Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

I found the following five types.

1. $(m,n)=(kn,n)$ for $n\ge3, k\ge1$.

2. $(m,n)=(hk,2k)$ where $h\ge1$ is an odd number and $k\ge2$.

3. $(m,n)=(m,3)$ where $m\ge4$ and $m\not\equiv0$ (mod $3$).

4. $(m,n)=(m,4)$ where $m\ge3$ is an odd number.

5. $(m,n)=(3,3)$.

I'm going to write that a regular $n$-gon can be inscribed inside a regular $m$-gon for each of these sets.

1. You can get a regular $n$-gon by choosing every $k$-th point of a regular $(m=)kn$-gon.

2. You can get a regular $2k$-gon by cutting every corner of a regular $k$-gon in an 'appropriate' way. Then, you get $(m,n)=(k,2k)$. Letting $k=hk^{\prime}$, you can get $(m,n)=(hk^{\prime},2hk^{\prime})$. Then, you can get $(m,n)=(hk^{\prime}, 2k^{\prime})$ by choosing every $h$-th point of a regular $(n=)2hk^{\prime}$-gon. Note that this type excludes the $h=2l$ cases, which are special cases of 1.

3. Take a point $P_0$ on an edge of a regular $m$-gon. Next, you go counterclockwise on edges then take a point $P_1$ such that $P_0P_1=x$. Again, take $P_2, P_3$ in the same way as above. Then, there exists a value $x$ such that $P_3=P_0$ if you make $x$ larger gradually. Now you get an equilateral triangle $P_0P_1P_2$. We need some more details in the case that $x$ doesn't increase. Note that points move continuously. Note type excludes the $m=3k$ cases, which are special cases of 1.

4. Take an axis of symmetry $AB$ of a regular $m$-gon. Next, take a point $P_0$ near a point $A$ and take a longitudinal rectangle $P_0P_1P_2P_3$ such that $P_0P_1\parallel AB$ (sorry for no figures). You can get a square when you get $P_0=P_1$ if you move $P_0$. Note that this type excludes the $m=2k$ cases, which are included in 1 or 2.

5. This type is obvious.

By the way, are these all sets? Though I've tried to find another type, I neither can find any set nor can prove that these are all possible sets. Then, here are my questions.

Question 1: Does there exist another set $(m,m)$ such that a regular $n$-gon is inscribed inside a regular $m$-gon ?

Question 2: If the answer for question 1 is yes, then could you show me how to prove that? If the answer for question 2 is no, then could you show me another set?

share|improve this question
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.