Geometric
Lévy C curve
Paul Lévy studied this curve in 1938. The rule is simple: replace each line segment with two equal segments meeting at a right angle, scaled by 1/√2. Iterating produces an intricate, plane-filling shape whose early iterations resemble a thick letter C.
At a glance
| Designer | Paul Lévy, 1938 |
|---|---|
| Hausdorff dimension | 2 (space-filling) |
| Self-similar copies | 2 copies at scale 1/√2 |
| L-system rule | F → +F−−F+ |
| Angle | 45° |
Why dimension 2
Two self-similar copies at scale 1/√2 give dim = log 2 / log √2 = 2. Despite the apparently curve-like construction, the Lévy C is space-filling: its image has positive Lebesgue measure.
Tapestry
Joining two Lévy C curves at their endpoints produces the “Lévy tapestry”, a closed region that tiles the plane under translation. This connects the C curve to the family of rep-tiles introduced by Solomon Golomb.
References
- Lévy, P., “Les courbes planes ou gauches et les surfaces composées de parties semblables au tout,” J. École Polytechnique, 1938.
- Mandelbrot, B., The Fractal Geometry of Nature, W. H. Freeman, 1982.
- Dragon curve · Koch snowflake
Try it
Run an interactive playground at /tools/lsystems.