It's a World Made of Triangles: Plato's Timaeus 53B-55C

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.