Showing posts from April, 2019

The formula that plots itself

Pt En < 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

Pocket maths: mathy broccoli

Pt En < 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 bigger Exhibit 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 ver