by Bruce E. Camber, 30 June 2017. Reviewed, 5 September 2025

Background:
1. Excerpt from a posting titled (January 2016),
On Constructing the Universe From Scratch.
2. Spheres are not simple: https://81018.com/sphere/

Jonathunder. This animated illustration is from Wikipedia. It demonstrates how spheres generate lines (lattice), triangles, and then a tetrahedron and octahedron. With that second layer of green spheres emerges the tetrahedral-octahedral couplet. The discipline, known as cubic close packing (ccp), deserves our attention. “The Kepler conjecture states that this is the highest density that can be achieved by any arrangement of spheres, either regular or irregular.

This conjecture was proven by Thomas C. Hales. (Wikipedia)

From a point, to a line, to a sphere, to sphere stacking, to a triangle, to a tetrahedron, and to the octahedron, it then becomes even more dynamic. Add the Fourier transform. Then, add as much information as we can from all the work of scholars ( most recently by scholars such as Smale and Milnor on attractors and repellers).

Also see:

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