The Inverse Problem

Given a fractal F, the inverse problem is to find affine transformations T₁, ..., T_n for which

F = T₁(F) U ... U T_n(F)

Remarkably, solving the inverse problem has only two steps:

1. Using the self-similarity (or self-affinity) of F, decompose F as F = F₁ U ... U F_n, where each F_i is a scaled copy of F. Because the transformations can involve rotations, reflections, and scalings by different factors in different directions, decomposition is not always as simple a task as it may seem at first. Here are some examples of more complicated decompositions. An additional problem is that decompositions never are unique. Here are some examples of different decompositions of the same fractal.

2. For each piece F_i, find an affine transformation T_i for which T_i(F) = F_i. (By "find an affine transformation" we mean find the r, s, theta, phi, e, and f values.)

Finding a decomposition and the corresponding transformations can be difficult. Keeping track of the translations for rotations and reflections causes special problems. Together with any smple graphics program, the web offers an elegant approach to mastering this skill.

* Bring up the homework web page on your computer.

* Place the mouse arrow over the fractal image whose IFS rules you wish to find, hold down the mouse button until the menu pops up, and select "copy this image."

* Now open your graphics program and paste the image. This is the target image.

* Paste another copy and use the graphics functions - scale, rotate, reflect - to cover one of the pieces of the target. You must be careful about the placement of the pieces under rotation and reflection, but the animations mentioned above seem to be good preparation for this.

* Continue until all the pieces of the target are covered.

* Test the transformations by running the IFS program.

For practice recognizing reflections and rotations, we consider two examples. Click on the small image to see a decomposition.

Here are some IFS problems for practice.

When the pieces are not scaled by such obvious amounts, we can find scalings and rotations by measuring distances and angles. Click on the picture for an example.

With a bit of thought, now we can find an IFS to generate the tree. The fern is trickier. Click each picture for the answer.

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