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Rogue Fractals ... bustin' the law!!

I knew it! There are exceptions to the rule of fractals!

Keywords: shape-filling curve, Hilbert curve, Koch fractal family, Dragon Curve, nature and fractals

Hi guys! In the last post, I said that:

"Fractals are not always self-similar, but they have non-integer dimensions"

Others say that:

"One exception proves the rule"

But why one? I found an infinity of exceptions, called shape-filling curves! Let me show you some of these exceptions:


The famous Hilbert's Curve! This is always a 1-D line (doesn't matter that it is bent) but when applied to the unreachable infinity of fractals, the line becomes a 2-D square! This seems fractal-like but the dimensionality is always an integer (1-D in a finite number of iterations, 2-D in an infinite)!

This fractal is always a 1-D line but forms a 2-D triangle in the infinite iteration of fractals. This is a special variation of the Koch fractal family (a "family" of fractals is a set of fractals that follow roughly the same iterative rules). Some members in the Koch family (to the sixth iteration) are as shown below:

This is the 160 degree version. Looks like an eroded plateau!

This is the 140 degree version. A lot less flat, but very sparse!

The 120 degree version.

The 100 degree version. Bearing more resemblance to our beloved Koch Island!

The 80 degree version. Uncanny resemblance! But hold your horses...


There! What did I tell you? Our Koch Island did finally come along! Formally, this is the 60 degree version.

The 40 degree version. Reminds me of a pentagram (five-pointed star).

The 20 degree version. Rather like a dystopian forest (do you see it too?).

The 10 degree version. As we get closer to the 0 degree case, the fractal becomes more tightly packed into a triangle. So in the 0 degree case, we would get something like the triangle shape-filling curve we saw above. In the 180 degree case, you would get nothing but a straight line.


But what do I mean by "the 160 degree version" or "the 80 degree version"? I mean that if you take the first iteration of each of these, you would get something similar to these three diagrams:

For each of these, the angle between the two orange lines determines the name of the fractal. For example, the leftmost one has an angle of 160 degrees between the orange lines and so its fractal is the "160 degree version". The rightmost one has an angle of 20 degrees and so its fractal is the "20 degree version".


The Title GIF at the top (the Dragon Curve (DC)) is, surprise, surprise, also a shape filling curve! But how is it a shape filling curve?



For each of these iterations, the blue area (the DC of the last iteration) is equal to the green area (the area added to the blue area to form the DC of the present iteration). Also, they are added perpendicularly to each other, so the side length of the new DC is sqrt(2) times the old one. So in the fractal dimension calculator, the fractal dimension of the DC is log[sqrt(2)](2) = 2(!). Therefore, the DC is a shape-filling curve.


Now I will debut a shape filling fractal of my own! Here it is:

The "Freedom of Speech" Fractal


How is this a shape filling curve? Well, let's look at its first iteration.

Each of those segments turns into a mini-version of the first iteration (the longer side counts for two segments). Using the fractal dimension formula from Part 4, the dimension is log[4](16) = 2(!). Somehow I have created a 2-D fractal from a 1-D iterative process!


Some further iterations of the Freedom of Speech fractal:


(2nd iteration)

(3rd iteration)

(4th iteration)


Further iterations take ages to draw using the programming interface I used for the ones above.


Here is the Freedom of Speech fractal in action:



So, should we consider these shape-filling curves as fractals? In a way, I think yes, as they, like other fractals, follow an infinite iterative algorithm to augment or diminish the dimensionality of an integer-dimensional manifold (line, polygon, polyhedron etc.). From another perspective, I think no, as they augment the dimensionality to an integer. Why is this "integer" fuss important? Well, one reason is to exclude non-fractalesque curves such as this one below:




This technically is a shape-filling curve as it is a line in a finite number of iterations but a square in the infinitieth iteration of fractals. But would you say that this is a fractal?


I don't know myself, but I guess I ask you two questions: "Does it matter whether a curve is augmented/diminished to an integer dimension for it to be a fractal?" and "Are curves such as the one in the video above really fractals?". This might seem a mere technicality that I am being pedantic about, but it is important for understanding the recursive nature of ... well, nature! Every generation affects the next, often in similar ways, and what with the millions of years of evolution (with hundreds of thousands of "iterations"), fractals could help us to realise the answers to the hardest and most obscure questions we could think about such as, "Why is nature so random and yet so productive, but at the same time, why does the orderliness of humanity limit our capability?". Either way, please share your thoughts and comments below and see you in my later posts!


P.S. Oh, and if you want to learn more about these shape-filling curves you can visit this site.

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