The three most important building blocks: sphere-tetrahedron-octahedron
- Construction: The whole, the parts, and pieces-and-plates
- The Octahedron Game
- The Octahedral Gap (the beginning of Quantum Fluctuations)
- The First Octahedron: Surrounded by Tetrahedrons
- Manufacturing: Perfect tetrahedrons and octahedrons
- References & Resources
1. Construction. 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 centerpoint (vertex) in the middle.
Notice the 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.
2. We made a game out of it! It’s a little crude, but it’s fun and it teaches the students about the interiority of the octahedron. Notice (image above) that octahedron has four hexagonal plates with different colors. We’ve given dozens of sets away. Most often those plates are outlined with red, white, blue, and black tape. The tape is the crude part, but it works. There are fourteen pieces inside that octahedron, six octahedra and eight tetrahedra. All fourteen pieces share the common center point.
The object of the game is to reconstruct the four plates as given when the pieces are emptied onto the table. Sounds simple. It’s not!
The competition can be between as few as two and as many five. Four seems ideal. Reconstructing the octahedron with the four plates as seen for some will be intuitive. After doing it a few times, we’ve had people put it back together in just under a minute while others have tried and tried but not succeeded. Use a three-minute timer with a 15-second warning just to keep the game moving so with four, each set of the competition takes no more than 12 minutes. Winners then compete, and eventually we have a winner for the class, then a winner for the grade, and then a winner for that semester. With extra sets on hand for those who get timed out, everybody eventually wins because now they have seen and felt the interiority of the octahedron and can readily apply it in other studies within the curriculum.
There are other games we’ve developed with the octahedron and tetrahedron that are very similar but not quite as much fun. We’ll be working on it.
4. The First Octahedron: The first octahedron emerges as an extension of tetrahedrons that are constantly emerging within cubic-close packing of equal spheres. You can actually watch the process!
5. Manufacturing: Perfect tetrahedrons and octahedrons. When we had our larger model of a tetrahedron with two generations of embedded objects, we began 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.
6. Applications. We are not sure how many doublings from the first octahedron to begin manifesting as a hexagonal ring of hexagonal 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.
Note: We began this article, Octahedron, on May 3, 2018. It is still a rough draft. It has its origins at MIT in 1979.
Last update: 11 July 2022
Richard Fitzpatrick (2006-03-29) http://farside.ph.utexas.edu/teaching/329/lectures/node54.html
Much more to come…