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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\…