Modeling with Fractals

Mathematics prerequisites and discussion:
Students should be familiar with basic concepts from geometry such as regular polygons (equilateral triangles and squares), properties of similar triangles/squares, calculating the perimeters and areas of triangular and square regions, use of the Pythagorean theorem to find the altitude of an equilateral triangle, and finding the midpoint of a line segment. In addition, students should have some background on geometric sequences and series. (However, you could introduce that topic as students work through this pull-out.) For Activity 4, students should also be familiar with logarithms so that they can solve the equation rd = n for d: d = log(n)/log(r) and then use their calculators to obtain an approximation for d.

This pull-out consists of four activities.

Activity 1 introduces students to fractals in nature, in mathematics, and in art. Then students draw fractal trees. They investigate the sequences formed by the number of tree branches and branch lengths drawn at each stage, which turn out to be geometric sequences.

In Activity 2 students draw by hand Sierpinski triangles. The perimeters and areas of each stage of their drawings also form geometric sequences. Students discover that as the stage number increases, the perimeter grows larger and larger while the area shrinks toward 0.

In Activity 3, students use GeoGebra to draw several stages of the Sierpinski triangle and Sierpinski carpet. With both of these fractals , perimeters grow with each stage of construction while areas shrink. Because of this property, the Sierpinski triangle and carpet have been used as antenna designs in wireless communication devices.

Activity 4 focuses on the problem of measuring coast line length. Using the Koch snow flake curve as an example, students discover that as more indentations are added to a representation of a coastline, its length grows. It turns out that coastline length is highly variable depending on the length of the measuring device used. Next, students are introduced to the concept of fractal dimension. They adapt this concept to estimating coastline dimension, which is a measure of the smoothness or ruggedness of a coastline. This activity concludes with a final challenge for students to apply what they have learned to create some fractal art.

The activities in this pull-out address the following standards from the Common Core State Standards for High School Mathematics: