Eight equilateral triangles on twelve edges, four meeting at each vertex. Plato gives it to air, the element that moves most easily and fills whatever it is put in. The octahedron is the dual of the cube, so the pair is a study in the same information seen from inside and from outside: eight faces against six, four meeting at a point against three.
Sacred Ratio
1:square root of 3 (insphere radius to circumsphere radius)
Construction
Two square pyramids joined base to base, each face equilateral. Euclid constructs it inscribed in a sphere at Elements XIII.14. Its vertices are the face centres of a cube and its faces the vertices of a cube, which is the duality made visible.