For example, the Sierpinski fractal is generated by the three affine transforms
A1: |
x_{1} = 0.5x_{0}+0.0y_{0} + 0.0 |
y_{1} = 0.0x_{0}+0.5y_{0} + 0.0 |
A2: |
x_{1} = 0.5x_{0}+0.0y_{0} + 0.5 |
y_{1} = 0.0x_{0}+0.5y_{0} + 0.0 |
A3: |
x_{1} = 0.5x_{0}+0.0y_{0} + 0.250 |
y_{1} = 0.0x_{0}+0.5y_{0} + 0.433 |
The small magenta, cyan, and blue squares displayed are the images of the boundary of
S
under the three affine transformations￼
A1,
A2 and
A3. Also, each
has 1/2 as its only eigenvalue. Moreover, they have (0, 0), (1, 0), and (0.5, 0.866), respectively, as their
attracting fixed-points.
Similarly, the Fern fractal is generated by using four affine
transformation including A4
A4: |
x_{1} = 0.0x_{0}+0.0y_{0} + 0.4987 |
y_{1} = 0.0x_{0}+0.3y_{0} + 0.0070 |
whose linear part has determinant zero. It maps
S to the vertical line
segment from (0.4987, 0.007) to (0.4987, 0.307). The linear part of
A4 has
eigenvalues are 0 and 0.3. Also, (0.4987, 0.01) is its attracting fixed-point.
For more details, see Introduction to Fractals and Chaos, Richard M. Crownover, Jones and Bartlett, 1995.