Let $T=\{[x_1,x_2,x_3,x_4] | x_1^2+x_2^2=1,x_3^2+x_4^2=1\}$
The map $r:[0,2\pi]^2 \rightarrow T$ given by
$ e(t)=[\cos(at),\sin(at),\cos(bt),\sin(bt)] $ be a parameterization of a curve on the set $T$, where $a,b \in \mathbb{R}$
What is the significance of the ratio $\frac{a}{b}$? What difference does it make rather $\frac{a}{b} \in \mathbb{Q}$ or $\frac{a}{b} \in \mathbb{R}-\mathbb{Q}$? My gut is telling me it has something to do with whether the map is onto or not. Like if $\frac{a}{b} \notin \mathbb{Q}$ then perhaps the map will be onto for some reason, but I don't know how to go about exploring this idea, much less make it rigorous.