Fractal Images
A fractal is something that looks the same no matter how close you get to it, such as a fern leaf: a small section, looks very similar to the whole thing. To generate a fractal fern requires up to four transformation matrices to derive the points at which to draw a point. A random transformation is selected and applied. The selection is not totally random, some matrices are more likely to be selected than others, with one only selected occasionally. The colour of the leaf is based on the x and y coordinates of the drawn point., and the process is iterated (repeated) 80,000 times, such that the choice of points approximates a fern pattern. In this example the matrix values 79, 89 and 98 are used. A fourth value could be added to the matrix, so as to achieve greater accuracy. The value is rarely called in its algorithm, so on the basis that it doesn't add much, it is omitted in this example. Fractals play an important role in computer-generated works. If an artist wants to create a realistic landscape on a computer, they might draw fields, trees, a nice blue sky, and maybe even a few sheep. However, it will never quite look real - the human eye can tell that there is something wrong with the picture; and owing to the fact that objects like trees are fractals in real life, they really need to be fractals in computer art. Computer art that includes fractals can be mistaken for photographic art, as people are unused to seeing objects, like trees, as fractals in anything other than photographs and in real life. Another method is to introduce some randomness so that all the trees are different.
Barnsley's fern illustrates the use of affine translations in an iterated function system (IFS) to create a fractal. In Table III.3 of Michael Barnsley's book, the IFS code for the four affine transformations for the Barnsley leaf is given as a table of values for the coefficients a, b, c, and d, the constants e and f and the probability percentage factor of p as follows:
| a | b | c | d | e | f | p |
|---|---|---|---|---|---|---|
| 0 | 0 | 0 | .16 | 0 | 0 | .01 |
| .85 | .04 | -.04 | .85 | 0 | 1.6 | .85 |
| .2 | -.26 | .23 | .22 | 0 | 1.6 | .07 |
| -.15 | .28 | .26 | .24 | 0 | .44 | .07 |
Delacroix included as a background to the struggle between Jacob and the Angel two large oaks with their fractal, self-similar structured tree branches. The Journal of Eugène Delacroix contains a description of fractal, self-similar structures including the recursive branching of trees:
Paris 5 August 1854
"When I have been drawing trees, I have often noticed that each separate branch is a miniature tree in itself; in order to see it as such one would only need the leaves to be in proportion."
