#
Fractint L-System True Fractals

##
A tutorial by William McWorter

* mcworter@midohio.net *

version 1.4

January 1997

Back to Tutorial Index

##
TRUE FRACTALS

Fractals are sets of points in space of such complexity that conventional
methods of measuring their dimension do not distinguish them from more
ordinary sets of points. For example, ** Koch's Snowflake **

(drawing of the Snowflake using KochCurve in
fractint.l)
is a curve, or a 1-dimensional set, according to conventional rules, even
at infinite order.

However, using the more involved notion of Hausdorff dimension, Koch's
Snowflake at infinite order has dimension 1.2618. Since Koch's curve looks
more like a frayed rope than a curve, this number seems a more reasonable
measure of its dimension; Koch's Snowflake looks more than a curve (dimension
1) but definitely less than an area (dimension 2).

Another famous fractal is Sierpinski's Carpet.

(drawing of Sierpinski's Carpet using the script below)

SierpinskiCarpet {
Angle 4
Axiom f
f=f+f-f-f-g+f+f+f-f
g=ggg
}

Visually, this fractal has more substance even though its conventional
dimension at infinite order is still 1. Its Hausdorff dimension, on the other
hand, is 1.8928, suggesting that Sierpinski's Gasket is just as it looks,
almost 2-dimensional.

When you experiment with constructing L-systems, your mistakes are likely to
be true fractals. Here is a pleasant error from a failed search for a new
spacefilling curve.

Pentigree {
Angle 5
Axiom F-F-F-F-F
F=F-F++F+F-F-F
}

(drawing of the Pentigree using the script below)

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