Editing Lévy C curve (section)

==L-system construction==
[[Image:Levy C construction.png|thumb|260px|First eight stages in the construction of a Lévy C curve]]
[[Image:Levy C Curve.svg|thumb|260px|Lévy C curve (from a L-system, after the first 12 stages)]]
If using a [[Lindenmayer system]] then the construction of the C curve starts with a straight line. An [[isosceles]] triangle with angles of 45°, 90° and 45° is built using this line as its [[hypotenuse]]. The original line is then replaced by the other two sides of this triangle.

At the second stage, the two new lines each form the base for another right-angled isosceles triangle, and are replaced by the other two sides of their respective triangle. 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. After ''n'' stages the curve consists of 2<sup>''n''</sup> line segments, each of which is smaller than the original line by a factor of 2<sup>''n''/2</sup>.

This L-system can be described as follows:

{|
|'''Variables''':||{{mono|F}}
|-
|'''Constants''':||{{mono|+ &minus;}}
|-
|'''Start''':||{{mono|F}}
|-
|'''Rules''':||{{mono|F → +F&minus;&minus;F+}}
|}

where "{{mono|F}}" means "draw forward", "+" means "turn clockwise 45°", and "&minus;" means "turn anticlockwise 45°".

The [[fractal curve]] that is the limit of this "infinite" process is the Lévy C curve. It takes its name from its resemblance to a highly ornamented version of the letter "C". The curve resembles the finer details of the [[Pythagoras tree (fractal)|Pythagoras tree]].

The [[Hausdorff dimension]] of the C curve equals 2 (it contains open sets), whereas the boundary has dimension about 1.9340 [http://mathworld.wolfram.com/LevyFractal.html].

===Variations===
The standard C curve is built using 45° isosceles triangles. Variations of the C curve can be constructed by using isosceles triangles with angles other than 45°. As long as the angle is less than 60°, the new lines introduced at each stage are each shorter than the lines that they replace, so the construction process tends towards a limit curve. Angles less than 45° produce a fractal that is less tightly "curled".