In his Timaeus, Plato constructs the regular polyhedra in a curious way. He composes the square face of the cube from four isosceles right triangles (half squares) and the equilateral triangular face of the tetrahedron, etc., from six right triangles with sides a, a ~/3, and 2a (half equilateral triangles). This procedure has, as far as we can see, found no satisfactory explanation by the commentators of the Timaeus. We propose to understand it as constructions for the duplication of the square and the triplication of the equilateral triangle. The same constructions provide us with what Plato calls the fairest bonds between segments a and 2a for the square and a and 3a for the triangle. This explains Plato's description of the original right triangles as the fairest ones. With respect to the triangles, Plato leaves open the possibility of finding fairer ones. In contrast to this, he declares the regular polyhedra to be beautiful in an absolute sense. In fact the regular polyhedra provide Plato with a significant example for his dialectics, and thus put themselves in a central position in his philosophy.