Learn about the interiority of the octahedron: The students learn about the most-simple, perfect configuration within an octahedron. There are fourteen pieces, six smaller octahedra and eight tetrahedra. All fourteen pieces share a common center point.

Notice in our image of that octahedron, some of fourteen pieces have edges that have color, most often we’ve used red, white, blue and black. Notice each color creates a hexagonal plate. The white and red plates are most readily identified in the image on the right. The blue and dark green plates are a bit obscured. The taping of pieces is still a bit crude at this point in time. This is the first generation of the game.

The object of the game is to reconstruct the four plates as given when the pieces are all 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 verbal 15-second warning (just to keep the game moving) so with four students, each set of the competition takes a little 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’re developing with the octahedron and tetrahedron that are very similar but not quite as much fun. We’re working on it!

Let us introduce the twenty-tetrahedral icosahedron in place of the five-tetrahedral cluster. The complexity and potential functionality of this cluster increases exponentially.

In May 2022 we began making a study of the cluster of fifteen sharing a common centerpoint (with the hexagonals within each octahedron) as if it would make an interesting gate within circuitry of the infinitesimal.

Our first iteration is above. The red line runs along the ridge then drops down to the centerpoint just above. For more, go to our studies of geometries. Another version.

Several pages are linked to this page. To return, the URL is https://81018.com/tot-2/ or use your “Back” button (top left arrow) within this desktop window. The URL is: https://81018.com/tot-2/

Observe the octahedron. Many students find it difficult at first. Do you see where the heavy dark lines converge just below the lighter yellow horizontal line. Do you see that center tetrahedron with the two heavy dark lines (black and a very dark blue)? In your mind’s eye follow all those edges back inside to centerpoint of the larger octahedron and the largest tetrahedron pictured here. Also, visit the image on this page.

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.

There are many references throughout this website to the octahedron. The first two images below are result of cubic-close packing of equal spheres (ccp). All links go to earlier studies.

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!

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.

From a conjecture about cannonball stacking:
In 1611 Johannes Kepler opened the door on a foundational relation in mathematics by addressing a difficult practical question about stacking cannonballs on the deck of a ship. The result, represented by this dynamic image on the left, shows the transition from circles to spheres to lines (lattice) to triangles to tetrahedrons to octahedrons. Here we begin to tile and tessellate the entire universe and encapsulate everything, everywhere throughout all time within 202 notations. Here symmetries go from simple to complex and appear set for action. Though the cannonball stacking problem appears inconsequential today, Thomas Hales introduced a series of proofs that have also open new dimensions within mathematics that includes his background work on the fundamental lemma, automorphic forms, unitary groups, and the stabilization of the Grothendieck–Lefschetz formula. A key page of the Symmetry discussion…

Feigenbaum constant: The doublings of the circles, then the spheres
are assumed to be a direct analogy to the emergence of these symbolic cannonballs. We’ll assume that the first circle emerges from the perfection of pi and the thrust of the universe, and we guess that the outlines of a sphere emerge with the next doubling. If so, then this dynamic image (top left box) can be replicated within six steps. 2-4-8-16-32-64. For the black and then the green, there are nine initial circles, then another nine to become spheres. That is 36 construction vertices. Possible? Symmetry discussion…

Honeycomb conjecture with Thomas Hales: A bold, creative mathematician, Hale’s work opens key doors to the foundations of the universe. Although still concerned with Kepler’s technical problem, this structure may have profound applicability to the deepest “real world” questions about the nature of space and time. So, we will pursue this line of inquiry as it is related to the “first generation” of the infrastructure of the universe. Go to the symmetry page…

Hexagonalsin octahedron: This image of an octahedron has six half-size octahedrons, one in each of the four corners and on the top and bottom. It has eight tetrahedrons in each of the eight faces. These objects evolve around a counterpoint that is also the center of four hexagonal plates shown here as red, white, blue and green. In the discussions of the the honeycomb, there appears to be no acknowledgement that these hexagonal plates are part of the tetrahedral-octahedral structure and that it emerges, as demonstrated within the dynamic image, from circles and spheres. It is easy to imagine these basic shapes replicating and morphing to create the Periodic Table of Elements. Symmetry page…

2010Olympicene molecule: An organic carbon-based molecule was synthesized with five rings (four benzene rings) to honor the 2012 London Olympics. What makes this especially significant is that in 2012 IBM researchers in Zurich captured this image using non-contact atomic force microscopy. Symmetry page…

There are many other pages that use this stacking image: [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], and [12].

Three levels of simple complexity: 1. Observe the little tetrahedron in the bottom left corner.

2. Notice that it is enclosed within a larger tetrahedron. Right beside that larger tetrahedron is a very colorful octahedron. There are two other larger tetrahedrons pictured on the top and on the right. There is a fourth larger tetrahedron in back corner not visible here. Every tetrahedron encloses four “smaller-sized” tetrahedrons and an octahedron.

3. Notice that our larger tetrahedron is enclosed by an even larger tetrahedron. This pattern repeats itself getting smaller and getting larger. Part of the complexity can be seen by observing the center octahedron. Notice the red, black and blue hexagonal plates. A white plate has been obscured in this image, yet it can be easily seen within this image of the octahedron. Each shares the common centerpoint.

4. Notice the octahedron in the middle whereby that center triangle is one of its eight faces. There are four faces that are the center faces of each side of the tetrahedron and there are four interior faces. The octahedron has a half-sized octahedron in each of its six corners and a tetrahedron in each of its eight faces.