I Precursor

Review the definitions from Chaos and go over the homework. Talk about the validity of the examples given by the students.

II Self Similarity

A basic property of fractal images is self-similarity. Using the example constructed yesterday (S) we will explain self-similarity.

Looking at S we notice that we can decompose this into 3 congruent figures call X, Y, Z.

Now notice that each of these figures is exactly 1/2 the size of S (it is not 1/3 because the middle region is empty).

That is to say if we magnify any of these 3 figures X, Y, Z, by a factor of 2, we obtain an exact replica of S. That is saying that S consists of 3 self-similar copies of itself with a magnification factor of 2.

This can be explained using vector components (see diagram A). We can also chop S up into 9 self-similar copies of itself, each with a magnification factor of 4, or 27 self-similar copies with a magnification factor of 8.

In general to see this for S (show this on the board):

Each vector of the smaller triangle is multiplied by 2, to map the smaller triangle back to the original. This is similar for any subdivisions.

Have the class attempt to do this for a square or a pentagon.

(Many possible answers) We could choose to break a square up into 4 self similar pieces and a magnification factor of 2. Show this on the chalk board.

III Fractional Dimension

Explain standard Euclidean dimensions with respect to the number of coordinate axis (i.e. (X) axis = 1D, (X,Y) axis = 2D, (X,Y,Z) axis = 3D etc..). Alternatively you could use the number of variables in a system of equations (i.e. x+y+z+w=2 is 4D).

Explain that these are integer dimensions.

Fractals 'live in' fractional dimensions, hence the name fractal. So we need a new way to define dimension. Mathematicians use:

The Hausdorff Dimension - If we take an object residing in Euclidean dimension D and reduce its linear size by 1/r (in each spatial direction), its measure (length, area, or volume) would increase to N=rD times the original. Where N= # of self similar pieces, r = magnification factor, D = dimension.

1 - Find a new formula for D (hint use Logs with respective log laws, the answer is very easy to deduce from the definition of Hausdorff Dimension).

2 - Use the formula deduced (D=log N/log r) to compute the dimension of S.

With the self similarity ratio computed previously we have, N=3 and r=2 which gives D=log3/log2 .

We could have also used N=9 and r=4 as deduced previously also which does not change our answer, or with any other self-similarity or magnification ratio.

Compute the dimension of a square and cube using your new concept of dimension.

Alternative exploration: There exists a relationship between Pascal's triangle (PT) and S, that is, if we look at PT (up to a certain level of course) and reduce this modulo n (where n is an integer), then PT resembles S. The key word here is resembles, because this reduced PT modulo n is not a fractal. It looks like S but is not it. It is an interesting relationship, which may be brought up and discussed if desired.