the idea comes from ancient philosophy, when philosophers could make claims like "everything is made of fire" or "everything is made of water" or "everything is made of triangles" and argue it reasonably. the platonic solids, also, are five in number, where the fifth is a dodecahedron. i think it was plato, among others, who also believed in a fifth element, aether, but i cant remember the details.

how one would demonstrate any of these relationships mathematically is beyond me.

I think it means that the writer doesn't really know what he is talking about.

And, by the way, there are five regular solids. The other is the dodecahedron. I imagine the writer would say that corresponds somehow to the "quintessence" some philosophers used to go on about. But that still doesn't mean this idea has anything to do with physical reality.

EDIT. By the way, in spite of all the mystical silliness that seems to surround it, personally, I think that the Soul of the World sculpture is a really cool idea.