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Pocket maths: mathy broccoli


This post is about showing you how mathematics is beautiful and how it occurs naturally in the world that is around us. In two previous posts (here and here) I talked about fractals. Today I am going to do the same thing, except now I will use broccoli as the example, instead of some weird set on the complex numbers!

Here's two pictures of broccoli:
which one is bigger? There's only two possible answers:
  1. Exhibit A is bigger
  2. Exhibit B is smaller
right? WRONGDon't be fooled like Joey! Options 1 and 2 are the same...
Going back to the matter at hand, which one is bigger? The right answer is exhibit A, but I don't really expect you to get that. The actual question is, how much bigger is A, when compared to B?

In fact, B was "removed" from inside A! But they both look like perfectly fine broccoli, right? This is one of the properties of fractals: self-similarity. Fractals usually exhibit this very interesting behaviour: you keep zooming in, and the things you see while you zoom in still look like the original one! (for example in this video, we are zooming in on the Mandelbrot set; by second 13, we get the same shape with which we started!).

I'll show you the two pictures again, but this time without the background removed: you can use it to get a relative scale of the two images.

This self-similarity can also be seen from a recursive point of view, by which I would define broccoli this way:
broccoli: a green vegetable composed of a stalk and smaller broccoli.

If you take a moment to think about it, this is exactly what is going on! Broccoli is the vegetable that has smaller versions of itself on top of a stalk.
Of course I went even further, and from within B I removed this:

At this point everything is so small, Mother Nature never bothered to finish up the details, and so there isn't much zoom that I could keep on doing.
Now let us zoom out. I started out with this:

then I extracted exhibit A, then from it I got exhibit B, and from it I got Broccoli Jr. Jr. Jr.. As a final image, I outlined for you the self-similar pattern of broccoli. Next time you eat broccoli, take the time to appreciate the fractals you are eating!


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