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
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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