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Impossible Schemes

By Peter Lloyd

After our geometry teacher taught us to bisect an arbitrary angle using only a compass and a straightedge, he informed us that to trisect an angle the same way is impossible. Even then, back in my sophomore year of high school, the concept of impossible raised my hackles.

I spent a good part of one semester trying in vain, of course, to trisect. My teacher had promised anyone an A for the year if they could accomplish the impossible.

Many years later I discussed angle trisection with a highly regarded mathematician. He introduced me to Pierre Wantzel's 1837 proof of the impossibility of trisection as defined, which I am still unable to understand.

Had I the Internet in high school, however, I might have stumped my teacher. He limited the tools we could use to “compass and straightedge,” but trisection can be accomplished with a compass and marked ruler.

Is it not fair to assume that a pencil, part of the compass, might be allowed to mark the ruler as well as the paper? And speaking of paper, you can accomplish trisection sans compass and straightedge using origami.

Impossible Beams
More recently I playfully discredited the geometric definition of a straight line as the shortest distance between two points. One day, before stepping outside, I wondered whether or not I should wear a coat. Rather than step outside, I checked the temperature on my mobile phone.

In a sense, the information I needed beamed who knows how far, along who knows how many straight lines, and back to my phone faster than it would have taken me to step a short distance out my door. Not exactly a refutation of the straight-line definition, but a curious challenge to some aspect of it.

Impossible Team
In 1976, when the Cincinnati Reds defeated the New York Yankees in four games, I watched the game on television just a mile or two as the crowflies from the actual game in Riverfront Stadium. The moment the Reds won, fireworks erupted in celebration over the stadium. I saw and heard the display simultaneously on my television a few seconds before the natural sound of the fireworks reached my ears!

The audio had flashed from microphones in the stadium, zigzagged across a web of towers and wires to network broadcasting facilities, possibly in New York, back to my local television station, and over the air to my receiver before the relatively sluggish speed of sound brought the same audio to my ears!

So if distance equals rate times time, then when time is approximately zero, distance approaches zero as well. The straight line across the Ohio River from Riverfront Stadium was not from my perspective the shortest distance in some crazy, twisted sense, which tickles the heck out of me.

Peter Lloyd is co-creator with Stephen Grossman of Animal Crackers, the breakthrough problem-solving tool designed to crack your toughest problems.

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