Lévy C curve

If using a Lindenmayer system then the construction of the C curve starts with a straight line.

So, after two stages, the curve takes the appearance of three sides of a rectangle with the same length as the original line, but only half as wide.

At each subsequent stage, each straight line segment in the curve is replaced by the other two sides of a right-angled isosceles triangle built on it.

It takes its name from its resemblance to a highly ornamented version of the letter "C".

It will need a set of two "rules" which are: Two points in a plane (the translators), each associated with a scale factor of 1/√2.

First eight stages in the construction of a Lévy C curve
Lévy C curve (from a L-system, after the first 12 stages)
Lévy C curve (from IFS, infinite levels)