Impossibility of Constructing a Regular Nine-Sided Polygon
Date: 04/07/98 at 23:01:23 From: fei su Subject: method to draw a regular 9 sides polygon Dear Dr. Math, My brother was given an easy method to draw a regular 9 sided polygon. His tools were only a compass and a ruler that has no units marked on it. Naturally, the regular 9 sided polygon is an approximate one judged by common sense. What he has been doing was proving it to be a real regular polygon, but he has not gotten the answer after spending more than a dozen years. I think that maybe there is no such answer. My question is: Is this drawing method useful? Is it worth it to find another easy method to draw a regular 9 sided polygon? Is there any value to spending time on this issue?
Date: 04/08/98 at 11:53:34 From: Doctor Wilkinson Subject: Re: method to draw a regular 9 sides polygon The problem is not one of practical importance, but it is certainly of great intellectual interest. Unfortunately for your brother, this problem is known to be impossible, so it would be better for him to work on something else. -Doctor Wilkinson, The Math Forum Check out our web site! http://mathforum.org/dr.math/
Date: 04/10/98 at 15:02:10 From: Doctor Wilkinson Subject: Re: method to draw a regular 9 sides polygon Hi. I think I may have given too brief a reply to your question about constructing a regular nine-sided polygon. I certainly do not want to discourage your brother or you from working on hard problems! First of all, when we say a construction problem with straightedge and compass is impossible, you may wonder how we can possibly know that. You may think that there is some trick that just hasn't been tried yet. The way we know some problems are impossible is that we can figure out just what kinds of things can and can't be done with straightedge and compass, and we can describe all this in terms of algebra. What can you do with a straightedge and compass really? You can make circles with a radius you've already got, you can draw lines between points, and you can get new points as the intersections of lines and circles or of circles and circles. Now the great French mathematician Rene Descartes discovered in the sixteenth century a way to make equations corresponding to geometric lines and curves. In particular, the equation of a straight line always has the form: ax + by = c and the equation of a circle has the form: (x - a)^2 + (y - b)^2 = r^2 where a, b, c, and r are constants, and x and y are the coordinates of a point on the line or circle. Finding the intersection of two lines, of a line and a circle, or of two circles thus is equivalent to solving a quadratic equation. So any construction that is possible with straightedge and compass is equivalent to an algebraic problem whose solution involves only addition, subtraction, multiplication, division, or extracting square roots. Using these ideas, Gauss was able to show around 1801 that the only regular polygons that could be constructed with straightedge and compass were those for which the number of sides was a product of a power of 2 and of distinct primes of the form 2^2^n + 1. For example, a regular polygon with 30 sides can be constructed because 30 is 2*3*5, and 3 is 2 + 1 or 2^2^0 + 1 and 5 is 2^2 + 1 or 2^2^1 + 1. But 9 is 3 * 3, and while 3 is a prime of the right form, the fact that you have 2 factors of 3 means it won't work. Gauss was the first to construct a regular polygon with 17 sides, which you can see is possible because 17 is 2^2^2 + 1. The primes of the form 2^2^n + 1 are known as Fermat primes, because Pierre Fermat, another Frenchman and a contemporary of Descartes, thought that perhaps all number of that form were prime. But in fact the only known primes of that form correspond to n = 0, 1, 2, 3, and 4. The next number of that form, 2^2^5 + 1, is not prime, so you couldn't construct a regular polygon with that number of sides even if you had time. There are a bunch of hard unsolved problems relating to these numbers. It isn't known whether there are any more primes among them, or whether an infinite number of them are prime, or whether an infinite number are composite. There are plenty of unsolved problems all through mathematics. Here's a webpage that will steer you to some of them: http://www.mathsoft.com/asolve/index.html Best wishes, and I apologize if my earlier reply was too short and too discouraging. -Doctor Wilkinson, The Math Forum Check out our web site! http://mathforum.org/dr.math/
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