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Showing posts from April, 2019

The formula that plots itself

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PtEn< change language By the end of this blog post I hope that you know how to make mathematical drawings and why the number $$N \approx 4.85845063618971342358209596 \times 10^{543}$$ is so special.

Given a function $f(x, y)$, how can you use it to make a drawing? Well, we just imagine the whole plane as a white, clean grid, and then we fill with black the squares at the positions $(x,y)$ such that $f(x,y) > \frac12$. In a way, it is as if the function $f$ is telling us whether to use white or black, i.e. to leave the square empty ($0$) or filled in ($1$).

(more rigorously, we divide the plane in unit squares, and we give each square the coordinates of its lower-left corner)

If we take, for example, $f(x, y) = x + y$, then square $(0,0)$ would be white because $f(0, 0) = 0 < \frac12$ but the squares $(0, 1)$ and $(1, 0)$ would be black because $f(0, 1) = f(1, 0) = 1 > \frac12$.

As another example, take $f$ to be this function: $$f(x, y) = \left\lfloor mod\left(\left\…

Pocket maths: mathy broccoli

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PtEn< change language 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:
Exhibit A is biggerExhibit B is smaller right? WRONG! Don'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 zoomi…