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It took over two thousand years for algebra to expose the limitations of the straight edge and compass, argues Keith McNulty, Analytics leader at McKinsey.
Most
of us have either painful or pleasurable memories of using a straight
edge and compass at school. Mine were always in little tin boxes, and
these simple instruments were our main connection with ancient Greek
geometry. For those minutes we used them, we would be constructing
shapes in almost exactly the way the Greeks did.
The only tools allowed in this game
Photo: Medium
At age 10, my teacher set the class a problem. He asked us if we could use only a straight edge and compass to construct a cube that had double the volume of a unit cube. He made it clear that we were only allowed to use a straight edge and compass — NOT a ruler — so we had no way of taking actual measurements.
Myself and a couple of other more math-nerdy classmates spent a long time in trial and error trying to find ways to do this — a few times we thought we’d found it but our teacher would immediately show that it was invalid. After we were exhausted with that problem, he set us another: given a unit circle, can we construct a square of the same area?
We were wasting our time of course, because ten years later as a Pure Mathematics undergraduate, I would discover proofs that it is impossible to double the cube or square the circle using just a compass and straight edge using a finite number of steps...
I personally find this link between abstract algebra and ancient Greek geometry both beautiful and inspiring. I hope you do too. If you are interested in playing around with it, you can try and prove some of the other impossible constructions. For example, while it was shown above how to bisect any angle, try to prove that it is impossible to trisect a 60 degree angle. (Hint: look for an equation that can express cos3θ in terms of cosθ and then see what follows from the logic in this article).
Source: Medium
