Thanks to its simplicity, the triangle is surely the king of polygons. As its name suggests, it has three angles and therefore three sides.
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For the familiar convex quadrilaterals, the classification criteria are, first, whether the sides are parallel, then whether their lengths are equal, and finally whether there are any right angles.
Two postulates in Euclid's Elements embody the ideal conception of the straightedge and compass inherited from Plato's realm of Ideas. The Alexandrian scholar built much of plane geometry—and the constructions he bequeathed to us—on these two postulates.
A symmetry appearing out of nowhere, an equilateral triangle emerging amid the trisectors of a completely arbitrary triangle: that is the whole "miracle" of Morley and his theorem. The many proofs of this beautiful property are marvels of geometry and trigonometry.
Regular polygons are particularly harmonious, thanks to their symmetries. At first glance, one might think they had long since yielded all their secrets. Yet the notion of duality reveals a wealth of surprises.
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