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Context: I'm trying to see which regular polygons can be built using points generated from a wallpaper group. I know that if a regular polygon has more than a certain amount of these vertices, some pairs of them will have to be related by a translation. If these pairs form certain impossible angles, I can arrive at a contradiction. However, these pairs can end up parallel, in which case I'm forced to answer the question in the title (copied below for convenience).

Question: For which regular $n$-gons are there two sides/diagonals with a rational ratio other than $1$?

From this answer, which takes care of the specific case side to diagonal, I can guess that answering this question will involve some knowledge on algebraic numbers, which I unfortunately lack. I know that regular hexagons have the aforementioned property (since their side to longest diagonal ratio equals $2$), so that all regular $6n$-gons have the same property too. But other than that, I'm completely lost on what to do.

EDIT: Meant rational, not integral. I hope this isn't an issue.

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  • $\begingroup$ Related: math.stackexchange.com/a/519115/15624 $\endgroup$ Apr 6, 2020 at 17:20
  • $\begingroup$ @AngelaRichardson Do you know if this argument could be adapted to my more general case? Alternatively, do you know where I could read the necessary theory to understand that solution? $\endgroup$
    – ViHdzP
    Apr 6, 2020 at 17:25
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    $\begingroup$ I'm don't know how to adapt the argument to the more general case. For an accessible introduction to Galois groups, I would recommend reading nrich.maths.org/1422. $\endgroup$ Apr 7, 2020 at 19:38

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