# Teacher Instructions

```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:

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!)

```