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… 

A Door on CCP (or FCC) and HCP is also opened by the Kepler conjecture. Cubic Close Packing also known as facecentered cubic (fcc) and hexagonal closepacked opens key discussions about the honeycomb conjecture, atomic packing factors (APFs), discrete translation operations, and crystal structure. All these faces of mathematics will be explored in an attempt to applied them within the base2 structure of the first 67 notations. Our 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. 248163264. 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… 

Hexagonals in octahedron: This image of an octahedron has six halfsize 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 tetrahedraloctahedral 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… 

2010 Olympicene molecule: An organic carbonbased 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 noncontact 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].