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Luis Ferroni
Luis Ferroni
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I start by stating the problem, which is already hinted in the title of the question. I do believe it is a research-level question.

Let us fix a regular $n$-gon with area $1$. What is the smallest possible area of a regular $m$-gon, so that it can cover the $n$-gon.

Of course, cover means to contain as a set. Of course, we can ignore the ill-defined scenarios, and fix $n\geq 3$ and $m\geq 3$. There is a number of situations on which it is possible to give a explicit formula (using sines and cosines) in terms of $n$ and $m$ of the desired smallest area.

Although this problem has a quite simple statement, it is not trivial, for example what the answer is for $n=5$ and $m=7$ or for $n=7$ and $m=5$. Notice that the question only asks for a covering, so that it is not required to the polygons to be concentric.

When $m=4$ I could get an answer for all even $n$, and I think I can also solve it for odd $n$. Also, the case on which $n\mid m$ is, I think, much easier since the intuitive configuration does work, and a formula can be given.

Any thoughts on this? Is there any hope to give a formula using sums and products or quotients of sines and cosines of ugly rational multiples of $\pi$ for arbitrary $n$ and $m$?

I start by stating the problem, which is already hinted in the title of the question. I do believe it is a research-level question.

Let us fix a regular $n$-gon with area $1$. What is the smallest possible area of a regular $m$-gon, so that it can cover the $n$-gon.

Of course, cover means to contain as a set. Of course, we can ignore the ill-defined scenarios, and fix $n\geq 3$ and $m\geq 3$. There is a number of situations on which it is possible to give a explicit formula (using sines and cosines) in terms of $n$ and $m$ of the desired smallest area.

Although this problem has a quite simple statement, it is not trivial, for example what the answer is for $n=5$ and $m=7$ or for $n=7$ and $m=5$. Notice that the question only asks for a covering, so that it is not required to the polygons to be concentric.

When $m=4$ I could get an answer for all even $n$, and I think I can also solve it for odd $n$. Also, the case on which $n\mid m$ is, I think, much easier since the intuitive configuration does work, and a formula can be given.

Any thoughts on this? Is there any hope to give a formula using sums and products of sines and cosines of ugly rational multiples of $\pi$ for arbitrary $n$ and $m$?

I start by stating the problem, which is already hinted in the title of the question. I do believe it is a research-level question.

Let us fix a regular $n$-gon with area $1$. What is the smallest possible area of a regular $m$-gon, so that it can cover the $n$-gon.

Of course, cover means to contain as a set. Of course, we can ignore the ill-defined scenarios, and fix $n\geq 3$ and $m\geq 3$. There is a number of situations on which it is possible to give a explicit formula (using sines and cosines) in terms of $n$ and $m$ of the desired smallest area.

Although this problem has a quite simple statement, it is not trivial for example what the answer is for $n=5$ and $m=7$ or for $n=7$ and $m=5$. Notice that the question only asks for a covering, so that it is not required to the polygons to be concentric.

When $m=4$ I could get an answer for all even $n$, and I think I can also solve it for odd $n$. Also, the case on which $n\mid m$ is, I think, much easier since the intuitive configuration does work, and a formula can be given.

Any thoughts on this? Is there any hope to give a formula using sums and products or quotients of sines and cosines of ugly rational multiples of $\pi$ for arbitrary $n$ and $m$?

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Luis Ferroni
Luis Ferroni
  • 1.9k
  • 1
  • 10
  • 28

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

I start by stating the problem, which is already hinted in the title of the question. I do believe it is a research-level question.

Let us fix a regular $n$-gon with area $1$. What is the smallest possible area of a regular $m$-gon, so that it can cover the $n$-gon.

Of course, cover means to contain as a set. Of course, we can ignore the ill-defined scenarios, and fix $n\geq 3$ and $m\geq 3$. There is a number of situations on which it is possible to give a explicit formula (using sines and cosines) in terms of $n$ and $m$ of the desired smallest area.

Although this problem has a quite simple statement, it is not trivial, for example what the answer is for $n=5$ and $m=7$ or for $n=7$ and $m=5$. Notice that the question only asks for a covering, so that it is not required to the polygons to be concentric.

When $m=4$ I could get an answer for all even $n$, and I think I can also solve it for odd $n$. Also, the case on which $n\mid m$ is, I think, much easier since the intuitive configuration does work, and a formula can be given.

Any thoughts on this? Is there any hope to give a formula using sums and products of sines and cosines of ugly rational multiples of $\pi$ for arbitrary $n$ and $m$?