Nice shapes of ideals of $\mathbb{Z}[i]$ from a (lattice) geometric point of view?

Question

If we draw the lattice for the ideal generated by $(2+i)$ in $\mathbb{Z}[i]$, and look at what is happening modulo $(2+i)$, we see a beautiful square, although it is rotated a little bit counterclock-wise.

I'm trying to draw an analogous picture for the ideal generated by $(4+3i)$ in $\mathbb{Z}[i]$, but unless I'm missing some points, there is no beautiful picture going on, just random points.

What am I supposed to get? Also, is there some reference to this kind of lattice-geometric approach to ideals of $\mathbb{Z}[i]$ and possibly another rings of integers?

Thanks a lot.

Possible duplicate of Ideals of $\mathbb{Z}[i]$ geometrically — Servaes, Jun 01 '16 at 20:50
Those are not random points: any ideal $\langle g \rangle \subset \Bbb{Z}[i]$ is a lattice with a nice square whose vertices are $0, g, ig, (i+1)g$. — Crostul, Jun 01 '16 at 20:52
I don't think this is a duplicate. I actually am the same guy who asked that one a while ago, and this one asks for a ideal that seems to be behaving differently — Shoutre, Jun 01 '16 at 20:54
I can't see your diagram, but I'm guessing maybe you need to zoom out a little bit more. — Robert Soupe, Jun 02 '16 at 12:25

score 4 · Accepted Answer · answered Jun 01 '16 at 21:20

Recall that in the complex plane, multiplication by a number $g$ is geometrically the composition of a dilatation of $|g|$ and a rotation of $\arg g$.

Since $\Bbb{Z}[i]$ is a lattice whose fundamental domain is the square of vertices $0,1,i, 1+i$, if you consider $g\Bbb{Z}[i]$ you will obtain a dilatated and rotated lattice, whose fundamental domain is the square of vertices $0,g,ig, (1+i)g$.

Nice shapes of ideals of $\mathbb{Z}[i]$ from a (lattice) geometric point of view?

1 Answers1