A. The geometric characterization of the simplest fractals is
selfsimilarity: the shape is made of smaller
copies of itself. The copies are similar to the whole: same shape but different size. 

B. More examples of
selfsimilarity examples, and variations including nonlinear selfsimilarity,
selfaffinity, and statistical selfsimilarity. Also, some fractal forgeries of nature. 

C. Initiators and Generators is
the simplest method for producing fractals. It is also the oldest, dating back 5000
years to south India. 

D. Geometry of plane transformations
is the mechanics of transformations that produce more general fractals by Iterated
Function Systems. 

E. An elegant application of plane transformations to growing fractals is
Iterated function systems. This method has been used in
image compression. 

F. Inverse problems finding the
transformations to produce a given fractal. This is a geometrical version of
Johnny Carson's "Karnak the Magnificent" routine. 

G. Random algorithm is another way to
render fractal images. Use the same transformations as with any other IFS, but apply
them to a single point, one at a time, in random order. What happens if we depart from
randomness? See the next section. 

H. Driven IFS a variation on the Random
Algorithm to test for patterns in data. We investigate patterns in mathematical sequences,
DNA sequences, financial data, and texts. 

I. Fractals in architecture African,
Indian, and European. Repetition across several scales is a theme common to many cultures,
developed independently so far as we can tell. 

Finally, fractals seem to be a an easy concept for kids. 
