20211016, 15:45  #1 
"Matthew Anderson"
Dec 2010
Oregon, USA
953 Posts 
17gon
Fermat Search dot org
Assuming F0 = 3, and that is the smallest Fermat number, F1 = 5, and F2 = 17. This implies that a regular 17gon can be constructed with pencil and compass and straightedge. See a YouTube video "The Amazing Heptadecagon (17gon )  Numberphile Brady Heron is usually the star of that channel This video was made in 2015. 17gon Good fun. Matt 
20211016, 18:15  #2  
Bamboozled!
"ð’‰ºð’ŒŒð’‡·ð’†·ð’€"
May 2003
Down not across
2^{4}·13·53 Posts 
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20211016, 18:53  #3  
"Rashid Naimi"
Oct 2015
Remote to Here/There
2^{2}×3×181 Posts 
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20211016, 18:57  #4 
"Viliam FurÃk"
Jul 2018
Martin, Slovakia
2×353 Posts 

20211017, 02:07  #5  
Feb 2017
Nowhere
2^{2}·1,279 Posts 
You might like Constructing 17, 257, and 65537 sided polygons
See also the references in Wolfram Mathworld's 257gon. The one by author Richelot, F. J. looks like it's right up your alley. In WARNING! 33MB PDF file! Dr. Euler's fabulous formula cures many mathematical ills, Notes to Chapter 1, we find Quote:


20211018, 05:05  #6 
"Matthew Anderson"
Dec 2010
Oregon, USA
953 Posts 
constructable polygons and Fermat numbers
Thanks for the useful comments everyone.
See the Wikipedia article on Constructable polygon. Assuming that there are only 5 Fermat primes, then every constructible polygon has a number of sides s with s = 3^e1 * 5^e2 * 17^e3 * 257^e4 * 65,537^e5. where e1, e2, e3, e4, and e5 are in the infinite set 0,1,2,... So polygons with number of sides like 9 (3*3) , 15 (3*5) , and 51 (17*3) are constructible. Enjoy. Matt 
20211018, 05:28  #7  
"Rashid Naimi"
Oct 2015
Remote to Here/There
2^{2}·3·181 Posts 
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I donâ€™t think that is quite correct. From your link: Quote:
3*2^n 5*2^n â€¦.. 

20211018, 06:16  #8 
"Matthew Anderson"
Dec 2010
Oregon, USA
953 Posts 
You are correct A1call.
A square is constructible. Also an octagon is constructible (8 sides). So any ngon with n=2^m is constructible. According to Wikipedia (constructible polygon article), there are infinitely many constructible polygons, but only 31 with an odd number of sides are known. 5 Fermat primes are known. Here I try to work out (unsuccessfully) Why 31  from article my combinatorics skills are not that good 3, 5, 17, 257, and 2557 (one Fermat prime each) [5 count] 3*5, 3*17, 3*257, 3*2557 then 5*17, 5*256, 5*2557 then 17*256, 17*2557 then 257*2557 (two Fermat primes each) [4+3+2+1=10 count] 3*5*17, 3*5*257, 3*5*2557 then 3*17*257, 3*17*2557 then 3*257*2557 (three Fermat primes each ) [3+2+1 = 6 count] 3*5*17*257, 3*5*17*2557, 3*5*257*2557, 3*17*257*2557, 5*17*257*2557 (four Fermat primes each) [5 count] 3*5*17*257*2557 (five Fermat primes for sides of this ngon)[1 count] So add 5+10+6+5+1 = 26 errrrr I seem to have missed five somewhere. It should be 31. Oh well, going to post anyway. oops the fourth Fermat prime is actually 65,537 so there is an error above in my workings out. to be clear, 2557 should be 65537. Who can show that there are only 31 known ngons that are constructible? Regards, Matt Last fiddled with by MattcAnderson on 20211018 at 06:41 Reason: didn't read before writing 
20211018, 06:30  #9  
"Rashid Naimi"
Oct 2015
Remote to Here/There
2^{2}·3·181 Posts 
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I too would like to have an expert weigh in. But I can tell you that a 9gon is not constructible AFAIK. Quote:
According to the 1st (correct statement) there can only be 5 known oddnumbered constructible regular polygons not 31. Unless we discover more Fermat primes. Last fiddled with by a1call on 20211018 at 06:44 

20211018, 06:48  #10 
"Matthew Anderson"
Dec 2010
Oregon, USA
3B9_{16} Posts 
Yes, an expert would help here.
Attached is an image from Wikipedia Constructible polygons. It enumerates all the odd numbers n such that that ngon is constructible. I counted the numbers in the file and there are 31 as there should be. So we can agree that there are 31 odd numbers n such that those ngons are geometrically constructible. Good night. Matt Last fiddled with by MattcAnderson on 20211018 at 06:49 
20211018, 06:52  #11 
"Rashid Naimi"
Oct 2015
Remote to Here/There
2^{2}×3×181 Posts 
Thank you very much Matt I stand corrected. So as long as the Fermat primes have a power of less than 2 then the polygon is constructible.
I did not know that. Ok, I think I can see how a 15gon can be constructed. I assume similar processes can be used for other combinations. 1/31/5= 2/15 So by centering the 2 angles you would get 1/15th of a circle on each side (or you can just bisect it). Last fiddled with by a1call on 20211018 at 07:50 