Independent Assignment. An in-depth study of the infinitely Complex world of Mandelbrot fractals. Patrick Robichaud. Ordered Chaos?. Purpose of This Project. What y ou’ll get out of it.
An in-depth study of the infinitely Complex
world of Mandelbrot fractals.
What you’ll get out of it
This project aims to help one understand fractals, and realize how deep and complex the field of fractals is. I chose the topic of fractals quite lightly, mainly because they are so fascinating and mysterious!
Information on fractals is found all over the internet, and software that renders fractals is also very helpful to explore them. However, now that I’ve completed my research, I realize that understanding the mathematics behind fractals is not for the faint of heart.
I don't pretend to understand fractals completely, but I have learned a great deal about them, and I will convey that information to my readers and teach them how to perform the basic calculations and draw a fractal.
Research on Fractals
In this project, I will be presenting the research I have done on the four main aspects of the vast field of fractals and the underlying chaos theory.
I will be summarizing the history and discovery of fractals; investigating the basic properties, categories, and features of fractals; and delving into more detail concerning the mathematical concepts and chaos theory behind fractals.
Finally, I will build upon this knowledgeby walking you through the steps to confect your very own fractal drawing by hand, with pencil and paper.
Discovery and Features of Fractals
The discovery of fractals was a tedious process that began in the 17th century, but took off in the 20th century, involving many key people. Many important concepts were discovered during this time that helped to understand why some features of fractals are present.
People and Places
The study of fractals and the underlying complex mathematics and Chaos theory goes back to times significantly before the invention of computers and fancy calculators, when computers were people, and mathematics behind it was only being discovered.
At that point in the 17th century, a few key individuals conducted research on self similarity and the geometry and equations that supported would eventually lead to the discovery of fractals.
However, it was not until the 19th century that significant discoveries were made, including the basic properties that make a fractal what it is, and drawings being released.
In the 20th century, however, is when the field began to take shape, especially because of the power of computers, allowing computations to be done much quicker than by hand, and also creating high-resolution drawings that really showed the extent of beauty and detail in fractals.
Research and Study…
The Mandelbrot Set, which is the most well known and in my opinion also the most beautiful fractal, was also the first to be discovered and researched extensively.
In the early 20th century, two men by the names of Pierre Fatou and Gaston Julia investigated the mathematical field of complex dynamics. Later in the century, a crude drawing of the set was drawn by Peter Matelski and Robert Brooks.
Finally in 1980 in the Watson Research center in New York, Benoit Mandelbrot was the first man to see a visualization of the set. He conducted extensive research on the quadratic polynomial aspect of fractals.
In 1985 particularly, the set and its underlying algorithm was popularized by AdrienDouady and John Hubbard. Many others contributed to the study of complex dynamics and abstract mathematics, but the Mandelbrot set has remained the centerpiece.
A fractal is….
A few characteristics are common to all fractals, essentially defining the mathematical field of chaotic structures.
Infinite Zoomable Detail
Power limitations aside, a fractal can theoretically be zoomed in for infinity, without any loss of detail or resolution. Additionally, no two parts will be completely identical.
Zooming into parts of a fractal will exhibit a strange feature called partial or complete self-similarity. A shape can be repeated many times within itself when zooming in.
Here are some more peculiar structures found in fractals, many of which are still not well understood.
Random Chaotic Information
Attractors and Orbits
Despite seeming random in nature, fractals are based entirely on a formula. Quite surprising when looking at the curls and such.
Mysterious shapes known as attractors and orbits, with special features including a center point, are found in fractals.
Geometric fractals have a clearly visible repetitive pattern and a generally uniform, simple geometric shape that does not exhibit chaotic behavior, only self-similarity.
The Seirpinski triangle consists of increasingly small inverted triangles cut out into the initial triangle, touching the border.
The Menger sponge is a block in which increasingly small squares are cut out in the center third of larger blocks.
Chaotic fractals don’t have clearly recognizable patterns, but upon closer inspection reveal self-similarity and
much more complex formations such as orbits.
The Mandelbrot set is a famous fractal that consists of repeating self-similar “circles”, while zooming in towards the left.
The Julia set is a variation of the equation for the Mandelbrot set, which in itself has many variations to speak of.
Chaos Theory is a major aspect in weather, land, and life patterns of creation, and the basis of many phenomena.
The formation of clouds and other weather patterns occurs based on still mysterious and unknown equations. However we can see an element of design.
The intricate patterns found in snowflakes are formed by ice crystals interacting with the upper atmosphere in complex ways, yet each is completely unique.
Electricity in lightning bolts interacts with air, causing it to split and turn to reach the ground in the most efficient manner. This happens at a molecular level.
Plant and animal life in particular also relates to chaos in the way it grows and how it reacts to certain conditions.
Specified information is shown in the growing patterns of leaves such as ferns, both in the direction of growth and the directions of the arteries that feed it.
The intricate patterns formed by river coast-lines may look uniform from far, but the coast extends infinitely when measured microscopic-ally at the sand level.
The spiral shape of certain seashells such as the nautilus models a pattern of squares of increasing sizes, called the Fibonacci spiral.
Scientific phenomena and the human body in particular has structures that exhibit organized formation patterns.
Our whole body can be considered fractal based. A good example is the blood vessels, which spread out to form a vast network.
The frost patterns produced by condensa-tion on cold windows is laid out in patterns that are both information rich and definite.
The pathways that electricity travels when going through solid objects like glass, create consistent pathways within that material.
Before continuing research any further, one must grasp the fundamental concepts that are critical to the field of fractals and chaos theory.
Standard Unit: i
Repeatedly multiplying the number I by itself, adding 1 to the exponent makes the base rotate or alternate between 4 values, I, -1, -I, and 1 (table on next slide).
Standard Form: (a+bi)
Use Pythagorean to find the magnitude of a complex number. Then you can more easily compare them.
Standard Equation: Z1=Z22+C
The iterative equation responsible for the Mandelbrot Set. The direction is shown.
Predicting the Unpredictable
In Lorenz’s experiment, Rounding off an initial computed value to a few decimal places gave a completely different final result after many iterations.
Draw a Cartesian Plane…
Start off by drawing a Cartesian plane with a scale of 0.5, extending to (2, 2i) in all directions. In the center of
the 9×9 grid should be located the origin (0, 0i).
The Distance from the Origin
For a point to belong to the set, the magnitude of Z must never exceed 2. Either it will continue fluctuating between 0 and 2 no matter how many times you iterate, in which case it belongs to the set and can be labeled with the appropriate color.
Otherwise it will eventually expand to infinity after a set number of iterations, in which case it does not belong to the set. The number of iterations is only important if you want to add shading, which shows extra visual detail that would not be visible otherwise.
Select and Solve a Point
Select the top left point (-2, 2i) to determine if it belongs to the set. First determine the magnitude of this number by squaring both the x and y values, adding them together, and finding the square root of their sum.
In this case the magnitude is greater than 2, approximately 2.82. The point does not belong to the set, and the iteration count is 0, since its value increases even before the first pass.
Point (-2, 2i) — Part of the set: No, Iteration count: 1, Color: Red
Second Point Example
Now solve for the center square, which is (0, 0). Substitute the x and y values into the equation, square both, add them together, and square root the result.
Since the initial value is 0, it can be assumed that even after infinite iterations, the Z will never increase beyond 2. Therefore the point is part of the set, and the iteration count is 0.
Point (0, 0i) – part of the set: Yes, Iteration count: 0, Color: Black
Third Point Example
Now solve for the point (-1.5, 0). This one is a bit trickier, since the magnitude is not 0, but it will be less than 2 on the first pass. We will have to repeat the iteration a few times to determine whether or not it is part of the set.
Since the magnitude is between 0 and 2, square the result of the previous iteration (-1.5, 1), by using the multiplication short-cut z2 = (x2-y2, 2xy). Then use the iterative equation again.
Third Point Iteration 2
Now we can simply add the x’s and y’s together, getting a complex number in (x, y) format as the result. Then we simply calculate the magnitude of that number to see if it is greater than 2.
Third Point Iteration 3
At last the point (-1.5, 1) has escaped the Mandelbrot set after a total of 2 iterations. Now that point can be plotted on the Cartesian plane as not part of the set, and the iteration count can also be recorded and used to color it in with the color chosen for that iteration count.
Point (-1.5, 1) — Part of the set: No, Iteration Count: 2, Color: Green
Finishing the Project
These steps can be repeated for any point on the Cartesian plane, no matter how precise it is. To complete this fractal, continue using the equation on each of the 81 points on the grid. As you calculate each point, record the iteration count and whether or not it is part of the set in a neat table, to simplify the task of coloring in the fractal.
In this case it is quite obvious which points escape the set, but it won’t always be that clear. When calculating points at high precision, an iteration limit must be set, which if exceeded, means that it will never reach infinity. This limit is the maximum number of times that the point will be iterated through the equation, after which the point is assumed to never increase. If a point forms a pattern or alternates between two values, it can be assumed that the point will continue the pattern indefinitely.
To Increase or Not To Increase?
Here is an example of two points whose magnitude fluctuates over the course of being iterated through the equation. The red point remains in the set under the threshold of 2, becoming part of the set. The blue point, on the other hand, surpasses 2 after many iterations and increases exponentially towards infinity immediately thereafter, making it not part of the set.
Draw the Results
Once you have calculated every point and determined:
A. whether or not the point belongs to the set, and
B. how many iterations the point has taken to escape,
You are ready to color in the plane. Choose a color for points that belong to the set, preferably black, and then a separate gradient for each quantity of iterations, from at least 0 to 3 iterations. Color in each square with its specific color, and you will have a the final product, a beautiful fractal of the Mandelbrot Set!
You’re on Your Own
If you want to further continue this project, here are some pointers. Draw a new grid with more points and a smaller scale, I recommend 17×17 with a scale of 0.25 or even 21×21 with a scale of 0.2, and solve for each point the using the same method as before. The amount of work increases exponentially in proportion to the amount of points, so be prepared to give up a lot of time!
We Need More Power
A computer has the power to increase resolution to millions of points, and render in a matter of seconds. It can also add colors and allow you to zoom in to nearly infinite detail. it would take you years to do this on your own; that is why I’m providing you with a program to do it for you, which you can download here: