Plato’s sacred geometry: In Euclidean geometry there are five Platonic solids. Each of them was associated with an element, and since there are five, one of these shapes were considered sacred by the old Greeks, and to know the shape, and to share that knowledge was punishable.
Platonic solids have clear definitions, to quote Wikipedia:
A Platonic solid is a regular, convex polyhedron with congruent faces of regular polygons and the same number of faces meeting at each vertex.
There are only five shapes that meets this criteria, though there are expanded versions.
Fire: tetrahedron. Tetra meaning four faces consisting of four triangles, and obviously the result is a pyramid:
Earth: a hexahedron is, as the name indicates, a shape with six faces, and therefore consists of six squares. A dice:
Air: the octahedron; is an eight-faced shape, and we are back at triangles:
Water: icosahedron is a shape with 20 faces, and hence made up of triangles:
I have saved the dodecahedron last. Dode- meaning 12, it is a 12-faced shape consisting of pentagons. It is not the one with most faces, but it is probably the most complex one, and it was seen as the sphere of the universe, the shape of which the cosmos itself is made. There is another point here, as the whole idea for us to see a 12-sided something as a sphere of anything a little odd. But consider: make a dodecahedron out of leather or fabric, and it is the one that most resembles a perfect sphere (footballs are not dodecahedrons, they have both pentagons and hexagons. So nothing very sacred about football. But we all new that.).
All this sacred la-di-da has, of course, reached the new-agers, and dodecahedrons can be found in a lot of their symbolism. To me, that is between mildly ridiculous and stupid. Though I am useless at math, I find the thought of Platonic solids, the ancient greeks, the patterns in nature, the building blocks of fractals delightful. The adventure to seek knowledge and enlightenment should never stop, and to wonder at nature does not mean there is a straight line from there to religion or quasi-religion, just because there are things we cannot yet understand, or just because something is ancient or pretty.
On an interesting note, Carl Sagan (all hail, master nerd!) wrote the book Contact where the drawings of a machine were transmitted to earth. The shape of this machine was a dodecahedron. Coincidence? I think not.
Maybe the most famous artist of our time to play with math and symmetry was M.C. Escher. As a child, I bought a book where you could build your own geometric models with his art. It was endless hours of fiddling with the cardboard inserts, glue and at times mind-boggling how the movable paper models could endlessly move. He must have been pretty obsessed by geometry, symmetry, and fractals.
I have made some dodecahedrons and polyhedra myself, that will be a post for another time, but here is a taste:
The odd thing is; I am not even a fan of symmetry. But I suppose beauty will not be denied.