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### Drawing a Fractal By Hand

An in-depth study of the infinitely Complex

world of Mandelbrot fractals.

Patrick Robichaud

Ordered Chaos?

Purpose of This Project

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.

Introduction to this Project

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.

History & Discovery of Fractals

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.

Discovery of the Mandelbrot Set

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.

Definition of a Fractal

A fractal is….

- A visual graph on a Cartesian plane of all points belonging to a given iterative equation.
- Forms a complex geometric pattern exhibiting self-similarity at many different scales.
- Increases in detail and resolution infinitely when zooming in to small scales.
- Possesses a non-integer dimension caused by the inclusion of the imaginary number i.
- Points belong to the set if they never increase to infinity (colored according to iteration count), don’t belong if they increase past 2 (colored black).

Characteristics of Fractals

A few characteristics are common to all fractals, essentially defining the mathematical field of chaotic structures.

Partial Self-Similarity

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.

Features of Fractals

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

Geometric fractals have a clearly visible repetitive pattern and a generally uniform, simple geometric shape that does not exhibit chaotic behavior, only self-similarity.

Seirpinski Triangle

Menger Sponge

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

Chaotic fractals don’t have clearly recognizable patterns, but upon closer inspection reveal self-similarity and

much more complex formations such as orbits.

Mandelbrot Set

Julia Set

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.

Fractals in Physics

Chaos Theory is a major aspect in weather, land, and life patterns of creation, and the basis of many phenomena.

Weather Patterns

Lightning Bolts

Snowflakes

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.

Fern Leaves

Coastlines

Conch Shells

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.

Fractals in Science

Scientific phenomena and the human body in particular has structures that exhibit organized formation patterns.

Window Frost

Blood Vessels

Electrical Pathways

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.

Conceptsto Understand

Before continuing research any further, one must grasp the fundamental concepts that are critical to the field of fractals and chaos theory.

- 1. Complex Numbers
- 2. Imaginary Numbers

- 3. Iterative Equations
- 4. The Chaos Theory

What is an Imaginary Number?

Standard Unit: i

- We’ve been taught that the square root of a negative number is impossible, and squaring a negative always gives a positive.
- But there is the possibility for the square root of a negative in many fields of science, therefore the number I was invented. It equals the square root of -1.
- It is a number that cannot mathematically exist, yet it is completely valid in many calculations, including signal processing and quantum mechanics.
- i squared is equal to a negative number 1, and i equals the square root of -1, which is undefined in traditional algebra. Try it on your TI calculator (2ndF, . period, squared).

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).

What are Complex Numbers?

Standard Form: (a+bi)

- Complex numbers are the combination of a real and imaginary value in order to form a “two-dimensional” number plane, representable on a Cartesian plane.
- The real part is represented by a regular number “a”, which defines the x-axis.
- The imaginary part “bi” is a real number multiplied by the imaginary number I, which defines the vertical axis.
- A real number’s magnitude is its absolute value, while an imaginary number’s magnitude is its distance from the origin.

Use Pythagorean to find the magnitude of a complex number. Then you can more easily compare them.

What are Iterative Equations?

Standard Equation: Z1=Z22+C

- An equation which is used to perform the same calculation on a number many times. The result is put in as a factor of the next iteration, and used to produce a new composite result.
- In the case of the Mandelbrot Set, the equation is Z1=Z22+C, where Z1 is the result, Z2 is initially 0 , and C is iterated value. After the first iteration, Z2 is substituted into Z1, then added to C.
- The result can either remain within a range after infinite iterations, belonging to the set, or increase to infinity.

The iterative equation responsible for the Mandelbrot Set. The direction is shown.

What is the Chaos Theory?

Predicting the Unpredictable

- A system with very sensitive dependence on initial conditions. A slight change in the initial system will cause a completely different, unrelated final result.
- Behaviors we consider random actually aren’t, but depend on very complex equations and interaction with the unpredictable environment.
- Natural phenomenon, such as weather, growth of plants, and fluid interactions behave the way they do based on chaos.

In Lorenz’s experiment, Rounding off an initial computed value to a few decimal places gave a completely different final result after many iterations.

By working out equations by hand and drawing

a basic fractal with pencil and paper, you will

learn through firsthand experience how the

previously explained concepts work in

much greater depth and detail.

Step #1: Preparation

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).

Importance of Magnitude

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.

Step #2: Calculation

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

Step #3: Calculation

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

Step #4: Calculation

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.

Step #5: Calculation

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.

Iterative Equation:

Step #6: Calculation

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

Step #7: Processing

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!

Continuing the Project

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!

Taking Advantage of Computers

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:

http://www.chaospro.de/cpro40.exe

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