Children and adolescents are often told by their parents about certain unsavory types they should avoid, lest they fall under an inappropriate influence.
The same can be said for fledgling math instructors.
Dr. Underwood Dudley, longtime DePauw University math instructor, warned students attending his lecture at Northeastern State University about a certain group of mathematicians known as “trisectors.”
Early on, students learn to bisect angles using a compass and straightedge. However, once bisected, the question arises as to how to trisect the same angle using the same method.
“You can’t,” said Dudley. “You can’t because it’s been proved you can’t. Pierre Wantzel, in 1837, proved you can’t trisect an angle with a simple compass and straightedge. That’s the beautiful thing about math; once it’s proven, it cannot change. Ever. That’s why people study math. The Pythagorean Theorem was absolutely true, remains so today, and will be true forever.”
Dudley said despite the proven theory, many mathematicians spend their lives trying to trisect angles using the Euclidean – compass and straightedge – method.
“If you leave this university and go out and teach geometry, don’t ever, ever bring up the fact that angles cannot be trisected using a compass and straightedge,” said Dudley.
“Because invariably, one of your students will make it their life’s work. Which is ridiculous, because no one will ever be able to prove that the sum of two even numbers equals an odd number.”
Which is not to say angles absolutely cannot be trisected. They can.
“The ancient Greeks found a way to do it without a straightedge and compass,” said Dudley. “All you need is a marked straightedge.”
From this point, the lecture devolved into a mathematician’s ultimate fantasy, as Dudley ran through all the methods by which angles can successfully be trisected – including Archimedes’ method, the Quadratix of Hippias, the spiral of Archimedes, a tomahawk, a carpenter’s square and a conchoid.
After observing the phenomenon of the behavior of “trisectors,” Dudley has made a habit of collecting their various works.
“I have the world’s largest collection – because I’m reasonably sure it’s the only collection – of trisectors,” said Dudley. “I have over 200. Take this one, for example, which a man spent over 12,000 hours working out.”
Dudley put up the 12,000 trisection on the large screen in the NSU NET building, which prompted gales of laughter from the large, but exclusively focused, crowd.
“As you can see, he came close, but no cigar,” said Dudley.
Dudley said the problem may lie squarely on the doorstep of math teachers.
“Often, mathematicians do not do what they should when they encounter a trisector,” he said.
“They should say, ‘Put it away! Burn it! Don’t you know it’s illegal?’ They should do whatever is necessary to keep them from thinking they can work the problem because it can’t be done.”
Dudley indicated the best way to get rid of a trisector is to ask for proof of the work.
“But this only gets rid of them temporarily,” he said. “They invariably come back with a proof, and the mathematician should end it here, by looking at the proof and pointing out the obvious error, which is always there.”
But in doing that, the instructor is simply sending the trisector back to the drawing board to come up with a more complicated, yet flawed, proof of the same work.
“This cycle often continues until the work becomes incomprehensible,” said Dudley. “So, the moral of the story is, do not talk to trisectors.”
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Don’t talk to those trisectors
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