The discovery of the fractal geometry of nature has caused mathematicians and scientists to think differently about how things in the universe are structured. Fractal geometry is now an important aspect of many scientific descriptions of natural phenomena, as well as an important research area in mathematics. It is not easy to put a simple answer to the question "What is a fractal?", partly because of the rich variety of objects and phenomena that are fractal. However, if there is a single unifying concept to all things fractal, it is the notion of self similarity and scaling. This lesson is intended to help students discover the concept of self similarity.

Preparation and Materials


Students will


This lesson fulfills portions of the following standards and curriculum guidelines:


Discussion of Similarity

Even if the students have heard of (the mathematical sense of) similarity before, they are likely not to remember it too well. What we need for this discussion is not the full technical definition with proportions, etc, but the concept that object A is similar to object B if one is a "zoomed in" version of the other. A reduction or enlargement with a copy machine is a good example that the students may already be familiar with. In any event, it is important that they understand what we mean by similarity in order for self similarity to have meaning, but it is also important not to get bogged down in defining similiarity precisely at this point.

Examination of Self Similar Items

Without bringing up the term "self similar" yet, show the students pictures of (or, ideally, actual physical examples of) objects that exhibit self similarity. Tell them that they should try to find some characteristic that is shared by all of the objects. Give them time to discuss this as a class or in groups and collect answers on the board.

Hints and Prompts

Possible prompts that could be introduced to the discussion include
telling them the answer
tell them that the characteristic you're looking for is called self similarity, and ask them to try to figure out what that might mean
visual prompt
mark one of the pictures with a box or circle around a small copy of the image and tell them to think about the indicated part of the picture
further prodding
if that's not enough, you could tell them to mentally enlarge that section of the picture (or sketch an enlarged version) and see what they notice about it
grabbing the students and shaking them
okay, not that, but do try to get them to see the idea for themselves before giving them the answer

Discussion of Self Similarity

Tell the students that the concept of self similarity and self simillar objects have recently caught the attention of biologists, chemists, meteorologists, artists, musicians, and t-shirt manufacturers. The reasons for this include

Creation of Self Similar Objects

The black and white images below were created with Snowflake. Students can use Snowflake to explore self-similarity interactively. Explain the basic use of the program without going into detail about the actual algorithm that generates the curves. Have them move the points around and observe what happens, and look for examples of self similarity. Have them save their favorite patterns, and allow each student to print out one favorite.

Writing Assignment

Ask the students to pretend that they have to explain self similarity to someone else their own age. In a short paper, have them describe the concept verbally as well as using illustrations (such as their printout or a sketch). When they have completed this assignment, have them read their own paper, pretending that they had never heard of self similarity before. What should they change to make it clearer?

Go to Snowflake Curriculum Resources

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Last Update: May 22, 2006
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Below are some computer-generated fractal images.