by Bruce E. Camber, July 20, 2020
Cubic-Close Packing of Equal Spheres. In 2016 when I first encountered Thomas Hales work, all the geometers of the world were in awe of his work. An historic mathematical puzzle was being solved. The physics community, on the other hand, hardly seemed to notice. For me, it was an introduction to cubic-close packing and I found Wikipedia’s animated illustration very helpful. I looked at those spheres as if they were the first spheres in the universe. At the Planck scale everything is a first. Those lines (lattice), triangles, and then a tetrahedron, and then three tetrahedrons surrounding a center triangle were all firsts. When the tetrahedrons fully enclosed an octahedron (also pictured below), it was a first and it is still a first. That relation is poetry. It inculcates Kepler’s The Harmony of the World as the “Music of the Spheres” (Harmonices Mundi). Yet, instead of going out to the planets, we go deep inside, right down into the Planck scale and the Fourier transform to hear the pure harmony of the spheres.
I am often critical of those folks who create neologisms to describe some aspect of their work. So it was with some surprise that I unwittingly made my first exception within my own work. My word combination, tetrahedral-octahedral couplet, seemed to describe the emergence; I did not know that it had never been used in the past. Yet, here there is an coordinated, necessary action. It feels like a couplet; it would naturally be a new rhyme, a new harmony, and possibly at some point, the beginnings of song or a harmony within the universe.
Couplet. That second layer of green spheres is where the very first tetrahedral-octahedral couplet emerges. It continues to emerge even at this moment.
Updates. We will always be in search of experts to guide us within disciplines where pi has a fundamental role. As a result sections of the article will be re-written and expanded. Is this the most logical path by which all of space and time becomes tiled and tessellated? Could this become a new science of the extremely small, the infinitesimal and the interstitial (the dance between the finite-and-the-infinite)?
The foundations of foundations. Also called the hypostatic, this exquisitely small is also the ideal, a domain of perfected states in space-and-time. We are beginning to engage all the research from Kepler to today, particularly the current ccp (hcp and fcp) research from within our universities, in hopes that we truly begin to understand the evolution of the most-simple structures.