Matrix transformations generate fractals by repeatedly applying one or more affine transformations to a starting shape or set of points. Each iteration creates smaller transformed copies, and repeated iteration produces the fractal's self-similar pattern.
An affine transformation can be written as
where is a and is a translation vector. The matrix can rotate, reflect, stretch or shrink the figure, while the vector moves the resulting image.
To generate a fractal:
For example, the Sierpiński triangle can be generated using three transformations:
| Transformation | Rule |
|---|---|
| Lower-left copy | |
| Lower-right copy | |
| Upper copy |
Here, is the identity matrix. Multiplication by halves every length, while the translation vectors place the three copies in different positions.
These three rules form an iterated function system. Because each linear part has scale factor , every transformation is a contraction: distances between points decrease with each application. After one iteration there are three copies; after two there are ; after iterations there are . The limiting set is independent of most reasonable starting shapes, because repeated contraction progressively removes the influence of the initial set.
The determinant explains the change in area. Since
each copy has one quarter of the previous area. Repetition creates detail at progressively smaller scales.
A common misconception is that the matrix alone controls the position of each copy. In fact, the translation vector is essential unless every copy remains centred at the origin.
This is AHL 3.9 Matrix transformations, so it is HL-only and may appear on any GDC-active HL paper. Show the transformation rule, apply it in the correct order, and use when interpreting the area scale factor.
