I saw the question asking about intervals and simple Cartesian products, but how can I visualize things like $S^1 \times S^1$, the Cartesian product of the unit 1-sphere? I understand that this is a Torus, but what should my thought process be here?
[Math] How to visualize the Cartesian product of sets
general-topologygeometry
Best Answer
A pair of angles $\theta$ and $\phi$ determine an element $(e^{i\theta},e^{i\phi})$ in $S^1\times S^1$, which in order they can be consider that give a position in the torus.
This is achieved mapping via $(e^{i\theta},e^{i\phi})\longrightarrow\left((2+\cos\theta)\cos\phi,(2+\cos\theta)\sin\phi,\sin\theta\right)$
In the picture, the blue point on the torus is specified by the two angles in green.