Assistance in recognizing & visualizing a specific topology

Question

I’m a bit out of my element asking math questions here, but hopeful someone can point me in the right direction. Thanks in advance!

Questions:

1. Do the two tables illustrated below each form a shape in math-topology? I’m thinking they both form a torus shape; some sort of “lumpy-torus” shape?

2. How would one go about graphically visualizing these table-shapes? Can anyone provide a little direction to a specific software/tool that would allow me to create a computer visual?

Answer 1

To reach something like what you're talking about, the most straightforward interpretation I can think of is to proceed along the following steps:

1. Treat each table as a group homomorphism from $\mathbb Z^2$ to an infinite cyclic group generated by $\sqrt a$ or $\sqrt b$.
2. Let $K$ be the kernel of this map; so on the left side, $K_a=\langle(1,1)\rangle$ and on the right side, $K_b=\langle(5,3)\rangle$.
3. Since $\mathbb Z^2$ embeds in the $2$-dimensional Lie group $\mathbb R^2$, each of $K_a$ and $K_b$ is automatically a (closed) subgroup of $\mathbb R^2$, so we can form the quotient group $\mathbb R^2/K_a$ or $\mathbb R^2/K_b$, which is another $2$-dimensional manifold with a different topology.

In that case, the result is a cylinder, not a torus. Ideally one would like to draw the original $\mathbb Z^2$ grid on this cylinder; I'm not aware of software that would do that. You could do it on actual paper, though; cut, rolled into a tube, and glued back together.