a simple way to create a square with twice the area of a given square.
the grey square below will be twice the area of the black square.
assume two adjacent sides of the black square are labelled a and b
so we can say the area of the black square is a2, since a and b are equivalent
we know a2 + b2 = c2 for some c, the greeks had deduced this themselves (as had other ancient cultures).
since we are dealing with squares, we can simplify this to a2 + a2 = c2
which simplifies to 2a2 = c2
so we know c = √(2a2)
so we can say that the area of the grey square is √(2a2) * √(2a2), which is 2a2
recall above that the area of the black square was expressed as a2, so our explanation is complete
this is elementary geometry, but i just find it fascinating that the greeks had such a simple yet obvious visual explanation that literally could be related by drawing in the sand with a stick. real elegance.
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last update 04/06/2007