The Mandelbrot Set

Abstract

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 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.

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.

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

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.

Algebra II

  • Numbers and Operations

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

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.

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.

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: Students 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

Key Terms

chaos

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 Z2+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)=Z2+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

  1. 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?

  2. 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.

  3. 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.

  4. 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.

  5. 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.

  6. 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.