Angle Trisection: Construction vs. Drawing
Date: 10/17/2001 at 15:31:36 From: Joe Maddox Subject: Trisecting an angle Has anyone ever divided an angle into three equal parts by construction? I have been told it has not been accomplished. Thank you, Joe Maddox, Ocala, FL
Date: 10/17/2001 at 17:14:04 From: Doctor Tom Subject: Re: Trisecting an angle Hi Joe, Using the standard methods of construction with a straightedge and compass, it can be proven that it is impossible to trisect an arbitrary angle. People who think they have solved the problem usually make one of two mistakes: 1) They trisect a particular angle that happens to allow a trisection. For example, anyone can trisect a 90-degree angle. 2) They do not understand the "rules" of straightedge and compass construction. For example, if you are allowed to make two marks on the straightedge to turn it into a sort of ruler, you can trisect any angle. But the offical rules do not allow this. - Doctor Tom, The Math Forum http://mathforum.org/dr.math/
Date: 10/17/2001 at 21:08:42 From: Joe Maddox Subject: Re: Trisecting an angle Thanks for the quick reply. I have trisected arbitrary angles up to 90 degrees. Don't laugh until you see it. Only a straightedge and a compass. No measuring. The drawing must be precise for the angle to be correct. Where could I submit my effort for confirmation? Thanks, Joe Maddox
Date: 10/17/2001 at 23:07:52 From: Doctor Peterson Subject: Re: Trisecting an angle Hi, Joe. I notice something in what you just said that indicates where you are misunderstanding the trisection problem. In the Dr. Math archives, at: Trisecting an Angle http://mathforum.org/dr.math/problems/henson11.21.96.html Dr. Tom listed two mistakes people commonly make (trisecting only a specific angle, or using the wrong tools). But there is another that is even more common: not recognizing what we mean by an exact trisection. When you say that "the drawing must be precise," you show that it is the drawing itself that you have been focusing on. But to a mathematician, the drawing itself is nothing. It is only a representation of something that really happens in an ideal world where lines have no thickness, and so on. In that world, we can prove that a construction is ABSOLUTELY exact; either it meets the precise point you claim, or it is a false construction. And since this is the world of the mind, ONLY proofs count. It doesn't matter how good a drawing you make, it proves nothing. So unless you can prove that your construction really works exactly, you have nothing to show anyone. And we know that you can't, because it has been PROVEN that such a construction can't be done. I've seen many constructions that come remarkably close, usually just because they are very complex; there is nothing at all impressive about a close approximation. Please don't waste your time on this, as so many people have. And see, from the Dr. Math FAQ: "Impossible" Geometric Constructions http://mathforum.org/dr.math/faq/faq.impossible.construct.html - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/
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