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July 1st, 2019, 02:49 PM   #1
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Regular convex polygon in another

What regular convex polygons of N sides, each equaling S, can fit into those of N-1 sides?

Last edited by Loren; July 1st, 2019 at 02:57 PM.
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July 1st, 2019, 03:20 PM   #2
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None of them can.
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July 1st, 2019, 04:43 PM   #3
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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?
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July 1st, 2019, 04:51 PM   #4
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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.)
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July 1st, 2019, 07:14 PM   #5
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Originally Posted by Loren View Post
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?
for all rotations about a common center?
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July 1st, 2019, 08:15 PM   #6
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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 $\displaystyle r_{outer}(N) < r_{inner}(N+1) \forall N >= 3$.
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July 2nd, 2019, 02:29 PM   #7
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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 N-gons will.
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