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Let $u_1,u_2,u_3,\dots$ be a permutation of the integers greater than $2$.

A unit circle is in a regular $u_1$-gon, which is a regular $u_2$-gon, which is in a regular $u_3$-gon, ad infinitum. Each polygon is as small as possible (they do not have to be concentric).

What permutation $u_1,u_2,u_3,\dots$ minimizes $R$, the radius of the outermost polygon?
And what is an approximation of that minimum $R$?

Could the optimal permutation just be $3,4,5,\dots$ ?

Based on page 7 of Tightly Circumscribed Regular Polygons, an upper bound for $R$ is $2\times 3.5809=7.1618$ (I multiply by $2$, because in the article the cetral circle has radius $1/2$.) I suspect $R$ could be much lower, maybe around $5$.

Related: Smallest regular $m$-gon covering a regular $n$-gon

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  • $\begingroup$ What is meant by the "radius" of a polygon? $\endgroup$
    – Gerry Myerson
    Commented May 14, 2023 at 2:44
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    $\begingroup$ @GerryMyerson By radius I mean the distance from the centre to a vertex. I've added a link to clarify. $\endgroup$
    – Dan
    Commented May 14, 2023 at 3:01
  • $\begingroup$ Do you expect the minimum to occur with non-concentric polygons? $\endgroup$
    – LSpice
    Commented May 14, 2023 at 10:58
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    $\begingroup$ @LSpice Yes, I do. If we assume the polygons have a finite number of sides, then any permutation would have some coprime consecutive polygons, in which case minimizing the outer one would (I think) mean that they are not concentric. On the other hand, in the following permutation the polygons are all minimized and concentric: $s(4),s(3),s(5),s(7),s(9),...$ where $s(k)$ means $k, 2k, 2^2k, ..., 2^\infty k$ but it can be shown that this permutation is not as economical as say $3,s(4),s(6)$ then $s(5),s(7),s(9),...$. $\endgroup$
    – Dan
    Commented May 14, 2023 at 14:23
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    $\begingroup$ Re, I understand what you mean geometrically, but $s(4), s(3), s(5), \dotsc$ is not a permutation of the integers greater than $2$ (because, for example, $3$ has infinitely many predecessors), and similarly for $3, s(4), s(6), s(5), \dotsc$. Do you mean to allow this situation? $\endgroup$
    – LSpice
    Commented May 14, 2023 at 15:22

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