I do not understand what is happening in the image below.
It claims that if the rotation is irrational then every orbit will be dense. But this seems wrong. If $\alpha=\pi/4$, for instance, then $0$ would be periodic, right?
2026-03-28 21:57:01.1774735021
rotations of a circle
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When they say "rotating by angle $\alpha$", they mean "rotating by angle $2\pi\alpha$". You can tell, because they use $\mathbb{R}/\mathbb{Z}$ to represent the circle.