Vesica Piscis

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The vesica piscis is an ancient Christian symbol  However, it was known to Euclid and Archimedes hundreds of years before.  It is a form in the shape of a lens produced by the overlap of two circles that have a common radius (the shaded area in the diagram.)  It has a most interesting relationship with square roots.  For the Greeks & Romans a square root simply meant the root of a square, i.e. the side of a square.  For example, a diagonal through a square is the square root of two, √2, which is the side of a square that is twice the area of the original square.  The square root of three, √3, is the side of a square that is three times the original square, and so on.  

In the diagram above, the √1 is the side of the unit square.  √2 is the diagonal through this square.  √3 is the vertical line through the vesica piscis.  √4, or 2, is represented by the vertical lines on either side of the vesica piscis and √5 is the long diagonal through the double square. The square roots of the trees in the painting below, are the results of artistic license and is quite a different matter.

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