Does present-day symmetry underlie the cosmology of Plato's Timaeus? A response to D.R. Lloyd.

In the Timaeus Plato uses apparently superfluous constructions to obtain the equilateral triangle and the square, the faces bounding the regular solids which underlie the four kinds fire, air, water and earth. Indeed, the constructions of the two faces involve six and four 'small' elementary triangles, respectively, whereas just two 'large' elementary triangles would suffice in both cases. Plato also says that the reason why he uses a particular scalene elementary triangle, which he calls the fairest (?????????), for the construction of the equilateral triangle is a long story (?????? ?????) to tell. Recently, Professor D. Robert Lloyd has proposed that present-day symmetry principles underpin both the superfluous constructions of the faces and the choice of the 'fairest' triangle. He claims that Plato may have well been inspired by symmetry-related criteria because in both the aforementioned constructions the apparently redundant number of elementary triangles and their arrangement are consistent with reflection and rotation operations brought about with respect to the proper symmetry elements of the two faces. He also emphasizes that it is only with these superfluous constructions, and the 'fairest' triangle which Plato adopts to construct the equilateral triangle, that the solids bounded by the two faces possess the correct symmetry. While I acknowledge that all these mathematical aspects entirely and indisputably apply to the plane and solid geometrical figures we encounter in the Timaeus, I argue that they motivated neither Plato's construction of the two faces, nor his choice of the 'fairest' scalene elementary triangle, including the 'long story' behind it. I shall attempt to demonstrate that the notion that symmetric figures remain invariant (or equivalent) by virtue of their immunity to change despite the dynamical character of the operations applied to them, such as reflection and rotation, conflict with the ontology, epistemology, and philosophy of mathematics of Plato's system. I also argue that the selection rules that Plato applies in the Timaeus to the transformation of the four kinds into each other conflict with the symmetry-group-based classification of the solids that shape those four kinds.