Step 3: Constructing Regular Polygons
Comments on Step #3:
Regular polygons are a standard topic in the geometry component of the middle school and high school curriculum. Although geometry books often discuss the construction of equilateral triangles, squares, hexagons and octagons, the general question of constructing regular n-gons for integers n > 2 is not considered. The purpose of this section is to discuss most of what is known about this fascinating subject. (By the way, although we will show in Unit 2 that a regular septagon (7-gon) cannot be constructed with compass and straightedge alone, a middle school student discovered the remarkably close approximation displayed on the Unit 2 home page!)
What do we mean by constructing a regular polygon with n sides?
For each postive integer n > 2, a regular polygon with n sides (or simply a regular n-gon) is a polygon in a plane whose vertices lie on a circle C in a plane and whose sides are congruent line segments. If A is the center of the circle C, then A is the center of the regular n-gon. The radius of the regular n-gon is the radius of the circle C. We also use the term radius to refer to any line segment joining the center of a regular n-gon to one of its vertices.
Our objective in this part of Unit 1 is to discuss the following problem:
Construction of the Regular n-Gon Problem:
Given a line segment AB and a positive integer n > 2, construct a regular polygon RP[n] with n sides such that A is the center of the polygon and |AB| is the common length of all the line segments joining the center A to any vertex B of RP[n].
In 1.3.1 Just Do It! we showed that, given a line segment AB, it is possible to construct a regular polygon RP, (i.e. an equilateral triangle) with radii of length |AB|.
In Problem 1 of Assignment 1.3, you are asked to contruct RP (a square) with a given radius. In Problem 2 of Assignment 1.3, you are asked to show that for each integer n > 2, RP[n] can be constructed with radii lengths equal to that of a given segment AB if and only if RP[2n] can be so constructed. Therefore, since an equilateral triangle RP and a square RP can be constructed with radii of a given length, so can RP, RP, RP, RP, and in general RP[n] for any postive n that is a multiple of 2 or 3 by a power of 2. The first value of n for which the construction of the corresponding regular polygon RP[n] is not included in the peceding list is n = 5. Problem 2 of Assignment 1.3 shows that the problem of constructing a regular pentagon RP is equivalent to constructing a regular decagon RP. This raises the problem:
Regular Decagon Construction Problem:
Given a line segment AB, can we construct a regular decagon RP or a regular pentagon RP with radius |AB| using compass and straightedge only?
The answer is yes, but, as we shall see, the construction is not nearly as simple as that of the cases discussed above. However, Greek geometers of Euclid’s time were well aware of a construction because Proposition 10 of Book IV of Euclid’s Elements provides a construction for an angle of 36 degrees, which is the central angle subtended by any side of a decagon. If we can construct the isosceles triangle BAP with vertex angle of measure of 36 degrees, then we can duplicate that tringle ten times to produce the decagon.
The following diagram describes the required construction:
But how do we know that the angle we have constructed has a measure of 36 degrees? You can verify this informally with Geometer's Sketchpad by completing the following
Which regular polygons are constructible with straightedge and compass?
More precisely, for which values of n is the Construction of the Regular n-Gon Problem stated at the beginning of this part solvable? The results that we have discussed so far show this problem is solvable for n = 3, 4 and 5 as well as any multiple of these values by a positive integer power of 2. The values of n between 3 and 20 not included in this list are: 7, 9, 11, 13, 17 and 19. Are the regular n-gons for these n values constructible or not?
One of the greatest mathematicians of all time, Carl Friedrich Gauss (1777 – 1855) found a ruler and compass construction for the regular 17 – gon (called a heptadecagon) and went on to show more generally that if p is a Fermat prime number (ie. a prime number p of the form
for some non-negative integer m, then the regular p-gon is constructible with compass and straightedge only. In 1837, Laurent Wantzel showed that if a regular p–gon is constructible for a prime p, then p must be a Fermat prime. His result was combined with the result of Gauss and some other lemmas to provide the following result:
The Gauss-Wantzel Theorem: A regular n-gon can be constructed with compass and straightedge if and only if:
1) n is a Fermat prime.
2) n is a power of 2.
3) n is the product of a power of 2 and distinct Fermat primes.
Thus, a regular 7-gon is not constructible because it is prime but not a Fermat prime, while a regular 9-gon is not constructible because 9 is the square of the Fermat prime 3. A 15-gon is constructible because 15 = (5)(3) and 5 and 3 are distinct Fermat primes. A regular 11-gon, 13-gon and 19-gon are not contructible because 11, 13 and 19 are primes but not Fermat primes.
The Gauss-Wantzel Theorem is too deep and difficult to prove here.