PYRAMID OF SIXES: Truncated Octahedron: 4-6-6 Archimedean Form: Faujasite Crystal.
Let us investigate the axes of symmetry that exist inside this form. There are 3 dominant axes, like the x-y-z axes in the cube, all at 90 degrees to one another. Each rectangular plane is in the ratio of 1:2 which relates this to both the musical Octave and the Mystery of Phi (since the Phi Formula: [1 + root5] divided by 2 is 1.618 or Phi, and this links to the diagonal of the 1×2 axial rectangles inside the Truncated Octahedron because the diagonal of each of these 3 rectangles is Root 5 or 2.236…).
Another special quality of this Archimedean Form is how well it packs or tiles itself in 3-dimensions, with no gaps. This is typical of Zeolite structures which are Truncated Octahedra.
Another crystalline structure that is Truncated Octahedrally is the Faujasite crystal used in the making of some batteries and in ArtificiaI Intelligence. Faujasite is a mineral group in the zeolite family of silicate minerals.
Leonardo da Vinci, the highly skilled artist, drew the Truncated Octahedron in his manuscripts, showing its transparent view or tilted perspective.
Many people are familiar with the giant geometrically designed Jungle Gyms that children and adults climb. If you look closely, many of them are in the configuration of the Truncated Octahedron!
I have in my collection of rare crystals a sample of Pure Gold and it distinctly shows some of the symmetry of the Truncated Octahedron by exhibiting a square face and a hexagonal face.
The form of the Truncated Octahedron essentially embodies the forms of both the square and the hexagon, and numerologically we could say it relates to 4 and 6. I found this beautiful symmetry of evolving numbers only using 4s and 6s (and 1s) which I call the Pyramid of 6s as the Number 6 is the dominant number here. Its only meant to be a wow factor highlighting Mathematics internal and hidden language of Beauty and Symmetry and how these Number Pyramids resonate with 3-dimensional geometries, with Phi and musical Octaves.
Concluding with one more inspiration from this Pyramid of Numbers, let us focus on the apex of the Pyramid, being the line: 16×4 = 64.I would like to show how any 2-Digit Number (like 64) being multiplied by any 1-Digit Number (like 4) can be done easily, quickly and mentally. I found this in a book called “The Trachtenberg Method” from a Swiss mathematician and I use this following method all the time for Speed Calculations.
We need to introduce the symbol of the Forward Slash “/” to partition the calculation and to allow a mental sliding movement.
With the 4, we multiply both of the 2 individual digits of “64” and write it like this:4 x 64 = 4×6 / 4×4 = 24/16 and the rule here is that we are only allowed one digit on the RHS (Right Hand Side) so the “1” must be carried over to the left or slid over to the left: giving a final answer of (24+1)/6 or 25/6 or 256.
Using only 4s and 6s, lets perform 3 other One-Line Rapid Mental Calculations
1- 6×44 =24/24 = 26/4 = 264
2- 4×44 =16/16 = 17/6 = 176
3- 6×66 = 36/36 = 39/6 = 396
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