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In the  Timaeus , Plato uses an apparently redundant number of two right-angled elementary triangles – six scalene triangles and four isosceles triangles – to construct, respectively, the equilateral triangle and the square, the two faces that bound the solids underpinning the four  genera  of the world. I argue that these constructions involve a circle circumscribed about each face, and that the notion of “circularity”, as well as the forms of uniformity and self-similarity may have inspired them. With Euclid’s  Elements  in mind, I propose that analysis of the two faces as a function of the parameters of the circles circumscribing them and of the two elementary triangles answers questions that have long puzzled commentators of the  Timaeus .

It’s a World Made of Triangles: Plato’s Timaeus 53B–55C