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Irregular Fractals
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Abstract

This lesson is designed to continue the discussion of fractals started in the , and lessons. Students are introduced to the notion of irregular fractals and given an idea of the difficulty involved in calculating the fractal dimension as outlined in the lesson.

Objectives

Upon completion of this lesson, students will:

  • have learned about irregular fractals and built a few
  • have practiced their pattern recognition skills
  • have practiced their plane geometry skills

Standards Addressed:

Student Prerequisites

  • Arithmetic: Student must be able to:
    • work with integers as scale factors and in ratios
  • Technological: Students must be able to:
    • perform basic mouse manipulations such as point, click and drag
    • use a browser for experimenting with the activities
  • Geometric: Students must be able to:
    • manipulate basic geometric shapes
    • work with angles

Teacher Preparation

Key Terms

fractalTerm 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
irregular fractalsComplex fractals whose dimension is often difficult to determine and in some cases is unknown
iterationRepeating a set of rules or steps over and over. One step is called an iterate
recursionGiven some starting information and a rule for how to use it to get new information, the rule is then repeated using the new information
self-similarityTwo 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:

  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 about fractals.
    • We are going to use the computers to learn about fractals, 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

  4. Guided Practice

  5. Independent Practice

    • Have the students try the computer version of the Flake Maker activity to investigate what sorts of interesting patterns and fractals can be generated.
    • Also have stundents practice calculating the fractal dimension of the fractals they generate.

  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 rearranged in several ways.

  • Do only one activity -- the flake maker makes more interesting fractal pictures.
  • Add the additional task of trying to build an image that looks like an actual object with flake maker. Some suggestions: mountain ranges, ocean waves, flowers, animals.
  • 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, Snowflake, to explore line deformation fractals more fully.

Suggested Follow-Up

After these discussions and activities, the students will have seen how complex fractals can be generated by generalizing the ideas for making regular fractals, introduced in the Geometric Fractals and Fractals and Chaos Game lessons. The next lesson, The Mandelbrot Set, is a cap-stone activity, designed to introduce the student to the most celebrated modern fractal object, the Mandelbrot set.

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