From the Big Board-little universe on the left, notice the range from Step 32 to 41. Nobody knows what is in there. The only thing to count is the expansion of vertices and the multiple of the Planck Length. With over a trillion vertices at step 40 (underlined in red), an imposing infrastructure could be created. From the Universe Table on the right, the left column (purple background) has five summary blocks from steps 21 to Step 65. Notice the vertex count within the first three summary blocks. Using two simple shapes, the tetrahedron and octahedron, we can perfectly tile the universe. Consider looking at the pictures just below the line. Also, within the octahedron are four hexagonal plates that can also tile the universe in 90 degree angles to each other. These domains appear to be as much about geometry as numbers.
Above Step 65, though we are all made up of the same types of particles and atoms, individuation is absolute. Trees are trees. Bears are bears. Humans are humans. Below Step 65, The structure of things can not be observed or easily measured, but there are over a quintillion vertices at step 60! Also, simple logic tells us that all these structures are somehow shared. How it merges into the human scale and then into our large-scale is a long-term research project and challenge. Our first principles, all based on simple geometries are about form and function, and geometry and number. These are: order/continuity, relations/symmetry and dynamics/harmony, all the preconditions of perfection and we will be looking for such evidence from the Planck Length to the Observable Universe. Our hope is that the Large Scale universe can shed some light on it.
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Geometries. With four vertices a tetrahedron could manifest. With six vertices an octahedron, with seven vertices a pentastar, and with eight vertices a tetrahedral-octahedral-tetrahedral (TOT) chain. As these structures grow, we can envision encapsulation. Four tetrahedron and an octahedron enfold within a tetrahedron. Structures build upon structures. With an octahedron six octahedrons and eight tetrahedrons are enfolded. Around the common center point, there are four hexagonal plates, all sloped in 90 degree angles to each other. Each can readily share edges with other hexagonal plates from abutting tetrahedrons and octahedrons such that the entire universe is tiled with each plate beginning with Step 4 and readily going to Step 206. Could this be the basis of a continuity, relationality, and dynamics?
Consider the tetrahedron: Four vertices. Within ten steps or doublings, there are 1024 vertices. With this growing number of vertices, any number of combinations can be imagined. One such combination could be as you see in this image. There is a tetrahedron with four half-sized tetrahedrons in each corner and an octahedron in the middle.
Consider the octahedron: Six vertices. Within just twenty steps or doublings there are over a million vertices and a much greater length (albeit meaningless to particles and atoms) such that an octahedron could conceivably shape those four hexagonal plates within it by the eight tetrahedrons and six octahedrons that can be most simply embedded within it. If you click on this image you will see a much larger, four-color version where you can see those hexagonal plates.
Consider the tetrahedral-octahedral-tetrahedral (TOT) chain: It begins with one octahedron and two tetrahedrons. With every two additional vertices that chains grows by either one octahedron or two tetrahedrons. Dubbed a TOT line, it can perfectly tile the universe with three dimensions. There are two cavities within each line where half-size TOTs can telescope within much like the rings of trees and vines.
Consider the pentastar: This object of just five tetrahedrons is a major study. Assumed to be a perfect construction until the work of chemists (F. C. Frank and J. S. Kasper, 1959) regarding the packing of complex alloy structures. Here we have the first instance of imperfection which is extended into the other basic structure called the icosahedron. And, for many applications, it is further extended into dodecahedral structures within the Pentakis Dodecahedron.