Constructing Polygons

Date: 06/03/98 at 10:26:43
From: Natalie Grant
Subject: Construction of polygons (decagon)

How do you construct a polygon in general? 

Example: a decagon. I knew how to do it, but I can't remember, and 
there are no book resources for this topic. So far I remember that 
there are two ways to choose from in order to construct a polygon in 
general. If you could please send me even one of these methods, I 
would be really grateful.

Date: 06/03/98 at 14:31:43
From: Doctor Wilkinson
Subject: Re: Construction of polygons (decagon)

I'm assuming you mean a regular polygon, and that you are talking 
about construction with a straightedge and compass. If you want to 
construct just any old decagon (say), all you need to do is pick any 
10 points on a circle and connect them up in order.

For regular polygons the problem is much more interesting, and it 
turns out that in fact there is no general construction, so you may 
not be remembering correctly. But the regular decagon can be 
constructed with a straightedge and compass, so you remembered 
correctly about that.

The problem of which regular polygons can be constructed with a 
straightedge and compass was solved by Gauss around 1800. He showed 
that it is possible to construct a regular polygon with n sides if and 
only if n is a power of 2 times a product of distinct prime numbers 
each of which is one more than a power of 2.  For example, 3, 5, 17, 
and 257 are all such primes, so it is possible to construct a regular 
polygon with 3, 4, 6, 8, 10, 12, 15, 16, or 17 sides, but not with 7, 
9, or 18 sides. The ancient Greeks knew how to construct regular 
polygons with 5, 10, and 15 sides, but Gauss was the first to show how 
to construct a regular polygon with 17 sides.

To construct a regular decagon, draw line segments from the center O 
of the pentagon to two consecutive vertices A and B and bisect the 
angle AOB.

To construct a line segment of length the square root of five, start 
with a line segment of length 1, draw a perpendicular to it at one 
end, and mark off a line segment of length 2. Then the hypotenuse will 
have length the square root of five by the Pythagorean Theorem.

-Doctor Wilkinson,  The Math Forum
Check out our web site! http://mathforum.org/dr.math/   

Date: 06/03/98 at 15:23:40
From: Doctor Rob
Subject: Re: Construction of polygons (decagon)

Actually, you cannot construct just any regular n-sided polygon; n 
must be of a very special form. Its prime factorization must be of 
this form:

    n = 2^e * 3^e0 * 5^e1 * 17^e2 * 257^e3 * 65537^e4

where n > 2, e >= 0, and e0, e1, e2, e3, and e4 are either 0 or 1.  
This means that you cannot construct a regular heptagon (n = 7) using 
compass and straightedge. The same can be said of n = 9, 11, 13, 14, 
18, 19, and infinitely many other values of n.

The method to construct a regular decagon (n = 10) is to construct a
regular pentagon (n = 5) inscribed in a circle, and then construct the
perpendicular bisectors of the five sides (these are also the 
bisectors of the five angles). The ten points where the five bisectors 
intersect the circle are the ten vertices of the decagon. (Five of 
them will also be vertices of the pentagon.)

To construct the regular pentagon, start with a circle with center O.  
Draw a diameter AOB, and draw the perpendicular bisector COD. Find the 
midpoint E of AO. With E as center, draw an arc with radius EC, 
intersecting the segment OB at F. Set your compass with radius CF.  
Now with center C, draw an arc intersecting circle O at G and J. With 
center G, draw an arc intersecting circle O at C and H. With center J, 
draw an arc intersecting circle O at C and I. Then connect C to G to H 
to I to J to C with line segments. CGHIJ is the regular pentagon, and 
side HI is parallel to the diameter AOB. If the radius of the circle 
is r, then the side of the regular pentagon is r*sqrt[(5-sqrt[5])/2].

The constructions of the regular 17-, 257-, and 65537-gons are much 
more complicated, but still possible.

For a table of regular polygons constructable with straightedge and 
compass whose angles are whole numbers of degrees, see the Dr. Math 
FAQ:

  http://mathforum.org/dr.math/faq/formulas/faq.regpoly.html   

-Doctor Rob,  The Math Forum
Check out our web site! http://mathforum.org/dr.math/