Math 150

Projects

Archimedes Trisection of the Angle

Dr. Wilson

 

The angle at A is one third of the central angle.

 

This construction is Archimedes' trisection of the angle by compass and straightedge. One thing which makes this construction remarkable is that in 1832, the French mathematician Evariste Galois proved that it was impossible to trisect an angle with a compass and straightedge. Every year, Mathematics Department chairs get trisections of the angle with compass and straightedge, and they throw them in the wastebasket because it is very well known that Galois proved that it can't be done. In fact, Galois' proof is more well known than Archimedes' construction.

Whenever someone proves that something can't be done, one should examine the proof for unconscious, limiting assumptions.1 The unconscious, limiting assumption here is that you can use a straightedge only to draw lines, and a compass only to draw circles. There many mathematicians who know about Archimedes' construction, but they explain it away by saying that it can't be done by compass and straightedge alone. You have to have a mark on the straightedge, and that doesn't make it a straightedge; it makes it a ruler. But, if you have a compass, you can make a mark on a straightedge. In Archimedes' construction, the compass is not used to draw circles, it is used to make a mark on the straightedge. In fact, you do not even have to mess up your straightedge by making a mark on it with the compass. It will suffice to simply set the compass opened to the radius of the circle on the straightedge and keeping one point on the straightedge at C, wiggle it around until you have found the line through C where one end of the compass is at a point (D) on the circle, and the other is at a point (A) on the line through O and B,

 

and this construction can be done using only a compass and a straightedge. I have found that it comes out about as accurately as any of my other compass and straightedge constructions.

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Math 100
Steve Wilson