Recently, I made a most startling discovery: someone else has been judging geometric shapes, and falsely publishing under my name!¹ I have been impersonated by a dastardly impostor!

After getting over the initial shock, I realized that my impostor actually made a good decision on what class of shapes to judge: fractals. Fractals are shapes where no matter how far you zoom into them, they always have some sort of structure – rather than just looking like a chunk of a line, plane, or volume (and so forth).

Previously, I’ve judged 2D, 3D, and 4D shapes. But, as my impostor failed to mention, fractals are another good category because they don’t neatly fit into a specific dimension.

For a 2D shape, the total amount of shape you have scales with the square (2^nd power) of the edge length. If you double every edge of a square, you wind up with a new square with 2² = 4 times as much area.

Similarly, for a 3D shape it scales with the 3^rd power. Doubling every edge of a cube creates a cube with 2³ = 8 times as much volume. A 4D shape it scales with the 4^th power, and so on.

But if you double all the edges of a fractal, it doesn’t always work how you’d expect. We’ll go over that as we see examples. As before, we’re judging these shapes as fine, lame, awesome, underrated, and overrated.

Fine: Sierpinski Triangle

My impostor started off with the Koch snowflake, saying it’s the first fractal people learn. But I think the first fractal people learn is actually the Sierpinski triangle.

There are many equivalent ways to construct a Sierpinski Triangle. One way is to start with an equilateral triangle, split it into 4 smaller equilateral triangles, and remove the upside-down center one. Then you take each remaining triangle and remove their center triangles and do the same process. Repeat this to infinity.

If you double the side length of a Sierpinski triangle, you get a bigger Sierpinski triangle made up of 3 copies of the old one. So its dimension (technically its “Hausdorff dimension”) is a number x such that 2^x = 3. Algebra students will recognize this as log₂(3) which is about 1.585.

What’s wild is that if you do a similar process with a tetrahedron, doubling each side gets you a bigger Sierpinski tetrahedron made of 4 copies of the old one. So its dimension is log₂(4) which is just 2 – we turned a 3D tetrahedron into a 2D shape just by cutting it up!

So the Sierpinski tetrahedron is cool. But the ordinary Sierpinski triangle is just a pretty typical fractal. I’d say it’s fine.

Lame: Cantor Set

I gotta say, basically all fractals are at least okay, but the Cantor set is definitely on the lamer side of okay. It’s like a 1 dimensional version of the Sierpinski triangle, or I guess technically a log₃(2) ≈ 0.6309 dimensional version. You take out the middle third of a line segment each time instead of the middle quarter of a triangle. The one thing it’s got going for it is that it contains as many points as an entire real line, but if you pick any random point the odds are 100% that it won’t be in the Cantor set. But this is also true of loads of other fractals with dimension between 0 and 1. So the Cantor set is lame.

Awesome: Mandelbrot Set

Yeah, yeah, I know it’s cliché, but the Mandelbrot set is iconic for a reason. Rethinking the idea of fractals and what counts as self-similarity? Being generated from complex numbers? Having a 2D boundary but also being a 2D fractal? Yep, totally deserves its status as the poster child of fractals. Plus, it is constructed out of the Julia sets which are also awesome, and the Feigenbaum constant which is cool too.

Overrated: Romanesco Broccoli

Sure, romanesco broccoli looks cool and is a fractal. But somehow people forget that regular broccoli is also a fractal! Romansco is just pointier! I suppose pointy fractals kinda look cooler, but I guarantee that if people were used to eating romanesco then regular broccoli would be the cool one.

Regular broccoli: Also a fractal!

Underrated: Dragon curve

My impostor seems to think that dragon curves are lame. This is not correct. Since at least one person thinks they are lame, and they are not lame, dragon curves are hereby declared to be underrated.

Have you ever gotten bored and started folding a piece of paper in half again and again? Next time, make all those folds in the same direction, and then start partially unfolding it, with each fold being a right angle. Guess what shape it makes? A dragon curve. This same construction is closely related to the Gray code, which is a way of counting in binary where you only need to change one bit at a time as you count up. It’s super useful in signal processing.

And the fact that the dragon curve can tile the plane also isn’t a coincidence. You can see the dragons tiling the inside of the dragon curve itself! The entire dragon curve is 2 dimensional, but its boundary has dimension of about 1.5236. Dragon curves are not lame and don’t let anyone tell you they are.

Coming soon: Preserved Fish, boss of New York City