Introduction to Fractals:
Infinity, Self-Similarity and Recursion

Abstract

This lesson is designed to get students to think about several of the concepts from fractals, including recursion and self-similarity. The mathematical concepts of line segments, perimeter, area, and infinity are used, and skill at pattern recognition is practiced.

Objectives

Upon completion of this lesson, students will have:

• seen a variety of line deformation fractals
• developed a sense of infinity, self-similarity and recursion
• practiced their fraction, pattern recognition, perimeter and area skills

Activities

This lesson introduces students to fractals through the following activities:

• Tortoise and Hare Race
• Cantor's Comb
• Hilbert Curve
• Another Hilbert Curve
• Koch's Snowflake

Standards

The activities and discussions in this lesson address the following Standards:

• Number sense, number operations, and number relationships
• Patterns, relationships and functions
• Geometry

Key Terms

This lesson introduces students to the following terms through the included discussions:

• indefinitely
• iteration
• infinity
• recursion
• Fibonacci Numbers
• fractal
• self-similarity

Student Prerequisites

• Geometric: Students must be able to:
  • recognize and sketch objects such as lines, rectangles, triangles, and squares
  • understand the concepts of and use formulas for area and perimeter
• Arithmetic: Students must be able to:
  • build fractions from ratios of sizes
  • manipulate fractions in sums and products
• 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
• pencil and paper
• copies of supplemental materials for the activities:
  • Tortoise and Hare Race Worksheet
  • Cantor's Comb Worksheet
  • Hilbert Curve Worksheet
  • Another Hilbert Curve Worksheet
  • Koch Snowflake Worksheet

Lesson Outline

Groups of 2 or 3 work best for these activities; larger groups get cumbersome. Working through one or two iterations of each curve as a class before setting the groups to work individually can cut down on the time the students need to discover the patterns. Plan on 15-20 minutes for each exploration.

1. Introduce the terminology:

   Initiator:

   The starting curve or shape

   Generator:

   The rule used to build a new curve or shape from the old one

   Iteration:

   The process of repeating the same step over and over

2. Describe the Tortoise and Hare Race to the students and ask them to speculate on who will win. Then have them run though several steps of the race, stopping when they think they see what is happening.

3. Have students run several steps of the Cantor's Comb. The students should look at the patterns made by the lengths of the segments and the total length. It may take drawing two or three iterations before the number pattern becomes obvious.

4. Repeat the previous exercise for the Hilbert Curve.

5. Lead a class discussion to clarify what "infinitely many times" means.

6. Repeat the previous exercise for Another Hilbert Curve, this time also asking students to discuss how a small change in the generator can lead to a large change in the final object.

7. Repeat the previous exercise for the Koch's Snowflake, this time also asking about patterns in the area enclosed as well as the length of the curve.

8. Lead a class discussion to introduce the formal idea of recursion.

9. Lead a class discussion to introduce the formal idea of self similarity.

Alternate Outlines

This lesson can be rearranged in several ways.

• Choose fewer of the activities to cover; for example, covering Cantor's Comb, the Hilbert Curve and the Koch Snowflake still allows for discussion of infinity, self-similarity and recursion.
• Have different groups of students do different activities and give group presentations.
• Leave out one or more of the concept discussions and focus on pattern recognition and fractions.
• Have the students draw several steps of each of the activities by hand before trying the computerized version. Graph paper and rulers would be needed for this. Plan on an additional 10-15 minutes per activity.
• Combine this lesson with the Geometric Fractals lesson, to give the students a well rounded picture of regular fractals, including a formal definition.
• If connected to the internet, use the enhanced version of the software, Snowflake, to explore line deformation fractals more fully.

Extensions

After these discussions and activities, the students will have seen a few of the classic line deformation fractals. The next lesson, Geometric Fractals, continues the student's initial exploration of fractals with those formed by repeatedly removing portions from plain figures such as squares and triangles.