Octahedron

Our initial study of the octahedron began with our study 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 a 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.

Tetrahedrons inscribe an octahedron

CCP-FCC-HCP
The dynamic unfolding of tetrahedrons and octahedrons

tetra-octa
Our first model of the tetrahedron (above) and our first model of the interiority of the octahedron (right) from 1997.  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 high school.

octahedron
Four bands of hexagonal plates around a centerpoint represented by yellow, red, white, and blue tape.

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.

moleculeWe 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.

Note:  Richard Fitzpatrick 2006-03-29 http://farside.ph.utexas.edu/teaching/329/lectures/node54.html