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On Puzzling.SE there's a puzzle called "Deusovi Honeypot" which states:

The angle of the scaled-down copy is an irrational part of $\pi$, so the angle will never repeat.

What is meant by "an irrational part"? I understand it as a part of $\pi$ that is irrational. This is where my mind turns left though because this implies that there are rational parts of $\pi$; which, I can conceptualize, but not make complete sense of with respect to there being irrational parts. To elaborate, if you take any substring of digits from the digits of pi, it becomes rational (I think) because it now terminates.

By definition, an irrational number is one that cannot be written in the form $\frac{n}{d}$ where $n$ and $d$ are integers, and $d$ is nonzero. As such, if I take any substring of digits from $\pi$ (let's say $14159$), it becomes rational because it can be written as $\frac{14159}{5} = 2,831.8$ which is clearly not irrational.


As such, I think I'm missing something crucial here. What does "$x$ is an irrational part of $y$" actually mean?

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    $\begingroup$ It's not standard terminology, but I guess it means an irrational multiple of $\pi$ i.e. an angle $\alpha$ such that $\frac{\alpha}{\pi} \in \mathbb{R} \setminus \mathbb{Q}$. For example $\alpha=\frac{3\pi}{5}$ is a rational multiple of $\pi$, while $\alpha=\frac{\pi}{\sqrt{2}}$ is an irrational multiple of $\pi$. $\endgroup$
    – dxiv
    Oct 14 at 1:00

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