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Q.E.D.

1 IntroductIon There are some mathematical objects whose beauty everyone is able to appreciate. The regular polygons and polyhedra are good examples these figures are surpassed in perfection only by the circle and the sphere. Then there is Pythagoras theorem, a cornerstone of the right angled worlds we build for ourselves, and perhaps the conic sections which describe the orbits of celestial bodies. Very few people appreciate more than some elementary aspects of mathematical beauty, much of it revealing itself only to mathe maticians in the study and creation of intricately crafted proofs, barely within the reach of the most highly trained human minds. As a mathematician, I declare that I have established the truth of a theorem by writing at the end of its proof the three letters Q.E.D., an abbreviation for the Latin phrase quod erat demonstrandum, which translates as what had to be proved. On the one hand, Q.E.D. is a synonym for truth and beauty in mathematics on the other hand, it represents the seemingly inaccessible side of this ancient science. Q.E.D. can, however, also be found at the end of some simple, striking, and visually appealing proofs. This little book presents a journey through a collection of these wondrous gems, exploring the ideas behind mathematical proof on the way, written for all those who are interested in the beauty of mathematics hidden below the surface.
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