Introduction to Fractals:
Infinity, Self-Similarity and Recursion
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
The fractals generated here all start with simple curves made from line segments. They
display the curiosities that intrigued the mathematicians looking at infinity at the turn of
the century. The Hilbert curves demonstrate that a seemingly 1 dimensional
curve can fill a 2-d space, and the Koch snowflake demonstrates that a 1-d curve can be
infinitely long and surround a finite area.
Upon completion of this lesson, students will:
- have seen a variety of line deformation fractals
- have developed a sense of infinity, self-similarity and recursion
- have practiced their fraction, pattern recognition, perimeter
and area skills
The activities and discussions in this lesson address the following
Understand patterns, relations, and functions
Use mathematical models to represent and understand quantitative relationships
- 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
- model and solve contextualized problems using various representations, such as graphs, tables, and equations
Apply transformations and use symmetry to analyze mathematical situations
Use visualization, spatial reasoning, and geometric modeling to solve problems
- describe sizes, positions, and orientations of shapes under informal transformations such as flips, turns, slides, and scaling
- examine the congruence, similarity, and line or rotational symmetry of objects using transformations
- draw geometric objects with specified properties, such as side lengths or angle measures
- use geometric models to represent and explain numerical and algebraic relationships
- recognize and apply geometric ideas and relationships in areas outside the mathematics classroom, such as art, science, and everyday life
Links to other standards.
- Geometric: Students must be able to:
- recognize and sketch objects such as lines, rectangles,
- understand the concepts of and use formulas for area and
- 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
Students will need:
- Access to a browser
- Pencil and Paper
- Copies of supplemental materials for the activities:
This lesson introduces students to the following terms through the included discussions:
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.
- 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:
Let the students know what it is they will be doing and learning today. Say something like this:
- Teacher Input
- Introduce the terminology:
- The starting curve or shape
- The rule used to build a new curve or shape from the old one
- The process of repeating the same step over and over
- Guidied Practice
- 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
- Have students run several steps of the
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.
- Repeat the previous exercise for the
- Lead a class discussion
to clarify what "infinitely many times" means.
- 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.
- Repeat the previous exercise for the Koch Curve, this time also asking about patterns in the area enclosed as well as
the length of the curve.
- Lead a class
to introduce the formal idea of recursion.
- Lead a class
to introduce the formal idea of self similarity.
- Independent Practice
- Allow the student's time to complete any or all of the worksheets that are provided
with the applets used during this lesson.
- Have the students try to draw a couple iterations of any of the fractals used during
this lesson and discuss why computers are useful when studying fractals.
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.
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
- 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.
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.