In the examples below, clicking on the image will open a PostScript slide show of the consecutive iterations of that particular fractal. The PostScript files are huge, so they have been placed in ZIP files. When you click on an image, choose to save the ZIP file to disk. Once it has completed, choose to open the file, and then double-click the PostScript file contained in the ZIP file to run the slide show

Note that if the first iteration is blank, it is not part of the slide show The first iteration is blank if the initial axiom contains no movement characters; that is, if the turtle doesn't move during translation of the axiom into an image.

The There
is more than one Lindenmayer grammar which will build this fractal.
The
one
used
here
illustrates
the purpose of having two different letters for movement, Conceptually the gasket can be produced by starting with a filled equilateral triangle and removing the inner triangle which is formed by placing its vertices at the midpoints of the original triangle's edges. (This part being removed is the largest white triangle in the image on the left.) This produces three smaller filled equilateral triangles. At each step, remove the inner triangle from every filled triangle. As this process is repeated indefinitely, the Sierpinski gasket is formed. |

Invented by German mathematician David Hilbert in 1891, the |

The |

This fractal is less known than those mentioned above, probably due
to its lack of mathematically interesting properties, but the |

Like the Willow, the |

Another visually appealing fractal. At any scale, each constituent
ring is an exact copy of the entire fractal. Note the surprising relative
simplicity of the transformation rule that produces the |

The The first iteration is simply a line. At each iteration, the middle third of every line is removed. As the iterations continue and the fractal becomes more "full of holes", the total length of the lines approaches zero, but the number of lines approaches infinity, conserving the complexity of the fractal. Note that specification of the rotation unit is unnecessary, as the turtle will never be told to rotate (neither the initial axiom nor either of the transformation rules includes a + or - character). |