# Angles and proportions occurring in L-system fractals

This is about properties of certain fractals defined by Lindenmayer systems, a.k.a. L-systems. Unlike “classical” fractals like Julia sets or the Mandelbrot set (the name “set” says it all), these fractals are not defined by calculus, but rather geometically, by iterating certain drawing procedures and resizing, which often results in a well-defined limit "curve." Some fractals, e.g. the well-known dragon curve, can be defined by very different L-systems.

I am mainly interested here in fractals where the self-similarity can be seen by decomposing shapes into smaller similar shapes (similar in the strict sense). Which angles can appear between similar shapes? E.g. for the Heighway dragon curve, it is easy to see directly from some of the L-system formulae defining it that these angles are $k\pi/8$.

Now I have encountered L-systems featuring angles which do not seem related to the one in the formula. E.g. for the following one, the angle between the biggest "hole" and the next-to-biggest one below it (colored blue) is about $70°$. (Note that it is smaller than $72°=2\pi/5$). This fractal has been obtained by the formula

Angle 8
Axiom F
F=+!F+!F++!F!+


Or the following one where in the "tail" on the left, subsequent copies of the rectangle-like parts have an angle of about $9°$ to $10°$ between them. This fractal can be obtained by the formula

Angle 8
Axiom FY
F=--
X=-XY-F
Y=-X++Y-


It is clear that in both cases above, the formulae do not give any hint about this angle. But my question is:

Can it be shown that the angles occurring in fractals of this kind are always rational?

Similarly, we can wonder about the ratio of sizes among the similar copies contained in the fractal, say the smallest one which is $>1$. Based on some "easy" examples, where this ratio can be derived from the formula, I'd like to ask:

Is this ratio always an algebraic number?

In your first example, I think the ratio is $$\arctan\frac{\sqrt{127+96\sqrt{2}\;}}{7} \approx 66.643774 \text{ degrees}$$ There is no reason to think this is a rational number of degrees. And: the second hole is exactly half the size of the largest hole!

These problems are fun...

Two of the methods to study self-similar sets are: the L-System (Lindenmayer System) and the IFS (iterated function system). To analyze this L-System, I first had to get the corresponding IFS. (Perhaps there are others who, to analyze an IFS, first convert it to an L-System...)

Here it is: The factor $d=2$, comparing the size of the whole to the size of one of the three parts, was found by trial-and-error. If you make $d$ too large or too small, then higher iterates of the picture either grow without bound, or shrink to zero. But factor $2$ seems to keep them about the same size. The one in the picture shows what it looks like when depth is odd. The whole thing is made up of three parts, similar to the original (the first and third reflected). The orientation at the exit is the same as the orientation at the entry.

For even depth, the picture looks like this: It looks exactly like an odd case, except rotated by a certain amount. (It turns out, rotated by that approximate 66 degrees described above.) And the exit orientation is rotated $-3\pi/4$ clockwise from the entry orientation.

Now, in the odd cases, the direction from entry point to exit point differs from the entry direction by some angle $\alpha$. In the even cases, the direction from entry pooint to exit point differs from a $-\pi/2$ direction by some angle $-\beta$.  So let's analyze the odd picture into its three parts (which are even pictures, some reflected). (This picture is rotated $-\pi/2$ since I want to use complex numbers to do my trigonometry: the entry direction is complex number $1$, represented as a vector pointing to the right.) To get from the entry point to the exit point of the whole thing (let's normalize so that this distance is $1$ unit), we go $1/2$ unit in direction $\beta$, then $1/2$ unit in direction $\pi/2-\beta$, then $1/2$ unit in direction $-\pi/4 + \beta$. So (starting at $0$) the exit point is $$w:=\frac{e^{i\beta}+e^{i(\pi/2-\beta)}+e^{i(-\pi/4+\beta)}}{2} .$$ This complex number should have modulus $1$ and argument $\alpha$. Solving polynomial equation $w\overline{w} = 1$ for $e^{i\beta}$, I get four solutions; $\tan \beta$ is a zero of $7X^4+24X^3-10X^2+8X-1$ the one that matches the picture is such that $\tan \beta \approx 0.14$ and $\beta \approx 8.03^\circ$. Putting this in the expression $w$, we take its argument to get $\alpha$. The result says $\tan \alpha\approx 0.27$.

The angle between the whole thing (which has the big lake) and the middle red part (which has the second lake) is $\gamma := \pi/2-\beta-\alpha$. Using the addition formula for tangents, I get $\tan \gamma$ as a zero of $49X^4-254X^2-47$. The zero that we want is $$\frac{\sqrt{127+96\sqrt{2}\;}}{7}$$ and $\delta$ is its arctangent.

So, to verify the $66$ degree answer: Here is a large picture on the right, then the same picture rotated $66$ degrees and shrunk by fator $1/2$ on the left. Two of the lakes are colored to help comparing them (for size and orientation). • that's impressive... and I'm sure you have had fun. Can you describe how you came up with that? – Wolfgang Jul 3 '14 at 6:34
• Great explanation. Thank you so much! I am just astonished that you used "trial and error" for finding the size factor 2. It would seem to me that along the lines of your analysis, it might be possible to find a general argument that this size factor is indeed always an algebraic number. What do you think? – Wolfgang Jul 7 '14 at 8:32

In a IFS implementation of a rep-n-tile an angle can be defined by cos ω = m/(2*sqrt(n)), m an integer. The possible rotations of elements of the rep-n-tile with respect to the tile depends on whether it is based on a rhombic, square or hexagonal lattice.

For a rhombic lattice the rotation can be ω or ω+π.

For a square lattice the rotation can be ω, ω+π/2, ω+π or ω+3π/2.

For a hexagonal lattice the rotation can be ω, ω+π/3, ω+2π/3, ω+π, ω+4π/3 or ω+5π/3.

Some irreptiles are based on the same grids. The angles are the same here (n is defined by the largest component). Other irreptiles are based on oblique grids. I don't understand how these behave in general, but in at least some of these angles are associated with the corresponding Perron number.

My impression is that if you relax the requirement that the curve fills the plane there is no general restriction on angles. For example, take the Koch curve. You can continually adjust this by widening the base and reducing the height of the middle pair of elements.