Iteration: An Introduction to the Mandelbrot Set

Barbara J. Ziegenhals and Patricia R. Jorgensen

Historical Notes

The study of fractal geometry is a very recent development in the field of mathematics. Benoit Mandelbrot, a Frenchman born in 1924, followed up on original work by a British scientist, Lewis F. Richardson (1881-1953), to find a way to measure border and coastline lengths accurately. His work showed that these boundaries cannot be measured because the smaller the unit used to measure the boundary, the longer the boundary. It was Mandelbrot who gave fractal geometry its name in the 1970's. The linguistic roots of the word fractal include the Latin fractus and frangere, the origins of which mean "to break." The Mandelbrot Set itself, however, is not a fractal.

The Map of the Mandelbrot Set

The Mandelbrot Set is mapped on the complex plane, where the horizontal axis notes the relative locations of real-valued numbers and the vertical axis notes the relative locations of coefficients of the imaginary number i. On this type of plane, called the c-plane, complex numbers of the form x + i y are plotted. For the student activities included here, an understanding of the c-plane is not necessary. The primary objective of this unit is to imbed a study of the Mandelbrot Set within the context of level-specific topics.

Overview

The process of iteration is essential to understanding the Mandelbrot Set. Iteration evaluates a defining function using an initial domain value called the "seed," and then uses the result of that evaluation as the domain value in the next evaluation of the same function and, as before, this result will be used as the next domain value. This process, carried out repeatedly, is known as iteration.

The major heart-shaped or cardioid region of the Mandelbrot Set and its piggy-back circular regions will be called "bulbs" in this set of exercises. The "mouth" of the cardioid is the dent where the lobes of the heart-shape come together. The "antennae" attached to bulbs lead to junctions from which other antennae radiate, much as spokes radiate from the hub of a wheel. Zooming in at various locations along these antenna-like paths, at depths to infinity, are found infinitely many mini-Mandelbrot Sets, similar in appearance but distinctly different from each other. Each of these shapes has unique characteristics which cannot be found in the same combination anywhere else within the map.

Navigating the Map

BULBS

The cardioid, the Mandelbrot Set's most prominent feature, appears to be symmetric with respect to the real axis; the imaginary axis passes through the cardioid somewhat to the right of "center." Circular bulbs of various sizes are found along the cardioid's perimeter. The cardioid and its individual bulbs can each be characterized by a number corresponding to the mathematical trait called the "period" or "cycle" of the points within the region, but this number cannot be used consistently to identify the relative sizes of the bulbs. In other words, bulbs identified as "3" or "period 3" are not all of the same size. Bulbs of the same period may appear to be similar, but no two are exactly alike.

Each period number is determined by the result of substitution into the equation f(z) = z²+c, where z represents a complex number and c is a constant. Starting with a seed value of z₀, this equation is evaluated and its result z₁ is then substituted into the equation f(z) = z²+c. The new result z₂ is again substituted into the equation, and the process is repeated over and over again. If the process eventually "sticks" at a single z-value, this z-value is called a fixed point and the bulb in which the originally substituted coordinates reside is called a bulb of period, or cycle, 1. If the repeated substitution eventually finds f(z) bouncing back and forth between two fixed points, then z resides in a bulb of period 2. Similar statements could be made for bulbs with periods greater than 2.

Consider a circle with radius r . Identify a seed or initial value for ø₀ such as 180•, and substitute it into the function f(ø) = 2ø. Successive iterations yield values 360•, 720•, 1440•, , all of which are located in the same place on the circle, namely 360•. This location is the single fixed point for this period 1 function. Using the same function with seed ø₀ =90•, the iteration results in values 180•, 360•, 720•, , which eventually is fixed at the same place, 360•. This is another example of a period 1 function. Using a seed value of 120• with the same function, however, produces 240•, 480•, 960•, 1920• , a sequence which simplifies to be 240•, 120•, 240•, 120•, , an example of a function of period 2. A period 3 function would be one that eventually results in the same three answers over and over again.

Each bulb which resides on the perimeter of the main cardioid has a rotation number, a fraction-like expression which identifies the bulb's relative location and its period. Moving around the cardioid's perimeter in a counterclockwise direction, the largest of the smaller bulbs between any two given bulbs has a rotation number with "numerator" equal to the sum of numerators of its two larger neighbors. The rotation number's denominator is formed in the same manner, using the denominators of the larger neighboring bulbs. Numbers in this type of sequence are known as Farey numbers. In this fraction-like number, the numerator represents the bulb's relative location from the mouth; the denominator represents the bulb's period number.

ANTENNAE: JUNCTIONS AND SPOKES

Coming off of each piggy-back bulb in a generally outward direction is an antenna-like path which leads to a junction of paths or to a tip. From each junction radiate a number of paths much like spokes of a wheel, and the number of radiating paths indicates the period of the bulb from which the antenna cluster originated. The originating bulb's rotation number is also revealed in the location of the shortest of the spokes. The antennae which extend from bulbs fall into one of two categories: those which lead to a junction or those which terminate at a tip. Miniature Mandelbrot Sets may be found by zooming in along antennae.

COLORS

Color is often used to illustrate different outcomes when point coordinates are substituted, one at a time, into the equation for the Mandelbrot Set. Because the mathematical outcomes that result from substitutions of the points within the bulb are all the same, the interior of the bulb is generally colored black. Similarly, all the regions that are single-colored identify coordinate pairs which produce mathematical outcomes that are similar to one another but different from the outcomes for points of a different color.

STUDENT ACTIVITIES

Several student activities are included in this unit. This material is suitable for use with students at or beyond the pre-algebra level. Activities involving iteration with real numbers are included in Parts A, B and C, and may be most suited to students at or above the pre-algebra level. Graphing selected regions of the Mandelbrot Set on the real coordinate plane is the topic of Part D, suitable for students of first-year algebra.

RESOURCES

Devaney, Robert L., Chaos, Fractals, and Dynamics. Menlo Park, CA: Addison-Wesley Publishing Co., 1990.

Briggs, John, Turbulent Mirror. New York: Harper & Row, 1989.

Seeds for Iteration

PART A:

"If I Plant It, Will It Grow?"

Objective: Generate data tables for the function

and compare outcomes for different seed values of x .

Complete each of the tables given below. Begin by letting your first value, called the seed, equal 1, noted x₁ = 1. Evaluate f(1) ; then let f(1) become your next x- value, noted f(x₁)= x₂ . Evaluate f(x₂ ), and then let f(x₂)= x_{3 .} Repeat this process of "iteration," using your calculator. Beside each table, describe the behavior of f(x) as the iteration progresses. Graph your f(x) sequence on the number line provided.

Example: Examine the table below prepared for the function .

(E) Iterate (n) x Seed: x = 1 1 1 Describe the behavior of the

2 2 sequence f(x) . _.

3 5 _.

4 26 _.

5 677 _.

Graph of f(x) -values: _< ^{2 5 26} _>

Now, it's your turn. Pro-seed!

(1) Iterate (n) x Seed: x = 1

1 1 Describe the behavior of the

2 4 sequence f(x) . _.

3 19 _.

4 _.

5 _.

Graph of f(x) -values: _< ⁴ _>

(2) Iterate (n) x Seed: x = 1

1 1 Describe the behavior of the

2 2 sequence f(x) . _.

3 4 _.

4 _.

5 _.

Graph of f(x) -values: _< ⁴ _>

(3) Iterate (n) x Seed: x = 1

1 1 Describe the behavior of the

2 -3 sequence f(x) . _.

3 5 _.

4 _.

5 _.

Graph of f(x) -values: _< ^-3 _>

(4) Iterate (n) x Seed: x = 1

1 1 Describe the behavior of the

2 -999 sequence f(x) . _.

3 _.

4 _.

5 _.

Graph of f(x) -values: _<^-999 _>

Caution: The seeds for the following problems are varieties other than "1"!

Your calculator's MODE should be "float" (floating decimal point).

(5) Iterate (n) x Seed: x = 0.5

1 1 Describe the behavior of the

2 0.25 sequence f(x) . _.

3 _.

4 _.

5 _.

Graph of f(x) -values: _< ^0.25 _>

(6) Iterate (n) x Seed: x = 0.2

1 1 Describe the behavior of the

2 0.04 sequence f(x) . _.

3 _.

4 _.

5 _.

Graph of f(x) -values: _< ^0.04 _>

(7) Iterate (n) x Seed: x = 0.1

1 1 Describe the behavior of the

2 0.01 sequence f(x) . _.

3 _.

4 _.

5 _.

Graph of f(x) -values: _< ^0.01 _>

(8) Iterate (n) x Seed: x = 0.3

1 1 Describe the behavior of the

2 0.09 sequence f(x) . _.

3 _.

4 _.

5 _.

Graph of f(x) -values: _< ^0.09 _>

Consider the iterations in problems #1-8 as you answer these questions.

(9) These eight problems illustrate two basic types of iterative behaviors. Briefly describe their differences and give each type a descriptive name.

(10) Create an original function to represent each type of iterative behavior you identified in problem #9, above. Complete a short iteration table and graph each function.

Seeds for Iteration

PART B:

"Hoe, Hoe, Hoe"

Objective: Investigate the "doubling function"

using iteration tables and number lines.

Complete each of the tables given below. Begin with the indicated seed. Use the iteration process to complete the charts. Determine whether f(x) "blows up," "shrinks," or "becomes fixed" as the iteration progresses. Graph your f(x) sequence on the number line provided.

(1) Seed: x = 1 (2) Seed: x = 0.5

Iterate (n) x Iterate (n) x

_{1 1 2(1) = 2 1 0.5 2(0.5) =}

_{2 2 2(2) = 4 2 2( ) =}

_{3 4 2(4) = 3}

_{4 4}

_{5 5}

This function _. This function _.

_{< > < >}

(3) Seed: x = 0.2 (4) Seed: x = 0.

Iterate (n) x Iterate (n) x

_{1 0.2 2( ) = 1 0}

_{2 2}

_{3 3}

_{4 4}

_{5 5}

This function _. This function _.

_{< > < >}

(5) Seed: x = -0.3 (6) Seed: x = 0.125

Iterate (n) x Iterate (n) x

_{1 -0.3 1}

_{2 2}

_{3 3}

_{4 4}

_{5 5}

This function _. This function _.

_{< > < >}

(7) Seed: x = - 0.1 (8) Seed: x = 5.7

Iterate (n) x Iterate (n) x

_{1 1}

_{2 2}

_{3 3}

_{4 4}

_{5 5}

This function _. This function _.

_{< > < >}

(9) Study the sequences that "blew up." What do they have in common? Name ten seeds whose orbits also blow up. Verify your claim for at least one of your selected seeds.

(10) Repeat the work required in problem #9, above, for sequences that "shrink."

(11) Which sequence among the problems in this activity is unlike the others? Identify and explain why it is different.

Seeds for Iteration

PART C:

"Cultivating the Circle"

Objective: Given different seeds, map orbits of the doubling function onto the circle to examine periodic behavior.

^{The Unit Circle}

In the figure above, each point on the circle represents the fraction of the circle's circumference that lies between the point and the circle's starting location. The circle above has center at (0,0) and a radius of 1 unit. All points on the circle may be named according to arc length, measured in a counterclockwise direction, which separates the point from (1,0). The location (0,1) represents ¹/₄ since it lies one-quarter of the way around the circle from the starting point. (-1,0) is half way around the circle; (0,-1) is three-quarters of the way around the circle. One (or ²/₂ or ⁴/₄ or ) complete revolution brings us back to the starting point. For this reason, the starting point is represented by both the numbers 0 and 1.

Now, let's make our new function f(ø) = 2ø by simply letting x be replaced by ø. where ø represents a fractional revolution about the circle.

The typical doubling function The doubling function

Seed: x = 0.5 Seed: ø = ¹/₂

Iterate (n) x Iterate (n) ø f(ø) = 2ø

1 0.5 2(0.5) =1 1 ¹/₂ 2(¹/2) = 1

2 1 2(1) = 2 2 1 2(1) = 2

3 2 2(2) = 4 3 2 2(2) = 4

_{graphs onto the number line. now graphs onto the circle}

Notice the iteration pattern for the circle: 1/2 --> 1 --> 2 --> 4 --> 8 --> 16 --->

Rename the numbers that coincide with the "start": 1/2 --> 1 --> 1 --> 1 --> 1 --> 1 --->

With the first iteration, the function begins at ø = ¹/₂ and takes us half-way around the circle to 1 since f(¹/₂ ) = 1. The next iteration, f(1), yields 2, sending us twice around the circle, again putting us at the location named "1." Then, the next iteration, f(2)=4, sends us around the circle four times, but we again end up at 1! Continued iteration of the function will result in multiple revolutions, always bringing us back to 1. If a function continually revisits the same point or points and never strays from its pattern once it has been established, then those locations are said to be fixed. The period of a function is the number of fixed points with which its orbit is associated. For this reason the orbit of f(ø)=2ø for the seed ø = ¹/₂ is said to be attracted to the single "fixed point" at 1, and the function has a period, or cycle, of 1.

Complete the following iteration tables, tracing the function's orbit on the given circle. List the iteration pattern, and identify the period for each function.

(1) Seed: ø = ¹/₄ (2) Seed: ø = ¹/₁₂

Iterate (n) ø f(ø) = 2ø Iterate (n) ø f(ø) = 2ø

1 ¹/₄ 2(¹/4) = 1 ¹/₁₂ 2(¹/12) =

2 2

3 3

4 4

5 5

6 6

7 7

_{. .}

(3) Seed: ø = ¹/₈ (4) Seed: ø = ¹/₃

Iterate (n) ø f(ø) = 2ø Iterate (n) ø f(ø) = 2ø

1 ¹/₈ 2(¹/8) = 1 ¹/₃ 2(¹/3) =

2 2

3 3

4 4

5 5

6 6

7 7

_{. .}

(5) Seed: ø = ¹/₁₀ (6) Seed: ø = ¹/₅

Iterate (n) ø f(ø) = 2ø Iterate (n) ø f(ø) = 2ø

1 ¹/₁₀ 2(¹/10) = 1 ¹/₅ 2(¹/5) =

2 2

3 3

4 4

5 5

6 6

7 7

_{. .}

(7) Seed: ø = 0

Iterate (n) ø f(ø) = 2ø

1 0 2(0) =

2

3

4

5

_.

(8) Which of the orbits in this exercise were (eventually) fixed?

(9) Which functions' orbits resulted in more than one fixed point?

(10) Do you think it is possible for a function to have 50 fixed points? n fixed points? Why or why not?

(11) Must a function have at least one fixed point? Justify your answer.

(12) Identify and verify at least 3 seed values of ø for which the tripling function f(ø) = 3ø is (eventually) fixed. Determine the period of this function for each of your chosen seeds.

Seeds for Iteration

PART D:

Tending the Bulbs

Objective: Through iteration, determine whether specific points are inside or outside the Mandelbrot Set.

Some functions, under iteration with a particular seed, blow up or shrink. Some are fixed or eventually become fixed. Others which have not been included in these exercises bounce erratically back and forth.

The equation f(z_n+1) = z_n² + c , where z represents a complex number, is the generator for the graph of the Mandelbrot Set.

Each point on the grid represents a seed value for the equation and the fate of its orbit determines whether that seed's location is inside, outside, or on the boundary of the Mandelbrot Set. Seed value locations which produce orbits that are (eventually) fixed reside within the set's boundary and are typically colored black. Seed value locations which produce orbits that blow up are outside the boundary. Seed value locations producing chaotic orbits that do not settle into one of the other two patterns form the boundary.

The Mandelbrot Set can be graphed in the real plane, however, by transforming the complex coordinates into their corresponding real plane partners.

(x, y) in the real plane will be denoted as (x² - y² + a, 2xy + b),

where x = a and y = b.

Selecting seed values for a and b , given the constraints above, iteration will produce an orbit whose behavior will determine where the seed lives --- inside, on the boundary of, or outside the set. Since each seed, or point, must be tested individually, and since there are an infinite number of points on a plane, this process can never be complete. It is possible to get a reasonably accurate picture of the set, however, by using calculators and computers to test a large number of points in a given area.

Use a calculator or computer to produce an iteration table for the following seeds. Draw a conclusion, based on the behavior of the orbit, regarding the point's location with respect to the Mandelbrot Set's graph.

(1) Seed: a = 0; b = 0

Iterate (n) (x, y) (x² - y² + a, 2xy + b), where x=a, y=b.

1 (0, 0) ( , )

2 ( , ) ( , )

3 ( , ) ( , )

4 ( , ) ( , )

(0, 0) lies the boundary of the Mandelbrot Set.

(2) Seed: a = -1; b = 0

Iterate (n) (x, y) (x² - y² + a, 2xy + b), where x=a, y=b.

1 (-1, 0) ( , )

2 ( , ) ( , )

3 ( , ) ( , )

4 ( , ) ( , )

(-1, 0) lies the boundary of the Mandelbrot Set.

(3) Seed: a = 1; b = 1

Iterate (n) (x, y) (x² - y² + a, 2xy + b), where x=a, y=b.

1 (1, 1) ( , )

2 ( , ) ( , )

3 ( , ) ( , )

4 ( , ) ( , )

5 ( , ) ( , )

6 ( , ) ( , )

7 ( , ) ( , )

8 ( , ) ( , )

9 ( , ) ( , )

10 ( , ) ( , )

(1, 1) probably lies the boundary of the Mandelbrot Set.

(4) Investigate the fate of orbits of at least two other seeds, given that -2 < a < 2 and -1.2 < b < 1.2. Justify your conclusion about the location of each tested seed with respect to the Mandelbrot Set.

(5) Design a cooperative activity to investigate the Mandelbrot Set. Include information about the investigation's purpose, how many people will work together on this project, and what activities will be undertaken to complete the task.

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