Thread: 17-gon
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Old 2021-10-18, 20:17   #23
R. Gerbicz
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"Robert Gerbicz"
Oct 2005

30478 Posts

Originally Posted by Dr Sardonicus View Post
You might like Constructing 17, 257, and 65537 sided polygons

See also the references in Wolfram Mathworld's 257-gon. The one by author Richelot, F. J. looks like it's right up your alley.
Still in high school's math camp we constructed a regular 17-gon.

Originally Posted by a1call View Post
Similarly for 3*17=51 gon
1/3-3/17= 8/51 of a circle. Then you can bisect 3 times to get 1/51 of a circle.
There is an elementary way to show that if you can make a regular m and n-gon [using straightedge and compass] and gcd(m,n)=1 then you can make a regular m*n-gon. Because making a regular k-gon is equivalent with constructing a 2*Pi/k angle.

So we can make a 2*Pi/n and 2*Pi/m angle.

We assumed that gcd(m,n)=1 so with extended Euclidean algorithm there exists x and y integers:

n*x+m*y=1 divide this equation by m*n

x/m+y/n=1/(m*n) multiplie by 2*Pi

x*(2*Pi/m)+y*(2*Pi/n)=2*Pi/(m*n), what we needed.
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