I am viewing the proof that $\mathbb{Z}[i]$ is a Euclidean domain, but I'm having a very hard time imagining it geometrically. Or in other words, showing that the size function exists by viewing the elements of that ring on the complex plane.
Proof:
We see that the elements of that ring form a square lattice in the complex plane and that if we write some $a$ in $\mathbb{Z}[i]$ as $re^{iθ}$ then $(a)$ is obtained from the lattice $\mathbb{Z}[i]$ by rotating through an angle θ and stretching though a factor of f. $a = 2+i$
We then see that for any complex number b, there's a point of the lattice $(b)$ whose square distance is less than $|a|^2$.
Can someone briefly explain the intuition behind this and how to imagine the idea (a) as squares on the lattice and see through that, that there exists a division algorithm for any two elements in $\mathbb{Z}[i]$?
Thanks so much