Barnsley fern

Like the Sierpinski triangle, the Barnsley fern shows how graphically beautiful structures can be built from repetitive uses of mathematical formulas with computers.

IFSs provide models for certain plants, leaves, and ferns, by virtue of the self-similarity which often occurs in branching structures in nature.

But nature also exhibits randomness and variation from one level to the next; no two ferns are exactly alike, and the branching fronds become leaves at a smaller scale.

V-variable fractals allow for such randomness and variability across scales, while at the same time admitting a continuous dependence on parameters which facilitates geometrical modelling.

The formula for one transformation is the following: Barnsley shows the IFS code for his Black Spleenwort fern fractal as a matrix of values shown in a table.

These correspond to the following transformations: Though Barnsley's fern could in theory be plotted by hand with a pen and graph paper, the number of iterations necessary runs into the tens of thousands, which makes use of a computer practically mandatory.

Crucially it does not reset exactly to (0,0) which allows it to fill in the base stem which is translated and serves as a kind of "kernel" from which all other sections of the fern are generated via transformations f2, f3, f4.