= Subgroups generated by circle rotations
{title2=$H_\alpha=\{e^{2\pi ik\alpha}:k\in\mathbb Z\}$}
Fix a real angle fraction $\alpha$, and repeatedly multiply by $g=e^{2\pi i\alpha}$ in the <circle group>. The generated <cyclic subgroup> is $H_\alpha=\{g^k:k\in\mathbb Z\}$. Equivalently, the angle fractions are the <fractional parts> of $k\alpha$, with one full turn identified with zero.
If $\alpha=p/q$ is a <rational number> in lowest terms, with $q>0$ and $p,q$ <coprime>, then $g^q=1$ and no smaller positive power is $1$. Hence $H_\alpha$ is a <finite cyclic group> of order $q$, consisting of equally spaced points. For $\alpha=2/7$, the successive angle fractions, starting at $k=0$, are
$$
0,\quad\frac27,\quad\frac47,\quad\frac67,\quad\frac17,\quad\frac37,\quad\frac57,\quad0.
$$
The <subgroup> is finite, so it is a <closed subgroup> of the <circle group>; it is also a <discrete subgroup>. It is not an <open subgroup>, since every ambient <neighbourhood> of any of its points contains an arc, rather than only these seven points.
If $\alpha$ is an <irrational number>, no nonzero <integer> $k$ satisfies $k\alpha\in\mathbb Z$, so the powers never repeat. The stronger statement is that $H_\alpha$ is a <dense set> in the <circle group>: every nonempty open arc contains a power $g^k$. Thus its <closure> is all of $\mathbb T$, while $H_\alpha$ itself is a countable proper <subgroup> and is not a <closed subgroup> or a <discrete subgroup>. See <irrational rotation> and <everywhere interval frequency under an irrational rotation> for the corresponding motion and its long-term frequencies.
For $\alpha=\sqrt2-1$, direct computation of the <fractional parts> of $k\alpha$ gives, to six decimal places,
$$
\begin{array}{c|rrrrrrrr}
k&0&1&2&3&4&5&6&7\\
\{k\alpha\}&0&0.414214&0.828427&0.242641&0.656854&0.071068&0.485281&0.899495
\end{array}
$$
For example, $\{3\alpha\}=3\sqrt2-4\approx0.242641$ and $\{5\alpha\}=5\sqrt2-7\approx0.071068$. These calculations specify exactly which points the diagram displays.
\Image[/rational-versus-irrational-rotations.png]
{title=Seven repeating rational-rotation points and the first eight and sixty-four irrational-rotation points}
{description=Labels give the power index $k$; the points are $e^{2\pi ik\alpha}$. The first panel uses $\alpha=2/7$, and the other two use $\alpha=\sqrt2-1$. Each finite sample is only an illustration: a finite plot cannot establish that the infinite <subgroup> is a <dense set>. Reproduce the diagram with \a[rational-versus-irrational-rotations.py][rational-versus-irrational-rotations.py].}
This example shows why algebraic generation and <closure> must be distinguished: one <irrational rotation> generates a countable <cyclic subgroup>, yet its <closure> is the whole uncountable <compact group>.
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