Objectives: To implement the N.C.T.M. standard by
1. developing a deeper understanding of dimension,
2. expanding the concept of Euclidean dimension to
the surprising world of fractal dimension,
3. relating the mathematical concepts of similarity
ratio and recursion to the topic of dimension,
4. extending applications of logarithms, area, volume,
and infinite series to fractal geometry,
5. connecting the mathematical models to art, nature,
and the real world.
Levels: Geometry, Algebra II, and "Precalculus"
Materials: 1. Overhead projector and transparencies 2. Worksheets 3. Scientific calculator 4. Aluminum foil 5. Play dough or clay(optional) Notes: 1. To derive the equation D = logN/logS let S = 1/r in the Hausdorff Equation NR^D = 1 2. The solution of the Skewed Fractal Web ( Sierpinsky Tetrahedron) involves the study of a regular tetrahedron and a regular octahedron. Teacher instructions: 1. To model the Menger Sponge you will need 27 cubes
each 2 inches on an edge.
2. To model the Fractal Skewed Web you will need 4 regular
tetrahedrons each 2 inches on an edge and 8 equilateral
triangles each 2 inches on an edge.
3. To enrich the exploration of the 3-D figures have each
student prepare 2 models each (cubes and tetrahedrons) as
homework the day before this lesson is presented or have
templates, scissors and tape for them to make the shapes
in class. (Recall all edges are 2 inches)
4. The Menger Sponge can be represented by a 3x3x3 cube
composed of 27 miniature cubes. Remove the 6 miniature
cubes in the center of each face and the miniature
cube in the center (7 total) to form the Menger Sponge
To compute the dimension, N (the number of miniature
cubes in the final figure) equals 20 and S (the scaling
factor) equals 3.
*Important* Before computing the fractal dimension of the Fractal Skewed Web, let the students discover a wonderful surprise. The Fractal Skewed Web can be explored with 8 regular tetrahedron Working in groups the students should quickly discover that the figure cannot be constructed with the 8 tetrahedrons. Allow time for the students to construct the missing middle piece with tape and the extra equilateral triangles. The missing piece is a regular octahedron with edge of 2 inches! A square base pyramid (half the octahedron) can be compared with 2 tetrahedrons. The students can experiment with clay or play dough to see how the volumes are related. After they make conjectures about the relations of the volumes, use the worksheets to verify mathematically. Volume of 1 regular octahedron = 4 * Volume of 1 regular tetrahedron. The dimension of the Fractal Skewed Web can be found by removing the center piece whose volume equal that of 4 tetrahedron. N (the number of miniature tetrahedrons in the final figure) = 4 and S (the scaling factor) equals 2. The fractal dimension of the 3-dimensional Koch Snowflakes is left to you as a challenge. (Good Luck!)