Justly called the most celebrated theorem of geometry, the Pythagorean proposition is probably the one bit of mathematics which millions of laymen will remember long after they have forgotten whatever other mathematics they may once have known. The theorem is notable first because of the rich historical associations with which it is attended; secondly, because of the amazing variety of proofs which have been given; and thirdly, because further exploration quickly leads to interesting and unexpected byways, such as the Golden Section, dynamic symmetry, logarithmic spirals, angle trisection, duplication of the cube, squaring the circle, determining the value of π, the concept of irrational number, regular and star polygons and polyhedra, theory of numbers, constructibility of angles and polygons, continued fractions, phyllotaxy, musical scales, Diophantine equations, Heronian triangles, and Pythagorean number lore.
Random packing of regular polygons and star polygons on a flat two-dimensional surface