Pi Day Countdown #3 – Pi and Plato

After witnessing the of his beloved mentor, Aristocle (Plato) the son of a prominent and wealthy Athenian family, leaves home and travels around the Mediterranean. On his travels he learns from the Pythagoreans and beginnings to see mathematics as an important part of Philosophy and beneficial in teaching young students logical thought.

Plato was said to have challenged his students with the problem “if given a circle with radius 1, construct a square with the same area as the circle”. His students were only allowed the tools of the time, a straightedge ruler and a compass.

The construction of the golden ratio. The distance from C to G is the golden Ratio
An Example the construction of the golden ratio using only a straightedge and a compass. The distance from point F to point D is the golden Ratio

Since the radius of the circle is 1, this means that the area is \pi. This means that one will have to construct a square with an area of \pi or a square with sides of length \sqrt{ \pi }. It is possible to construct values such as square roots, however it turns out to be impossible to construct \pi.

The construction of the square root of 3, notice that the distance from point B to C is equal to square root of 3
The construction of the square root of 3, notice that the distance from point B to C is equal to square root of 3

Try as any of his students may, they could never find a way to construct \pi. Today the “squaring the circle” problem has been proven to be impossible, meaning that with only a compass and a straight edge, one can never construct a line of distance \pi.

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