Fractals in the Classroom

What exactly is a fractal? Traditionally, students learn about the familiar forms of sym metry: reflection, translation, and rotation. Intuitively, fractals are symmetric with respect to magnification. A magnifi cation of a small part of the fractal looks essentially the same as the entire picture. More formally, frac tals have the property of self-similarity?that is, a fractal is any shape that is made up of smaller copies of itself. Self-similarity is what distinguishes frac tals from most conventional Euclidean figures and makes them appealing. Do fractals hold the same characteristics as other Euclidean objects? Fractals offer much to explore for even very young students. In the past, we have taught fractal geometry to students of various ages and abilities, introduc ing the idea of self-similarity to kindergarten and middle school children through workshops tailored to each of these age groups. Further, we have led in-depth investigations with nonmajors in liberal arts courses and have gone into even more detail with mathematics majors in upper-level geometry courses. Fractal geometry offers teachers great flex ibility: It can be adapted to the level of the audience or to time constraints. Although easily explained, fractal geometry leads to rich and interesting math ematical complexities.