Can a Regular Polygon with 17 Sides be Constructed Using Only Compass and Ruler?

[Mafer]

I want to draw a polygon with 17 sides by using only the compasses and an unscaled ruler...however, many times i tried, I just managed to draw a polygon with 23 sides...any site?

 

[AlphaNumeric]

Are you sure you end up with a 23 sides regular polygon? Because I was under the impression that 23 sides wasn't possible. Certainly it's not listed here.

 

[shmoe]

23 sided polygon is not constructible. An n-sided reguar polygon is constructible if and only if n is the product of distinct fermat primes (primes of the form 2^(2^m)+1) and some power of 2. Known options for fermat primes are currently only 3, 5, 17, 257, and 65537.


I haven't checked if this works for a 17-sided construction, but I'm inclined to trust in Conway:

http://mathforum.org/kb/message.jspa?messageID=1078422&tstart=0

 

[Mafer]

I'm sure it's a 23 sided...because I drew the things about twenty times and all end up with 23 and not 17...
anyone can tell me what's wrong? Is there something I might forgotten?

 

[VietDao29]

I'm sure it's a 23 sided...because I drew the things about twenty times and all end up with 23 and not 17...
anyone can tell me what's wrong? Is there something I might forgotten?

Maybe it would be good that you show us your steps, and we may verify it for you. :)
Ther may be something wrong in your approach.

 

[Mafer]

I drew a cirlcle and a horizontal line,AB,through the centre...and i made a straight line, Z, 90 degress from the AB...and divide the line(the ones on above the horizontal line) into 4, marked the 1/4 as C. from C I draw a line towards A and from there, I divide the degrees between Z and AC into 2where, I marked D. and from there, I divide it one more times, and marked and E. from C, I draw another point F which have the same distance as CD. then, I divide AF into 2. the centre is drawn a line straight up(90 degress), and marked the point that cross the circle, G. using compasses, I measure from A to G. With the same distance, I draw another point from G...and another...and another...but arround the third circle to mark, I stop and there are many empty and unmarked. Using the two dots besides, i divide them into two and marked a point. When all points are made, there...23 sides!
If you do not understand what I mean, please tell me...that's because here, in my year, I don't learn MAths in English, and the government had only changed the education policy years before for those in first year...so I am the last secong batch who study Maths and Science in non-English...please forgive me, and it's a pleasure if you want to teach me how to improve my English in MAths and Science...because according to the new policy, my A-level(which is GCE or STPM, here, or matriculations) and education after this...all MAths and Science will be in English...:(

 

[robert Ihnot]

shmoe: I haven't checked if this works for a 17-sided construction, but I'm inclined to trust in Conway.

It was Gauss who discovered the whole matter of construction beyond that of the Ancient Greeks. Discovering the constructability of the 17 sided regular polygon, he said this was the first progress on the problem in 2000 years. He made this discovery by 19, and was not even a decided "math major" at the time.

Constructability is defined as employing a straight edge and compass. From this we can do the four ordinary functions of arithmetic, and we can find SQUARE roots. We can not, as a general rule, find cube roots and thus trisect an angle. (However Archimedes was able to do this employing a paper strip.) The question then of constructability is thus very narrowly defined, but it allows us to translate a plane geometry construction into an algebraic equation. Thus, for example, we can determine

$$cos(36) =\frac{1+\sqrt5}{4}$$ which allows us to construct the 5 sided pentagon.

Gauss did publish in Disquisitiones Arithmeticae, where you can find it, the algebratic form of the side of the 17-sided Heptadecagon, but he did not carry this form to completion. It is complicated, and has little practical use.

The construction is partly shown: http://www.geocities.com/CapeCanaveral/Lab/8972/lessons/Heptadecagon.html

 

[Mafer]

then how did the same wat...could developed into a 23 sided polygon?

 

[robert Ihnot]

You said you made a straight line, 90 degrees from AB? YOu did not say where it touched, if it did, AB. Maybe you could include a picture somehow, it's been done.