Old Memories About Trisecting Angles
This is just some remembering. When I was in middle school (about 12 or 13 years old), I was convinced I could find a way to trisect an angle. Of course, this is known to be impossible, by an argument that I now (but didn’t, at the time) understand. That didn’t stop me from trying, though. Here’s the construction I spent most of my time on:
I started out with an angle, which is the two outermost lines on the given drawing. Using the compass, I marked off points equally far along the sides. I then constructed their midpoint. Then I drew three overlapping circles centered at those three points, as shown. The intersection points of these circles furthest from the vertex of the angle were used to draw my “trisecting” lines.
Of course, this doesn’t really trisect an angle, since it’s a construction with a compass and straight edge, and no such construction can possibly trisect an angle. But if you didn’t know that, there are a few sanity checks you might perform.
- Does it appears to work? Yes, it does, as it turns out. In fact, I performed this construction fifty or sixty times, and measured the result with a protractor. Each time, it was close enough to trisected that I chalked up the difference to human error. (Having just performed the construction now using the Kig software, to capture the image above, I can now see that the result is visibly off for extremely small angles; but of course these are the ones that I was least able to check without reasonable computer software at the time.)
- Does it work for cases that are easily calculated? Yes! A few easy examples: If you start with a 180 degree angle, you get back 60 degree angles… perfectly trisected. A zero-degree angle is vacuously correct. It’s a slightly more involved argument, but the construction also trisects a 90 degree angle correctly.
At this point, one can’t be blamed for getting a little excited. We have a construction that gives a division of an angle into three parts. The parts always seem to look equal, and are always close enough to be within reasonable measurement error. It works for the cases we’ve checked so far. One can’t be blamed for shifting gears a little here, gaining a little confidence, and looking a little less for a counter-example, and a little more for a proof.
Of course, those familiar with the impossibility of this task will be looking for the case where we start with a 60 degree angle. And, of course, it turns out not to work. In particular, the “trisected” angle works out to about 19.1066 degrees. Close, but not quite.
So what am I really constructing? Well, something pretty ugly. I’m leaving out the math (basically, just pick out a couple triangles, and apply known triangle relations such as the law of sines and thelaw of cosines), but, the answer in Maxima notation (and simplified as best as Maxima can) is -asin((sin((3*%theta-5*%pi)/6)*abs(sin(%theta/2)))/sqrt(-2*sin(%theta/2)*cos((3*%theta-5*%pi)/6)+sin(%theta/2)^2+1)). Yeah, wow. But how could we have fooled ourselves into thinking that mass of complicated stuff is actually theta / 3? Well, take a look at Maxima’s plot of this angle versus theta (that starting angle):
Ifthe construction were correct, that graph would be a straight line passing through (3,1). As it turns out, it may be somewhat difficult to tell the difference in this graph! So while this construction is wrong, it’s also remarkably close.