Spheres look simple; they are not.

Sphere to tetrahedron-actahedron couplet
Attribution: I, Jonathunder

Excerpt from a posting titled,
On Constructing the Universe From Scratch

This animated illustration is from Wikipedia. It demonstrates how spheres generate lines (lattice), triangles, and then a tetrahedron. 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)

As a point, to a line, to a sphere, to sphere stacking, to a triangle, to a tetrahedron, and to the octahedron. Then it becomes even more dynamic. Add Fourier’s transforms. Then, add as much information as we can from the work of Smale and Milnor on attractors and repellers.

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