Marching Squares

June 23rd, 2018

An animation based on Mathologers great video on Deadly Marching Squares. This shows a nice visual way of describing the sequence of pairs $$m$$ and $$n$$ such that $m^2 + m^2 = n^2 \pm 1 .$ Starting with $$m=1$$ and $$n=1$$, we can build up the sequence with a recurrence relation. Can you figure it out from the animation?

The solution to this problem gives us an interesting intertwined recurrence relation \begin{align} m &\mapsto m + n\\ n &\mapsto 2m + n \end{align} giving use two sequences, one for $$n$$ and one for $$m$$. \begin{align} n &= \{1,3,7,17,41,99,239,577,\ldots\}\\ m &= \{1,2,5,12,29,70,169,408,\ldots\} \end{align} . And we can check that indeed \begin{align} 1^2 + 1^2 &= 1^2 + 1\\ 2^2 + 2^2 &= 3^2 - 1\\ 5^2 + 5^2 &= 7^2 + 1\\ 12^2 + 12^2 &= 17^2 - 1\\ &\vdots \end{align} .

Interestingly, if you look at the sequence you get by diving terms in $$n$$ by those in $$m$$, you get consecutive approximations to the continued fraction for $$\sqrt{2}$$! In other words \begin{align} \frac{3}{2} &= 1 + \frac{1}{2}\\ \frac{7}{5} &= 1 + \cfrac{1}{2+\cfrac{1}{2}}\\ \frac{17}{12} &= 1 + \cfrac{1}{2+\cfrac{1}{2+\cfrac{1}{2}}}\\ &\vdots \end{align} which are better and better approximations of $$\sqrt{2}$$. This comes from the fact that $$\sqrt{2}$$ is an irrational number. This is why we get the alternating $$\pm 1$$ after every term. You can watch Mathologers full video for a neat proof involving the marching squares which inspired these animations, as well as further details. Essentially, being off by $$1$$ and no more reflects the fact that continued fractions give us the best possible approximations to irrational numbers.