Constructing PolygonsDate: 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/ |
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