Exploring fractal geometry with children

School Science and Mathematics, Feb 1999 by Vacc, Nancy Nesbitt

The intent of this article is to heighten elementary school teachers: teacher educators', and teachereducation researchers' awareness of possible applications of fractal geometry with children and, subsequently, to initiate a discussion about the appropriateness of including this new mathematics in the elementary curriculum. Activities for exploring children's thinking and understanding relative to (a) describing, identifying, and measuring fractal patterns and (b) the attribute of self-similarity in some fractal sets are presented, as are recommendations for future research.

Question: How are popcorn, a fern, vapor clouds from the Apollo lift-off, and a rose alike?

Answer: Each is a fractal set.

Only 2 decades have passed since the concept of fractals was introduced by Benoit Mandelbrot (1977).Yet, during that period of time, fractal geometry has become well established as a new mathematics that has many applications in such areas as chemistry, physics, biology, and geography. As a result, educators are being challenged to examine ways in which the study of fractals can be addressed most effectively across the K-12 curriculum.

The term fractal was coined by Mandelbrot in 1975 from the Latin word fractus, meaning "to break" (Pietgen, Jurgens, & Saupe, 1990). In simplest terms, fractals refer to highly irregular or fragmented spatial patterns that cannot be described by terms from classical geometry. As Mandelbrot (1977) indicated, "Some fractal sets are curves, others are surfaces, still others are clouds of disconnected points, and yet others are so oddly shaped that there are no good terms for them in either the sciences or the arts" (pp. 1-2). Thus, with the introduction of fractals, objects that formerly could not be defined geometrically because of their irregular or fragmented patterns can now be described mathematically by shape.

Yet, in the elementary mathematics curriculum, discussions about shape continue to focus on objects with classical geometric shapes only. This behavior is not unexpected, given the "newness" of fractal geometry. However, it seems that, as with classical geometry, some basic concepts of fractal geometry could be introduced to young children, including shape, self-similarity, and measurement. Accordingly, the intent of this article is to heighten the reader's awareness of possible applications of fractal geometry in the elementary school and, subsequently, to initiate a discussion about the appropriateness of including this new mathematics in the elementary curriculum.

Examples of Fractals

Fractals can be divided into two categories: natural fractals and mathematically structured fractals.

Natural Fractals

One does not have to look very far to find examples of natural fractals in the everyday environment of children. Food alone provides excellent illustrations of shapes that could be classified using the fractal sets identified by Mandelbrot: kale and curly leaf lettuce (curves), batter-fried chicken and pizzas (surfaces), whipped cream and mashed potatoes (clouds), broccoli florets (disconnected points), and popcorn (undefinable odd shapes). Examples of other natural fractals include coastlines and curvy rivers (as illustrated on wall maps), flowers such as a rose or carnation, tree branches, rock formations, mountain ranges, seaweed and other aquatic plants, coral, and parts of the human anatomy such as curly hair, veins, and intestines.

Mathematically Structured Fractals Included in this category are simulations of fractal patterns that are computer generated. Popular examples of these simulated fractals are the Mandelbrot set, Koch snowflake, and Julia sets. These and other fractals are available through different sites on the World Wide Web, such as the following:

http://www.ncsa.uiuc.edu/Edu/Fractal/Fractal_Home.html

http://www.kcsd.kl 2.pa.us/-projects/fractal/pics.html

http://www.geocities.com/CapeCanaveral/2854/mandelbrot.html

http://www.geocities.com/CapeCanaveral/2854/julia.html

http://www.geocities.com/CapeCanaveral/2854/gallery.html

Through explorations of these websites, children (and teachers) are able to gain an appreciation of the beauty of the fractals displayed, even though they lack knowledge of the specific mathematics underlying the construction of the fractals. However, for those interested in exploring the generation of mathematically structured fractals, Thomas (1989) has provided a good introduction to the dimensionality of fractals. Included are an explanation of how the Koch snowflake is made and suggestions for creating fractals using Logo. Information for building the Mandelbrot and Julia sets is included in some of the Web sites cited above. Also, Pietgen et al. ( 1990) provides an excellent overall introduction to fractals for the classroom.

Activities for Introducing Basic Concepts of Fractal Geometry

The following activities have been used with elementary school children in a professional development school in central North Carolina. The activities are recommended as starting points for exploring children's thinking and understanding in fractal geometry. The activities specifically address the tasks of describing, identifying, ancmeasuring fractal patterns and exploring the attribute of self-similarity in some fractal sets.