The following discussions and activities are designed to lead the students to explore the
Mandelbrot Set. This lesson is designed as a capstone activity for the idea of fractals started in the
Infinity, Self-Similarity, and Recursion,
Geometric Fractals, and
Fractals and the Chaos Game lessons. Students are introduced to the notion of a complex number and function iteration in
order to motivate the discussion of Julia sets and the Mandelbrot set.

This lesson is best implemented with students working individually. Allow the students at least 30
minutes to explore each computer activity.

Objectives

Upon completion of this lesson, students will:

have learned about fractals and built a few

have investigated Julia sets and the Mandelbrot set

have been introduced to complex numbers and function iteration

Standards Addressed:

Grade 10

Geometry

The student demonstrates an understanding of geometric relationships.

The student demonstrates conceptual understanding of similarity, congruence, symmetry, or transformations of shapes.

The student demonstrates a conceptual understanding of geometric drawings or constructions.

Grade 6

Geometry

The student demonstrates conceptual understanding of similarity, congruence, symmetry, or transformations of shapes.

Grade 7

Geometry

The student demonstrates conceptual understanding of similarity, congruence, symmetry, or transformations of shapes.

Grade 8

Geometry

The student demonstrates conceptual understanding of similarity, congruence, symmetry, or transformations of shapes.

Grade 9

Geometry

The student demonstrates an understanding of geometric relationships.

The student demonstrates conceptual understanding of similarity, congruence, symmetry, or transformations of shapes.

The student demonstrates a conceptual understanding of geometric drawings or constructions.

Number and Quantity

The Complex Number System

Perform arithmetic operations with complex numbers.

Represent complex numbers and their operations on the complex plane.

Grades 6-8

Algebra

Represent and analyze mathematical situations and structures using algebraic symbols

Grades 9-12

Algebra

Understand patterns, relations, and functions

Geometry

Specify locations and describe spatial relationships using coordinate geometry and other representational systems

Use visualization, spatial reasoning, and geometric modeling to solve problems

Algebra II

Numbers and Operations

Competency Goal 1: The learner will perform operations with complex numbers, matrices, and polynomials.

Geometry

Data Analysis and Probability

Competency Goal 3: The learner will transform geometric figures in the coordinate plane algebraically.

Geometry and Measurement

Competency Goal 2: The learner will use geometric and algebraic properties of figures to solve problems and write proofs.

Integrated Mathematics III

Geometry and Measurement

Competency Goal 2: The learner will use properties of geometric figures to solve problems.

Integrated Mathematics IV

Number and Operations

Competency Goal 1: The learner will operate with complex numbers and vectors to solve problems.

Technical Mathematics I

Geometry and Measurement

Competency Goal 2: The learner will measure and apply geometric concepts to solve problems.

Technical Mathematics II

Geometry and Measurement

Competency Goal 1: The learner will use properties of geometric figures to solve problems.

Secondary

Algebra II

AII.06 The student will select, justify, and apply a technique to solve a quadratic equation over the set of complex numbers. Graphing calculators will be used for solving and for confirming the algebraic solutions.

AII.17 The student will perform operations on complex numbers and express the results in simplest form. Simplifying results will involve using patterns of the powers of i.

AII.6

AII.17

Student Prerequisites

Geometric: Students must be able to:

recognize basic geometric shapes

Arithmetic: Student must be able to:

work with integers as scale factors and in ratios

perform basic operations, including squaring

Algebraic: Students must be able to:

work with simple algebraic expressions and functions, such as linear and quadratic expressions

graph ordered pairs of points on the cartesian coordinate plane

Technological: Students must be able to:

perform basic mouse manipulations such as point, click and drag

use a browser for experimenting with the activities

Teacher Preparation

Access to a browser

Copies of supplemental materials for the activities:

Chaos is the breakdown of predictability, or a state of disorder

escapees

A complex number is an escapee of a Julia Set if its orbit, a sequence of complex numbers generated by successive iterations of a given function, is unbounded.

fractal

Term coined by Benoit Mandelbrot in 1975, referring to objects built using recursion, where some aspect of the limiting object is infinite and another is finite, and where at any iteration, some piece of the object is a scaled down version of the previous iteration

Julia set

The set of all the points for a function of the form Z^{2}+C. The iterations will either approach zero, approach infinity, or get trapped

Mandelbrot set

Discovered much later than Julia sets, it is generated by taking the set of all functions f(Z)=Z^{2}+C, looking at all of the possible C points and their Julia sets, and assigning colors to the points based on whether the Julia set is connected or dust

prisoners

A complex number is a prisoner in a Julia Set if its orbit, a sequence of complex numbers generated by successive iterations of a given function, is bounded.

self-similarity

Two or more objects having the same characteristics. In fractals, the shapes of lines at different iterations look like smaller versions of the earlier shapes

Lesson Outline

Focus and Review

Remind students what has been learned in previous lessons that will be pertinent to this lesson
and/or have them begin to think about the words and ideas of this lesson:

Does anyone remember what a fractal is?

What are some fractals that we have looked at thus far?

Objectives

Let the students know what it is they will be doing and learning today. Say something like this:

Today, class, we are going to learn to calculate complex number functions and see how these
fumctions lead to the creation of fractals such as the Julia set and the Mandelbrot set.

We are going to use the computers to learn about complex number functions, but please do not
turn your computers on until I ask you to. I want to show you a little about this activity
first.

Teacher Input

Lead a class
discussion on two variable functions (which can also be introduced as complex number functions).

Lead a class
discussion on function iteration and julia sets.

Guided Practice

Have the students try the computer version of the
Function Iterator activity to investigate two-variable function iterations and prisoners and escapees.

Have the students try the computer version of the
Julia Set activity to investigate what sorts of interesting fractal patterns are possible from the
boundaries of prisoner sets.

Lead a class
discussion on how the Mandelbrot set is built from Julia set behavior.

Independent Practice

Create a set of patterns for the students to find within the Mandelbrot set.

Have the students try the computer version of the
Mandelbrot Set activity to investigate what sorts of interesting fractal patterns are possible by zooming in
on parts of the set.

Closure

You may wish to bring the class back together for a discussion of the findings. Once the
students have been allowed to share what they found, summarize the results of the lesson.

Alternate Outline

This lesson can be enhanced in several ways. However, cutting out any of the discussions or
activities would limit the student's understanding of the ideas behind the Mandelbrot set.

Add the additional task of trying to find an image that looks like an actual object.

Have a contest in which the students are asked to find the most interesting image, with a
panel of teachers or the entire class being the judge. (Have the students print out their
images so that a display can be set up.)

If connected to the internet, use the enhanced version of the software,
The Fractal Microscope, to explore the Mandelbrot set more fully.

Suggested Follow-Up

After these discussions and activities, the students will have seen how the Mandelbrot set is
built for the simple case of quadratic functions. This set has many number-theoretic properties
which can be explored. For future reading on this complex and beautiful topic see:

Michael Barnsley, Fractals Everywhere, Academic Press 1988.

Benoit Mandelbrot, The Fractal Geometry of Nature, W. H. Freeman 1982.

H.-O. Peitgen and P. H. Richter, The Beauty of Fractals, Springer-Verlag 1986.