The most-simple construction of the octahedron is with six octahedrons, one in each corner, and eight tetrahedrons, one in each face. Those fourteen objects all share a common vertex in the middle.
There are four hexagonal plates crisscrossing each other. Pictured here is a plate defined by red tape, another defined by white tape, and another defined by blue tape (partially obscured). A plate defined by the dark green is also partially obscured.
Our initial study of the octahedron began with our earliest studies of the tetrahedron. It was such a surprise to find the octahedron making up the center of every tetrahedron. And, it was extremely satisfying to find all fourteen objects sharing that common center point and to see that all of these centerpoints were also center of four hexagonal plates. Following our simple logic, when we observe graphene as a single atom thickness, it is always so much more. It is the manifestation of a plate from a period-doubling bifurcation.
It was not until we had our larger model of a tetrahedron with two generations of embedded objects did we begin to see the plates of triangles, squares and hexagons. To make these models required developing molds to manufacture thousands of perfect octahedrons and tetrahedrons. These models were all part of our teaching Platonic geometries in a New Orleans high school.
We are not sure how many doublings from the first octahedron would we then have fracturing of these plates. Our guess is somewhere between the 67th notation (or doubling) and 87th depending on the element being made.
Here, for example, is the hexagonal plate defining a single cell.
This article was begun on May 3, 2018. It is still a rough draft. It has its origins at MIT in 1979.