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.
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
The activities and discussions in this lesson address the following
NCTM standards:
Algebra
Understand patterns, relations, and functions
- represent, analyze, and generalize a variety of patterns with tables, graphs, words, and, when possible, symbolic rules
- relate and compare different forms of representation for a relationship
- identify functions as linear or nonlinear and contrast their properties from tables, graphs, or equations
Represent and analyze mathematical situations and structures using algebraic symbols
- develop an initial conceptual understanding of different uses of variables
- use symbolic algebra to represent situations and to solve problems, especially those that involve linear relationships
Geometry
Specify locations and describe spatial relationships using coordinate geometry and other representational
- use coordinate geometry to represent and examine the properties of geometric shapes
Use visualization, spatial reasoning, and geometric modeling to solve problems
- use visual tools such as networks to represent and solve problems
- recognize and apply geometric ideas and relationships in areas outside the mathematics classroom, such as art, science, and everyday life
Links to other standards.
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 such as Netscape for experimenting with
the activities
Teacher Preparation
Students will need:
- Access to a browser
- Copies of supplemental materials for the activities:
Key Terms
This lesson introduces students to the following terms through the included discussions:
Lesson Outline
This lesson is best implemented with students working individually.
Allow the students at least 30 minutes to explore each computer activity.
- 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 Outlines
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 further 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.
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