Transformation Model

The Platonic Bodies are not only presented as such in the “Timaeus”, but also a theoretical model of their composition, and potential transformations are given. While the bodies themselves represent more a kind of concept for structure and arrangement, triangles constitute the underlying constructive elements, so to speak their “atoms”. Of course not arbitrary triangles, but two special categories of triangles, which may be determined by being the most beautiful ones of all.

There is no deeper explanation how this theory comes off and on what it is based. But the annotations reveal that the beauty of a shape is mainly relying on its symmetry and nativeness. Therefore the following two classes of triangles have been elected: isosceles triangles, where the cathetus build a right angle, and right-angled triangles, where the length of the smaller cathetus is the half of the hypotenuse.

Four of the isosceles triangles may be arranged to build a square, while six of the right-angled triangles may be arranged to build an equilateral triangle. So the base areas of up to four of the Platonic Bodies can already be compiled – the dodecahedron remains excluded, since its base area consists of a regular pentagon.

Based on that concept the number of the individual, elementary triangles belonging to each body may be calculated:

tetrahedron24 right-angled trianglesfire
hexahedron24 isesceles trianglesearth
octahedron48 right-angled trianglesair
icosahedron120 right-angled triangleswater

A Platonic Body may now be disassembled into its elementary triangles, to be compiled to a new and different one. Looking at the table above shows the possibilities and relations of transformation. But it shows also that the cube (hexahedron) is taking an exceptional position, since it can’t be converted into one of the other three bodies, nor is it possible to build the cube out of the others. A square can’t be build from the right-angled triangles, while an equilateral triangle can’t be build from the isosceles triangles, too.

The dodecahedron is also taking an exceptional position, since it is the only one compiled by regular pentagons. It is excluded from the depictions, but a subordinate clause is mentioning it to be the constructive element of the planet Earth. May it be a hint to the grid of Christ?

Whatever the way you look at it might be – we have here a mind model depicted in detail, but not explained. Even the youngest text analysis are not picking it up, although ancient Greece is well known for its outstanding knowledge in the field of geometry. Can it really be that such a precise presentation is simply inconsiderable?