= Solution
Use the elementary irrational-rotation density fact: if $\alpha$ is irrational, $\{n\alpha\bmod1:n\in\mathbb Z\}$ is dense in the circle. One proof uses the <pigeonhole principle> on fractional parts of $0,\alpha,\ldots,N\alpha$. Two lie within $1/N$; taking their difference and, if needed, its negative yields an integer multiple of $\alpha$ with fractional part $\delta\in(0,1/N)$. The multiples $0,\delta,\ldots,\lfloor1/\delta\rfloor\delta$ form a $1/N$-net of the circle and are themselves integer multiples of $\alpha$ modulo one. Taking $N$ arbitrarily large proves density.
For any target coordinates $(u,v)$ modulo one, points of the line with parameter $t=u+n$ have first coordinate exactly $u$ modulo one and second coordinate $\alpha u+n\alpha$ modulo one. The density fact approximates $v$ arbitrarily well. Since products of small coordinate intervals form a neighbourhood basis of the <torus>,
$$
\boxed{\overline{\pi(L)}=T^2.}
$$
Nevertheless the point $\pi(0,\alpha/2)$ does not lie in $\pi(L)$. Such an equality would require a line parameter $t=n\in\mathbb Z$ and $\alpha(n-1/2)\in\mathbb Z$, contradicting irrationality. Hence
$$
\boxed{\pi(L)\ne T^2.}
$$
This is the <irrational winding of the torus>: dense need not mean surjective.
Back to article page
