July 1st, 2019, 02:49 PM  #1 
Senior Member Joined: May 2015 From: Arlington, VA Posts: 435 Thanks: 28 Math Focus: Number theory  Regular convex polygon in another
What regular convex polygons of N sides, each equaling S, can fit into those of N1 sides?
Last edited by Loren; July 1st, 2019 at 02:57 PM. 
July 1st, 2019, 03:20 PM  #2 
Global Moderator Joined: Dec 2006 Posts: 20,823 Thanks: 2160 
None of them can.

July 1st, 2019, 04:43 PM  #3 
Senior Member Joined: May 2015 From: Arlington, VA Posts: 435 Thanks: 28 Math Focus: Number theory 
Then, does it make sense to ask: What regular convex polygons of N sides, each equaling S, can fit into those of N+1 sides? 
July 1st, 2019, 04:51 PM  #4 
Member Joined: Jun 2019 From: USA Posts: 69 Thanks: 27 
That does make sense, and my guess would be all of them. As N increases, the angle defining one side gets narrower ($\displaystyle 2\pi/N rad$). If the side length remains constant, the circumscribed circle gets larger. Of course, we do have to check to make sure no vertices will poke out. It might be worth plotting up to N=10 or 20 to see what they look like. (Edit: I got curious and plotted it. Not exactly a proof, but enough for me to put money on it.) Last edited by DarnItJimImAnEngineer; July 1st, 2019 at 05:19 PM. 
July 1st, 2019, 07:14 PM  #5 
Senior Member Joined: May 2015 From: Arlington, VA Posts: 435 Thanks: 28 Math Focus: Number theory  
July 1st, 2019, 08:15 PM  #6 
Member Joined: Jun 2019 From: USA Posts: 69 Thanks: 27 
Now that would be easier to (dis)prove. Find the expression for the radii of the inscribed and circumscribed circles (simple with trig functions) and see if $\displaystyle r_{outer}(N) < r_{inner}(N+1) \forall N >= 3$.

July 2nd, 2019, 02:29 PM  #7 
Member Joined: Jun 2019 From: USA Posts: 69 Thanks: 27 
A regular triangle will not spin freely inside a square of the same side length. Nor will a square rotate in a pentagon. The pentagon and higher Ngons will.


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convex, polygon, regular 
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