractal dimensions denote the complexity

of a fractal, where a higher number

corresponds to a higher level of intricacy.

Since the patterns are too simple to fill an

area, yet too detailed to be described as a

line, fractals can have a non-integer

dimension between 1 and 2, or sometimes

even higher. A dimension D can be

calculated by comparing the number of self-

similar segments n created as the result of

each change of scale s by the equation

n = 1 s = 1/2 n = 4 s = 1/3 n = 9

D = log 𝑛

log1

𝑠

=log 4

log1

1/2

= 2 D = log 𝑛

log1

𝑠

=log 9

log1

1/3

= 2

n = 1 s = 1/2 n = 2 s = 1/3 n = 3

D = log 𝑛

log1

𝑠

=log 2

log1

1/2

= 1 D = log 𝑛

log1

𝑠

=log 3

log1

1/3

= 1

Lines:

Squares:

D = 𝒍𝒐𝒈 𝒏

𝒍𝒐𝒈𝟏

𝒔

D = log 𝑛

log1

𝑠

= log 4

log1

1/3

= 1.262

D = log 𝑛

log1

𝑠

= log 16

log1

1/9

= 1.262

Method 1:

Four reflective ornaments were

stacked in a pyramid formation

without having the mirrored cube

to create reflections off of an

ordered boundary. Red, green and

blue light were shone into three of

the openings, while the fourth

opening was used to observe the

fractal pattern created by the

iterated reflections.

Method 2:

A single reflective ornament was

hung from the top of the cube. The

cube was made from reflective

paper rather than mirrors. Openings

were made in the top 4 corners so

the same colors of light could be

shone in three openings, while the

remaining was used to capture the

reflected pattern. Blood vessels in lungshttp://fractalfoundation.org/OFC/OFC-1-2.html

Pulmonary systemhttp://americancrackpot.blogspot.com/2013/10/chap

ter-5-formally-structured-mind.html

The brainhttp://urbanshakedowns.wordpress.com/2010/12/0

4/the-fractal-patterns-of-nature/

DNA structurehttp://publications.nigms.nih.gov/thenewgenetics/chapter1.h

tml#c1

River networkhttp://world.mathigon.org/Fractals

Nautilus shell Lighteninghttp://originalbeauty.wordpress.com/2009/02/27/fractals-in-nature/

Structure of universehttp://www.mpagarching.mpg.de/galform/

millennium/

Galaxieshttp://fractalfoundation.org/OFC/OFC-1-

7.html

“Where chaos begins, classical science stops. For as long as the world has had

physicists inquiring into the laws of nature, it has suffered a special ignorance

about disorder in the atmosphere, in the turbulent sea, in the fluctuations of

wildlife populations, in the oscillations of the heart and the brain. The irregular

side of nature, the discontinuous and erratic side---these have been puzzles to

science, or worse, monstrosities.” -James Gleick, Chaos: Making a New Science.