Lévy's C-curve

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Lévy's C-curve
The Lévy's C-curve is a self-similar fractal.

Contents

Basic Description

Like the Harter-Heighway Dragon Curve, the Lévy's C-curve is constructed by starting with a single straight line.

We then use this line as the hypotenuse for an isosceles right triangle (45-45-90). Then, the original line is removed, leaving the two legs of the newly created triangle. This process is illustrated below, with black being the initial lines and blue being the constructed lines.


We continue this process of using lines as bases for constructing isosceles right triangles and then removing the original lines. Again, we use black for the lines constructed before the current step and blue for the lines constructed during this step.

The first eight iterations of this process are shown below.

Image:260px-Levy C construction.png

The twelfth iteration of the curve already has the same general shape as the main image on this page.

The curve itself is created by iterating this process infinitely. The main image is created by using a computer to generate thousands of iterations and then adjusting lighting effects.

A More Mathematical Explanation

Number of Sides

For the first iteration, we see that we end up with two lines, so the number of s [...]

Number of Sides

For the first iteration, we see that we end up with two lines, so the number of sides is two.

After the second iteration, each of those two lines as turned into two more, so there are four sides.

This pattern of doubling continues, and so the number of sides of the Lévy's C-curve for any degree of iteration (k) is given by N_k = 2^k\,.

Length of Sides

Looking at the first few iterations, we can see how the length of the sides changes. Assuming that the original base has length one, we can see that the two legs of the triangle must have length  \ell_1= \frac{1}{\sqrt{2}}=\frac{1}{2^{1/2}} , since we must have that  1^2=\ell_1^2+\ell_1^2=2\ell_1^2

At the second iteration, we have  \ell_1 as the hypotenuse of our right triangle, so:

 \left(\frac{1}{\sqrt{2}}\right)^2= \ell_2^2+\ell_2^2=2\ell_2^2

 \left(\frac{1}{2}\right)\left(\frac{1}{2}\right)=\ell_2^2

 \ell_2=\frac{1}{2}=\frac{1}{2^{2/2}} .

For the third iteration, we now have  \ell_2 as the hypotenuse of our right triangle and we now have

 \left(\frac{1}{2}\right)^2= \ell_3^2+\ell_3^2=2\ell_3^2

 \left(\frac{1}{4}\right)\left(\frac{1}{2}\right)=\ell_3^2

 \ell_2=\frac{1}{\sqrt{8}}=\frac{1}{2^{3/2}} .

We can now see that the length of a side for each iteration is given by  \ell_k=\frac{1}{2^{k/2}}


Length of the Curve

The length of the curve at any iteration is simply the number of sides for that iteration times the length of each side. Therefore the length for any iteration k is  (N_k)(\ell_k) (2^k)\left(\frac{1}{2^{k/2}}\right)=2^{k/2} .

As the curve is iterated infinitely, we see that the total length approaches infinity since  \lim_{k\rightarrow \infty} 2^{k/2}=\infty

Fractal Dimension

The fractal dimension of Lévy's C-curve can be calculated as follows.

We have already found that  N_k = 2^k , so the N in the formula for fractal dimension is 2.

Since the length of the sides is given by  \ell_k=\frac{1}{2^{k/2}} , we see that each side is scaled down by a factor of  2^{1/2} from the previous iteration. So we have that e in the formula for fractal dimension is  2^{1/2} .

 D = \frac{\log N}{\log\left({e}\right)} = \frac{\log 2}{\log 2^{1/2} }=\frac{1}{1/2}=2.

Angle

The Lévy's C-curve iterates with a 90 degree angle but variations of the curve can be created by changing this angle.




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Future Directions for this Page

-A good animation of a Lévy C curve being drawn. -A picture of a Lévy C curve variation with an angle of 60 degrees.




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