Define the torus $$(S^1)^3=\left\{ (e^{2\pi ir_1},e^{2\pi ir_2},e^{2\pi ir_3});\ \ r_1,r_2,r_3\in \mathbb R\right\}$$ and consider the dense subgroup $$Q:=\left\{ (e^{2\pi ir_1},e^{2\pi \alpha r_1},e^{2\pi ir_3})\right\}$$ where $\alpha\in \mathbb R\setminus\mathbb Q$.
Now consider the projection $$p:(S^1)^3\to (S^1)^2$$ $$ (e^{2\pi ir_1},e^{2\pi ir_2},e^{2\pi ir_3})\mapsto (e^{2\pi ir_1},e^{2\pi ir_3})$$ I would like to visualize the restriction $p|_Q$. I mean an illustration figure with the following components: The space $Q$ and the fiber $(p|_Q)^{-1}(y)$ as a dense subset of the circle $S^1$ and the torus as the base. Any good pictures!
Suggestion: Since $Q\cong \mathbb R\times S^1$ and the factor $\mathbb R$ makes $Q$ dense, then neglecting $S^1$ allows us to correspond $Q$ to a skew (dense) line in the base?? Does this help to draw a nice picture?