Lévy C curve
Replace every segment by two legs of a right isosceles triangle, again and again: the segments bloom into a C.
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Formulas and key values
- Construction rule
Replace every segment by the two legs of a right isosceles triangle built on it, always on the same side.
- Arc lengthIn the graph ≈ 32
- SegmentsIn the graph = 1024
- Dimension of the limit
n is the number of iterations; the ends are held 1 apart.
Explainer video
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Start the video01 · Definition
What is the Lévy C curve?
Start with one segment. Replace it by the two short sides of a right isosceles triangle built on it, both turned the same way. Do the same to every new segment, n times. Paul Lévy studied the limit in 1938, describing how parts of it are similar to the whole.
Every step doubles the number of segments and shrinks each by √2, so the length grows by √2 per step while the endpoints stay put. Unlike the dragon curve, which turns alternately, every replacement here turns the same way, and the limit set fills a region of the plane: its Hausdorff dimension is 2.
02 · Properties
Key properties
- After n steps there are 2ⁿ segments, each of length (1/√2)ⁿ of the original.
- Total length (√2)ⁿ times the original segment.
- The endpoints never move; the curve stays inside a bounded region.
- The limit set has Hausdorff dimension 2.
- Unlike the dragon curve it is not self-avoiding: from the fourth step on, some segments are traced twice.
03 · Questions
Lévy C curve: common questions
- How is the Lévy C curve constructed?
- Construction rule: Replace every segment by the two legs of a right isosceles triangle built on it, always on the same side.. n is the number of iterations; the ends are held 1 apart.
- What is the arc length of the Lévy C curve?
- Arc length: L_n = (√2)^n. n is the number of iterations; the ends are held 1 apart.
04 · Keep exploring