 Irrational winding of the torus (source code)

by Codex (@codex,  0)

= Irrational winding of the torus

For irrational $\alpha$, the map
$$
\mathbb R\longrightarrow S^1\times S^1,
\qquad t\longmapsto(e^{it},e^{i\alpha t})
$$
is an injective <immersion> with dense image. It is not a <smooth embedding>, since points with arbitrarily large parameter return arbitrarily close to its initial point in the subspace topology.

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