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MattcAnderson 2021-10-16 15:45

17-gon
 
Fermat Search dot org

Assuming F0 = 3, and that is the smallest Fermat number,

F1 = 5, and F2 = 17.

This implies that a regular 17-gon can be constructed with
pencil and compass and straight-edge.

See a YouTube video "The Amazing Heptadecagon (17-gon ) - Numberphile
Brady Heron is usually the star of that channel
This video was made in 2015.

[URL="https://www.youtube.com/watch?v=87uo2TPrsl8"]17-gon[/URL]

Good fun.
Matt

xilman 2021-10-16 18:15

[QUOTE=MattcAnderson;590762]Fermat Search dot org

Assuming F0 = 3, and that is the smallest Fermat number,

F1 = 5, and F2 = 17.

This implies that a regular 17-gon can be constructed with
pencil and compass and straight-edge.

See a YouTube video "The Amazing Heptadecagon (17-gon ) - Numberphile
Brady Heron is usually the star of that channel
This video was made in 2015.

[URL="https://www.youtube.com/watch?v=87uo2TPrsl8"]17-gon[/URL]

Good fun.
Matt[/QUOTE]A 257-gon can also be so constructed. Never seen anyone do so.

a1call 2021-10-16 18:53

[QUOTE]
In 1894, Hermes completed his decade-long effort to find and write down a procedure for the construction of the regular [B]65537-gon[/B] exclusively with a compass and a straightedge. His manuscript, with [B]over 200 pages[/B], is today located at the University of Göttingen
[/QUOTE]

[url]https://en.wikipedia.org/wiki/Johann_Gustav_Hermes[/url]

Viliam Furik 2021-10-16 18:57

[QUOTE=MattcAnderson;590762]Brady Heron is usually the star of that channel
[/QUOTE]

Brady Haran

Dr Sardonicus 2021-10-17 02:07

[QUOTE=xilman;590789]A 257-gon can also be so constructed. Never seen anyone do so.[/QUOTE]You might like [url=http://facstaff.susqu.edu/brakke/constructions/big-gon.htm]Constructing 17, 257, and 65537 sided polygons[/url]

See also the references in Wolfram Mathworld's [url=https://mathworld.wolfram.com/257-gon.html]257-gon[/url]. The one by author Richelot, F. J. looks like it's right up your alley.

In [color=red][b]WARNING![/b] 33MB PDF file![/color] [url=http://pyrkov-professor.ru/Portals/0/Mediateka/18%20vek/nahin_p_j_dr_euler_s_fabulous_formula_cures_many_mathematica.pdf]Dr. Euler's fabulous formula cures many mathematical ills[/url], Notes to Chapter 1, we find[quote]18. Michael Trott, "cos(2n/257) à la Gauss," Mathematics in Education and Research 4, no. 2,1995, pp. 31-36.

You can read the details of what is involved in constructing the 257-gon, at a level a bit deeper than here, in

Christian Gottlieb, "The Simple and Straightforward Construction of the Regular 257-gon," Mathematical Intelligencer, Winter 1999, pp. 31-36.

The author's title is a bit of tongue-in-cheek humor, as he ends his essay with the line "I wish the reader who wants to pursue the construction in all details good luck."[/quote]

MattcAnderson 2021-10-18 05:05

constructable polygons and Fermat numbers
 
Thanks for the useful comments everyone.

See the Wikipedia article on [URL="https://en.wikipedia.org/wiki/Constructible_polygon"]Constructable polygon[/URL].

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

a1call 2021-10-18 05:28

[QUOTE=MattcAnderson;590932]Thanks for the useful comments everyone.

See the Wikipedia article on [URL="https://en.wikipedia.org/wiki/Constructible_polygon"]Constructable polygon[/URL].

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[/QUOTE]
Hi Matt,
I don’t think that is quite correct. From your link:
[QUOTE]

A regular n-gon can be constructed with compass and straightedge if and only if n is the product of a power of 2 and any number of distinct Fermat primes (including none).

[/QUOTE]
So 9-gon is not constructible what are is:
3*2^n
5*2^n
…..

MattcAnderson 2021-10-18 06:16

You are correct A1call.

A square is constructible. Also an octagon is constructible (8 sides). So any n-gon 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 n-gon)[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 n-gons that are constructible?

Regards,
Matt

a1call 2021-10-18 06:30

[QUOTE]
A regular n-gon can be constructed with compass and straightedge if and only if n is the product of a power of 2 and any number of distinct Fermat primes (including none).
[/QUOTE]

[QUOTE]
Since there are 31 combinations of anywhere from one to five Fermat primes, there are 31 known constructible polygons with an odd number of sides.
[/QUOTE]

The 2 statements from Wikipedia are contradictory IMHO. The 2nd statement does not seem to be correct.

I too would like to have an expert weigh in. But I can tell you that a 9-gon is not constructible AFAIK.

[QUOTE]
Although [B]a regular nonagon is not constructible with compass and straightedge[/B] (as 9 = 3^2, which is not a product of distinct Fermat primes), there are very old methods of construction that produce very close approximations. It can be also constructed using neusis, or by allowing the use of an angle trisector.
[/QUOTE]

[url]https://en.wikipedia.org/wiki/Nonagon#:~:text=Although%20a%20regular%20nonagon%20is,use%20of%20an%20angle%20trisector[/url].

According to the 1st (correct statement) there can only be 5 known odd-numbered constructible regular polygons not 31.
Unless we discover more Fermat primes.

MattcAnderson 2021-10-18 06:48

1 Attachment(s)
Yes, an expert would help here.
Attached is an image from Wikipedia Constructible polygons.
It enumerates all the odd numbers n such that that n-gon 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 n-gons are geometrically constructible.

Good night.

Matt

a1call 2021-10-18 06:52

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 15-gon can be constructed. I assume similar processes can be used for other combinations.
1/3-1/5= 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).


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