Plato in two dimensions

Plato attaches a crucial role to plane surfaces in the Timaeus, the dialogue bearing his 'physical/chemical' theory of the universe. Combinations of two elementary triangles produce the equilateral triangle and the square, which are the faces bounding the tetrahedron, the octahedron, the icosahedron and the cube, that is, four of the five "Platonic" solids that shape the types fire, air, water and earth, respectively. Surfaces are a key factor not only in the very first stage of the coming into being of the universe and in the formation of its solid contents, but also in its phenomena. Indeed, it is through resolution into, and recombination of, elementary triangles that the particles of the four types incessantly inter-transform into each other (fire, air and water) and/or change their size (including earth). I propose that the phrase "in two dimensions" applies also to the question whether (1) Plato assumed that mathematical objects are the ultimate principles of the universe, or (2) there is another dimension that better accounts for his theory. I shall argue for the second dimension.