1. Introduction to Fractals

Here we introduce some basic geometry of fractals, with emphasis on the Iterated Function System (IFS) formalism for generating fractals.

In addition, we explore the application of IFS to detect patterns, and also several examples of architectural fractals.

First, though, we review familiar symmetries of nature, preparing us for the new kind of symmetry that fractals exhibit.

A. The geometric characterization of the simplest fractals is self-similarity: 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 self-similarity examples, and variations including nonlinear self-similarity, self-affinity, and statistical self-similarity. 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.

A. The geometric characterization of the simplest fractals is self-similarity: 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 self-similarity examples, and variations including nonlinear self-similarity, self-affinity, and statistical self-similarity. 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.