## MRCM Fractal Generator

The MRCM Fractal Generator is based on the concept of the Multiple Reduction Copy Machine described in the book, *Fractals for the Classroom*, by Peitgen, Jürgens, and Saupe published by Springer-Verlag in 1992. Clicking the link will download a copy of the web page to your computer. To get started, simply find the download on your computer (labeled MCRM Fractal Generator 2) and open it.

*Please note that this is “bare-bones” software. It is not optimizedfor tablets or mobile devices, and it is not designed to save data.*

To generate a fractal, enter the coordinates of an initial image and a set of coefficients a, b, c, d, e, and f for the transformation equations:

X = ax + cy + e

Y = bx + dy + f.

Each transformation will create a copy of the initial image, modifying it by some combination of translations, rotations, reflections, similarities, or shears. To form an image of a fractal, each transformation must include a *reduction* in size. You may scale by different amounts in the horizontal and vertical directions.

Create a new image in Stage 1 by transforming the initial shape in Stage 0. The values 1, 0, 0, 1, 0, 0 for a, b, c, d, e, and f represent the *identity* transformation, which leaves the shape unchanged. Suggestions for exploring the effects of the coefficients:

Begin by allowing

*a*and*d*to have other values between 0 and 1. (This ensures that they*reduce*the shape in size.)Next explore non-zero values of

*e*and*f.*These involve translating (moving) the shape.Finally explore the effects of changing the values of

*b*and*c*. (These are more complicated, but they are needed for rotations.)

Use multiple transformations to create more than one copy of the image.

Stage 2 will apply the same set of transformations to your image in stage 1. Stage 3 will transform the image in Stage 2, etc. By using multiple sets of transformation coefficients, you create multiple images at each stage, resulting in an image that becomes more and more complex.

**Notes**

To create a rectangle, enter coordinates for a pair of opposite vertices.

You can simulate individual points by using very small, unfilled rectangles.

If you use a large number of transformations, image creation for the later stages may sometimes be slow. In rare cases, the image may not be created at all.

**Sample:**

To create a *Sierpinski Triangle*, use the following three sets of coefficients for a, b, c, d, e, and f.

0.5 0 0 0.5 0 0

0.5 0 0 0.5 0.5 0

0.5 0 0 0.5 0.25 0.5

You may use any initial image you like! Experiment with different possibilities. The top transformation creates a half-sized copy of the original image in the lower left corner. The middle transformation creates a half-sized copy of the original image in the lower right corner. The bottom transformation creates a half-sized copy of the original image in the upper middle. Each successive transformation does the same thing to the previous image.