# FRACTALS

by

## Richard G Krutchkoff

richard@krutchkoff.org

## FRACTALS:

Fractals are pictures generated by mathematical equations using an iterative plotting process. Some Fractals are exhibited in the complex plane and others are in the usual two dimensional plane. Fractals often exhibit self-similarity, which means, if you magnify a small piece of the picture you will find a picture looking like the original. If the resolution was high enough you could keep repeating this, forever, and still see a picture similar to the original. The small pictures you see below are catalog pictures and if you click on a picture you will see a larger copy of the Fractal which you can right click on and copy. (They are 8.5X11 inches but only at web resolution 72 dots/inch. If you need 600 dots/inch just e-mail me and I will send it to you.)

The Iterative function System fractals are usually 8.5x8.5 unless the width had to be made wider to fit the fractal. If you need to contact me to obtain a printed copy of the Fractal make note of the catalog number and your preferred orientation.

For a lecture on FRACTALS CLICK HERE.

## COMPOSIT FRACTALS:

These are Composite Fractals made up of one or more fractals superimposed on one another.

## ITERATIVE FUNCTION SYSTEMS:

These fractals are obtained by using a number of pairs (at most 100) of affine [they don’t send thing to infinity] linear equations. Each pair of equations takes a point and obtains another point putting a dot at that point in the color assigned to the pair of equations. A starting point, usually (0,0), begins the plotting system, which could go on forever, but is usually stopped after 10 million points are plotted. Most of these plots exhibit clear self-similarity, while in others it is not so clear. I have always believed that Iterative Function Systems were the mathematics of genetics.
For a lecture on ITERATIVE FUNCTION SYSTEMS CLICK HERE.

## LETTERS AND NUMBERS:

The L and N series are Iterative Function System Fractals. If you want to use these in constructing words or the like, feel free to crop, resize and change the colors as needed.

## 3D FRACTALS:

3D Fractals are really just sculptures made from parts of fractals. I have several of these but some would be difficult to duplicate.

## 3D FLOWER FRACTALS:

A One unit 3D flower fractal looks like figure A. It can be hung on a pin or placed on a table and rotated. Since the colors are hard to distinguish, only one of the layers are shown here.

Figure C indicates the color sections, backgroung (a), leaf sections (b,c,d). If none of the existing flowers appeals to you CLICK HERE. for a color chooser and find your own colors. Send me the 3 numbers, Red, Green, and Blue associated with each of a,b,c and d.

## POLAR COORDINATE FLOWERS:

These may look like fractals but are actually polar coordinate graphs of equations. The numbers ending in an F can be made into flowers like those above.

For each of the flowers, or for the 2D graphs, I can create a copy with any colors you want. CLICK HERE. for a color chooser and find your own colors. Send me the 3 numbers, Red, Green, and Blue associated with each of the colors you want to replace.

### MISCELLANIOUS:

The rest are some miscellanious stuff I made for special purposes.