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Dan
Dan
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edited title; added 60 characters in body
Source Link
Dan
Dan
  • 3.5k
  • 9
  • 43

All the regular $n$-gons are nested rightlytightly around a unit circle. How to order them to minimize the outer radius, and what is that minimum radius?

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 radiusradius 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

All the regular $n$-gons are nested rightly around a unit circle. How to order them to minimize the outer radius, and what is that minimum radius?

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

All the regular $n$-gons are nested tightly around a unit circle. How to order them to minimize the outer radius, and what is that minimum radius?

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

Source Link
Dan
Dan
  • 3.5k
  • 9
  • 43

All the regular $n$-gons are nested rightly around a unit circle. How to order them to minimize the outer radius, and what is that minimum radius?

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