Name____________________________________

MGMT 2240 Spring2016

Chaos Theory Summary Project

The purpose of studying fractals and chaos in this course is to give students experience with non-linear dynamical systems—mathematical representations of real-life phenomena in

which small changes in initial conditions can lead to large differences in outcomes.

This Chaos Summary project counts as 5% of the course grade.

I. Complete this 14-page handout “MGMT 2240 Project: Introduction to Fractals” by filling in any blanks and constructing the iterations of the Koch Curve, Peano Curve and Sierpinski’s Carpet and finding fractal dimension. A video entitled “2240 Introduction to Fractals handout” is posted to help you complete the handout.

II. Watch the videos “2240 Chaos Theory Part 1” and “2240 Chaos Theory Part 2”.

https://s3.amazonaws.com/connect-prod.mheducation.com/connect/prod/connect_files/1416200163860/2240_ChaosTheory_Part_1.mp4?AWSAccessKeyId=AKIAJCFKFIKXITPCNAJQ&Expires=1470366593&Signature=ggZQIVmj79uajg4RZyId%2BAO%2Fnmk%3D

https://s3.amazonaws.com/connect-prod.mheducation.com/connect/prod/connect_files/1416199054863/2240_ChaosTheory_Part_2.mp4?AWSAccessKeyId=AKIAJCFKFIKXITPCNAJQ&Expires=1470366741&Signature=RxkgBfRq5mY3CkxOAVtl1nDMJLM%3D

III. Read “Chaos Theory Brief History.”

IV. Read “Chaos Theory, Financial Markets, and Global Weirding.”

V. Read “Stock Market Chaos Theory.”

VI. Write a 2-3 page single-spaced summary of what you have learned about chaos and fractals. Your summary should be written in your own words and should include topics such as determinism, Butterfly Effect, iterations, fractal, chaos and self-similarity. Your summary should include at least one original example (from your own experience and observation) illustrating fractal behavior and/or chaos in some aspect of business.

VII. Submit your completed 14-page handout to your instructor. Submit your 2-3 page single-spaced summary through Canvas. This project is worth 5% of your course grade.

Introduction to Fractals

A mathematician named Benoit Mandelbrot noticed that nature exhibited certain patterns that classical geometry couldn’t produce.

"Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line."(Mandelbrot, 1983).

Based on his observations, in 1975 Dr. Mandelbrot defined a fractal as "a rough or fragmented geometric shape that can be split into parts, each of which is (at least approximately) a reduced-size copy of the whole."1 The property that Dr. Mandelbrot noticed is called self-similarity.

Self-similarity means that an object has a certain shape or pattern that is repeated over and over again on a smaller and smaller scale. In mathematical fractals, these patterns repeat infinitely many times. Mathematical fractals are infinitely self-similar; small details of the structure of a mathematical fractal viewed at any scale repeat elements of the overall pattern. Usually, there is some type of rotation or reflection involved in the repetition of the pattern.

Below are a few examples of self-similarity occurring in nature.

( Fern ) ( Frost )

( Romanesco Broccoli )

This project will focus on mathematical fractals that are created by a beginning shape (initiator) and a set of instructions (generator rule) for altering the shape. The set of instructions is repeated (iterated) in an endless recursive process.

2.1 The Koch Curve

The Koch curve is one of the earliest fractals introduced. For the Koch curve, we begin with an initiator that is a line segment. We’ll define the length to be 1.

Iteration 0: Length: 1

The generator rule is as follows: divide the straight line segment into thirds, remove the middle third and replace it with two line segments, each having length equal to one-third of the original line segment, as shown. Note that the two segments inserted form two sides of an equilateral triangle (the base side is removed).

Iteration 1: Length: _________

Repeat the process to form Iteration 2 by applying the generator rule to each line segment found in Iteration 1. Notice that each of the line segments from Iteration 1 is replaced by four line segments, each of which is one-third the length of the line segments in the previous iteration.

Iteration 2: Length: ______________________

Circle the scaled-down copies of Iteration 1 that were created by Iteration 2.

How many copies of Iteration 1 were created? _______________________

Draw Iteration 3:

Iteration 3:

Length: ________________________

Circle the scaled-down copies of Iteration 2 that were created by Iteration 3.

How many copies of Iteration 2 were created? _______________________

Circle the scaled-down copies of Iteration 1 that were created by Iteration 3.

How many copies of Iteration 1 were created? _____________________

Iteration 5 is shown below.

Find the length of Iteration 5: ____________________________________

The Koch curve fractal becomes more and more detailed with each iteration, having more and more sharp corners on a smaller and smaller scale. It becomes difficult to distinguish the changes in further iterations due to the smallness of the scale.

If the process were continued indefinitely, what would be the total length of this curve?

________________________________________________________________

2.2 The Peano Curve

Another fractal, called the Peano curve, is also formed using a line segment as its initiator.

Iteration 0: Length = 1

The generator rule for the Peano curve is the following: divide each line segment into thirds. Construct two squares each having the middle third of the line segment as one side.

Iteration 1: Length of Iteration 1: ______________

Draw Iteration 2 by applying the generator rule to each line segment in Iteration 1. Use a different color for each application of the generator rule on each line segment.

Iteration 2:

How many copies of Iteration 1 were created by Iteration 2? _________________________

What is the length of each of the scaled-down versions of Iteration 1 created? _________________

Find the total length of Iteration 2:_____________

You should have gotten the following:

Iteration 2: Total length of Iteration 2: 32= 9

Iterations of the Peano curve quickly become very laborious to draw. (Try drawing Iteration 3 using Iteration 2 above!)

Notice in the next two iterations that the figure begins to look more and more like a square region and that more and more points inside the square region are being covered by the curve.

Iteration 3: Iteration 4:

Total length of Iteration 3:_______________________ Total length of Iteration 4:________________

It has been proven that this curve, if iterated indefinitely, will completely fill the square region.

2.3 Sierpinski’s Carpet

Fractals can also be created starting with figures such as square regions or triangular regions instead of line segments.

One such fractal is Sierpinski’s Carpet. The initiator for this fractal is a square region.

Iteration 0: Area = 1

The generator rule is the following: Take each square region and divide into nine equal smaller square regions. Remove the middle square region.

Iteration 1:

How many new, smaller square regions were created by the iteration (don’t count the middle region that got removed)? ________________________

What is the area of each of the smaller squares? _____________

What is the total area of Iteration 1? Area = ____________________

Now repeat the generator rule on the eight remaining smaller square regions, dividing each into nine equal square regions and removing the middle square region.

Iteration 2:

How many copies of Iteration 1 were created by Iteration 2? _________________________

What is the area of each of the scaled-down copies of Iteration 1 created by Iteration 2? _______________________________

What is the total area of Iteration 2? ______________________________

Repeat the generator rule to create Iteration 3 of Sierpinski’s Carpet. Use a different color in each iteration to color the removed square regions.

Total area of Iteration 3 = ______________________

After repeated iterations, with each iteration removing more and more center squares of area, the remaining figure is full of holes and the resulting figure contains less and less area.

Question: What would be the total area of Iteration 20? ____________________________________________

What percent of the original area remains after 20 iterations? ______________________________________

If the process were repeated indefinitely, find the area of Sierpinski’s Carpet: ____________________

The resulting fractal image formed after many iterations is shown below. Notice again the self-similarity of the image.

http://commons.wikimedia.org/wiki/File:Sierpinski_carpet.png

Cell phones and other wireless devices use antennas that are shaped like an iteration of Sierpinski’s Carpet. Since the antenna has a similar structure at different scales—self-similarity—it is able to receive wavelengths of varying sizes, eliminating the need for multiple antennae and reducing the overall size of the device.

2.4 ThePeano Curve Revisited

Another property of fractals is that small changes in the generator rule can produce significant changes in the appearance of the resulting fractal.

For example, let’s revisit the Peano curve, still starting with a line segment for the initiator, but this time we will remove the middle third of the original line segment as we generate the iterations.

Iteration 0:

Length = 1

Iteration 1:

Length = _____________

Draw Iteration 2 using the grid provided.

Remove the middle line segment by erasing completely.

Compare this iteration to Iteration 2 of the Peano curve

found previously.

How does the removal of the middle third of each line

segment affect the look of the resulting iterations?

______________________________________________________________

______________________________________________________________

Iteration 2:

Again, the process quickly becomes extremely tedious and time-consuming with further iterations.

Iteration 3: Iteration 4:

The fractal curve, made up of line segments, will eventually become Sierpinski’s Carpet, which was constructed by removing square regions.

Two completely different ways of creating the same fractal: one which removes regions from a two-dimensional object (a square region) and one that adds line segments to a one-dimensional object (a line segment).

2.5 Fractals and Fractal Dimension

Most students are familiar with geometrical definition of dimension known as Euclidean dimension.

Basically, lines and line segments are one-dimensional _________

( 3-dim ) ( 2-dim ) ( 1-dim )square regions and planes are two-dimensional,

and solid cubes and space are three-dimensional.

But what does that mean? What exactly is meant by the dimension of a geometric object?

Consider a square region.

What is the dimension of a square region?

________________________________

If each side of the square region were divided into equal lengths, and these lengths were used to define smaller square regions, how many of those smaller square regions would be needed to fill the original square region?

N = ____________________ when

If each side of the square region were divided into equal lengths, and these lengths were used to define smaller square regions, how many of those smaller square regions would be needed to fill the original square region?

N = __________________________ when

In general, if each side of the square region were divided into equal lengths, and these lengths were used to define smaller square regions, how many of those smaller square regions would be needed to fill the original square region?

N= _________________________________________________ for any given .

What part of the formula above represents the dimension of the square region?

____________________________________________________

Now consider a solid cube.

What is the dimension of a solid cube? _____________________________________

If each edge of the solid cube were divided into equal lengths, and these lengths were used to define smaller solid cubes, how many of those smaller solid cubes would be needed to fill the original solid cube?

N = ______________________________ when

If each edge of the solid cube were divided into equal lengths, and these lengths were used to define smaller solid cubes, how many of those smaller solid cubes would be needed to fill the original solid cube?

N = ______________________________ when.

In general, if each edge of the solid cube is divided into equal lengths, and these lengths were used to define smaller solid cubes, how many of those smaller solid cubes would be needed to fill the original solid cube?

N = ______________________________ for any given .

What part of the formula above represents the dimension of the solid cube?

______________________________________________________________________

In general, the dimension d of a geometric object is related to the values of r and Nby the formula:

__________________________________ where d is the dimension, r is the number of equal line segments each side was divided into, and N is the total number of smaller regions needed to fill the original region.

What would you guess is the dimension of the Koch curve? Why? ____________________________________

_________________________________________________________________________________________________________________

To find the dimension of a fractal such as the Koch curve, the notions of N and r must be adjusted somewhat. The values of and are determined in the following manner:

( Iteration 2 ) ( Iteration 1 )

Compare Iteration 1 to Iteration 2.

How does the length of each line segment in Iteration 2 compare to that of Iteration 1? _________________________________________________________________

Letr be the number of smaller line segments in Iteration 2 needed to make a line segment congruent to those found in Iteration 1.

For the Koch curve, r = _________________________.

Let N be the number of copies of Iteration 1 formed by Iteration 2.

For the Koch curve, N = ________________ .

Substituting these values into the formula involving dimension, what is the relationship among N, r and d for the Koch curve? _____________________________

Solve the equation above for d. d = _________________________________________

In general, can be solved for d by taking the logarithm of both sides of the equation.

= _________________________

This formula gives the dimension for fractals based on the values of N and r.

For the Koch curve, N = _______________ and r = ________________________

Dimension of the Koch curve: _____________________________

Does this surprise you? How close was your guess?

What would you guess is the dimension of Sierpinski’s Carpet? Do you think the dimension will be larger or smaller than the dimension of the Koch curve? Why? _____________________________________________________________________________

_____________________________________________________________________________

To find the dimension of Sierpinski’s Carpet:

Compare Iteration 1 and Iteration 2of Sierpinski’s Carpet to determine the values of and r:

Remember: r is the number of smaller line segments in Iteration 2 needed to make a line segment congruent to those found in Iteration 1 and

N is the number of copies of Iteration 1 formed by Iteration 2.

r = _______________________

N = ______________________

d = _____________________________

Fractal dimension has turned out to be a powerful tool. Now mathematicians are able to measure forms which were previously immeasurable such as mountains, clouds, trees and flowers. Fractal dimension indicates the degree of detail or “crinkliness” in the object and how much space it occupies between the Euclidean dimensions. http://people.math.umass.edu/~mconnors/fractal/dimension/dim.html

References

1. Mandelbrot, B.B. (1982). The Fractal Geometry of Nature. W.H. Freeman and Company. ISBN 0-7167-1186-9.

2. Definition of fractal found at http://www.thefreedictionary.com/fractal

3. Picture of fern fractal (used on first page) found at http://en.wikipedia.org/wiki/File:Fractal_fern_explained.png

4. The software used to draw iterations of the Peano curve can be found at http://www.shodor.org/interactivate/activities/AnotherHilbertCurve/

5. Cynthia Lanius, A Fractals Unit for Elementary and Middle School Students http://math.rice.edu/~lanius/fractals/self.html

6. Photo of Sierpinski Carpet antenna can be found at http://csdt.rpi.edu/african/African_Fractals/applications6.html

7. Discussion of fractal dimension can be found at http://www.vanderbilt.edu/AnS/psychology/cogsci/chaos/workshop/Fractals.html

8. Interactive site for calculating fractal dimension http://www.shodor.org/interactivate/activities/FractalDimensions/