6. NOTE: The TOT as a tiling would be the largestbut simplest possible system that spatially connects everything in the universe. Yet, even with just octahedrons and tetrahedrons, it is also exquisitely complex; we'll see the beginnings of that complexity with the many variations of R2 tilings (two dimensional) within this initial R3 tiling (three dimensional).
7. And, The Universe Is Still Expanding^{4} Thus, the TOT would also be expanding every moment of every day opening new lines instantaneously. One might say that the TOT line is the deepest infrastructure of form and function. Perhaps some might think it is a bit of a miracle that something so simple might give such order to our universe.
8. The purpose of this article is to begin to introduce why we believe that this could be so.
9. Notwithstanding, we acknowledge at the outset that our work is incomplete. By definition tilings are perfect and the TOT tiling is the most simple. In our application these tilings logically extend from the within the first doubling to the second doubling to all 201+ doublings necessarily connecting all the vertices within the universe.
10. In earlier articles we observed how rapidly the vertices expand.^{5} Yet, that expansion may be much greater once we understand the mathematics of doublings suggested by Prof. Dr. Freeman Dyson,^{6} Professor Emeritus, Mathematical Physics and Astrophysics of the Institute for Advanced Studies in Princeton, New Jersey. We are still working on that understanding.
11. We are taking baby steps. It is relatively easy to get a bit confused as to how each vertex doubles. The first ten doublings will begin to tell that story.
12. And, of course, we are just guessing though basing our conclusions on simple logic.
13. THE MOST SIMPLE TILING. Using very simple math  multiplying by 2  the first tetrahedron could be created in the second doubling (4 vertices). Then, an octahedron might be created in the third doubling. That would require six of the 8 vertices. The first group of a tetrahedraloctahedraltetrahedral chain requires all eight. Today we are insisting on doubling the Planck Length with each notation and to discern the optimal configurations. By the fourth doubling, there could be 16 vertices or six tetrahedrons and three octahedrons. At the fifth doubling (32 vertices), we speculate that the TOT extends in all directions at the same time such that each doubling results in the doubling of the Planck Length respective to each exponential notation.
14. We Can Only Speculate. We can only intuit the formfunctions of this tiling as it expands. And, yes, within the first 60 or so notations, it seems that it would extend equally in all directions. With no less than two milliontrillion vertices (quintillion), using our simple math of multiplying by 2, we will see how that looks and begin to reexamine our logic. Again, this tiling is the most simple perfection. And although we assume the universe is isotropic and homogeneous, there is, nevertheless, a center of this TOT ball, Notations 1, 2 and 3.^{7}
15. That center even when surrounded by no less than 60 layers of notations is still smaller than a fermion or proton. This model uniquely opens up a very smallscale universe which for so many historic reasons has been ignored, considered much too small to matter.
16. Nevertheless, it seems to follow logically that this TOT tiling is in fact the reason the universe is isotropic and homogeneous.^{8}
17. Key Evocative Question from the History of Knowledge and Philosophy: Could this also be the Eidos, the Forms, about which Plato had been speculating? Could this be the domain for cellular automata that John von Neumann, Alan Turing, and others like Steve Wolfram have posited? Here we have an ordering system that touches everything and may well be shared by everything. Within it, there can be TOT lines that readily slide through larger TOTs. There could be any number of cascading and layering TOTs within TOTs.^{9} (A new image is under development with at least ten layers. A link will be inserted as soon as we have it.)
18. A SECOND GROUP OF TILINGS. Within the octahedron are four hexagonal plates, each at a 60 degree angle to another. Each of these plates creates an R2 tiling within the TOTs that is carried across and throughout the entire TOT structure.
19. There are no less than four plates (R2 tilings) of triangles and six plates of squares. Along with those hexagonal plates, all begin to extend from within the first ten notations from the Planck Length, and then, in theory, extending right to the edges of our expanding universe.
21. It gets even more fascinating. Another plate can be represented by the hexagon with various configurations of triangles around it. Only by looking at our clear plastic models can we actually see these different R2 tilings.
22. We discover other tilings by focusing on plates that are joined in ways that combine across tetrahedrons and octahedrons to make an even greater diversity of plates. We have just started that study and we are getting help from other school teachers.
23. Jo Edkins, a teacher in Cambridge, England made our study even more dramatic by adding color in consistent patterns throughout the plate We can begin to intuit that there could be functional analysis based on such emphases.^{10}
24. Edkins also introduced other possible R2 tilings. Each was a challenge to see if we could find each of her plates within our octahedraltetrahedral models. We believe we can find most of her tilings within the models; and when we cannot, we will first look at what we are calling, "R2.5 plates." Some of these are asymptotic plates, quite irregular and undulating plates. Some are perfect.
25. Within the Wikipedia article on Tessellation (link opens a new window), there is an image of the 3.4.6.4 semiregular tessellation. We stopped to see if we could find it within our R3 TOT configuration. It took just a few minutes, yet we readily found it! One of our next pieces of work will be to highlight each of these plates within photographs of our largest possible aggregation of nesting tetrahedrons and octahedrons.
26. Here the square base of the octahedrons couple within the R3 plate to create the first manifestation of the cube or hexahedron. We will also begin looking at the very nature of set theory, category theory, exponential objects, topos theory, Lie theory, complexification and more.^{11}
27. Obviously there are several R2 tilings within our R3 tiling. How do these interact? What kinds of relations are created and what is the functional nature of each? We do not know, but we will be exploring for answers.
28. A THIRD TILING BY THE EXPERTS. Turning to today's scholars who work on such formulations as mathematical jigsaw puzzles, I found the work of an old acquaintance, John Conway. In 2011 with Professors Yang Jiao and Salvatore Torquato (all of Princeton University), they defined a new family of threedimensional tilings using just the tetrahedrons and octahedrons.^{12}
29. We are in the process of studying the ConwayJiaoTarquato (CJT) tiling. It is not simple. Notwithstanding, conceptually it provides a second R3 tiling of the universe, another way of looking at octahedrons and tetrahedrons. Here are professional geometers and we are still attempting to discern if and how their work fits into the 205+ base2 notations. And, we are still not clear how the CJT work intersects with all of the R2 tilings, especially the four hexagonal plates within each octahedron.
AS ABOVE, SO BELOW
30. It takes on a new meaning within this domain of the veryveryvery small. Fine structures and hyperfine structures? Finite and infinite? Delimited infinitesimals? There are many facets  analogies and metaphors  from the edge of research in physics, chemistry, biology and astrophysics that can be applied to these mathematical and geometric models.
31. From where do these expressions of order derive? "From the smallest scale universe..." seems like a truism.
32. This entire domain of sciencemathematicsandphilosophy perhaps should be known as hypostatic science, rather loosely interpreted as "that which stands under the foundations of the foundations."
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Notes & Work Areas:
1. This article is linked from many places throughout all the articles and documents. It is a working document and still subject to the frequent updates.
2. This patent is for embedding a TOT line within a TOT line. There are two triangular chambers through the center; and for the construction industries, we are proposing four sizes to compete with rebar, 2x4sto2x12s, and possibly steel beams.
The Patent Number: US 8.769.907 B2, July 8, 2014 is fully disclosed at the WordPress website, http://octet12.wordpress.com/
Also, on a somewhat personal note, although we call it a TOT line it is hardly a line by the common definitions in mathematics; it's more like Boston's MBTA Orange Line. Now here is a real diversion. Thinking about Charlie on the MTA in the Boston Transit (a small scale of the London Underground or NYC Transit), this line actually goes places and has wonderful dimensionality, yet in this song, it is a metaphorical black hole. Now, the MBTA Orange Line is relatively short. It goes from Oak Grove in Malden, Massachusetts to Forest Hills in Jamaica Plain, a part of Boston where I was born.
3. Classroom discussion on December 19, 2011 in metro New Orleans high school where we went inside the tetrahedron by dividing in half each of the edges and connecting those new vertices. We did the same with the octahedron discovered inside that tetrahedron and did the same process with it. We divided the edges of these two objects in half about 110 times before we finally came into the range of the Planck Length. We then multiplied each edge by 2 and connected those vertices. In about 100 notations we were somewhat out to the edges of the Observable Universe. We are still learning things from this basic construction.
4. Where is the Good in Science, Business and Religion is located in several places on the web, however, it was first published on September 2, 2014 within a LinkedIn blog area. The chart was first used in another blog, "Is There Order In The Universe" which was published on June 5, 2014.
Please be patient. We are actively attempting to fix these number references today.
5. The Concept of the Expanding Universe is part of the concept of the Cosmological Principle (metric expansion of space) that resides deep within the concept of the Observable Universe.
5. As of this writing, there does not appear to be any references anywhere within academia or on the web regarding the concept of counting the number of vertices over all 205+ notations. Using the simplest math, multiplying by 2 (base2), there is a rapid expansion of vertices. Yet, it can also be argued that vertices could also expand using base4, base6, and base8. That possible dynamic is very much part of our current discussions and analysis. It is all quite speculative and possibly just an overactive imagination.
6. We have made reference to Prof. Dr. Freeman Dyson's comments in several articles. If he is correct, his assumption adds so many more than a quintillion vertices, it gives us some confidence that everything in the universe could be readily included as a whole. Within this link to fifteen key points, the Dyson reference is point #11.
7. If the Planck Length is a vertex from which all vertices originate, and all vertices of the Universe in some manner extend from it, the dynamics of the notations leading up to particle physics (aka Particle Zoo) become exquisitely important. Questions are abundant: How many vertices in the known universe? What is the count at each notation? Do these vertices extend beyond particle physics to the Observable Universe? In what ways are the structures of the elementary particles analogous? In what ways are the periodic table of elements analogous? What is the relation between particle physics and these first 60 or so notations? Obviously, we will be returning to each of these questions often.
8. Isotropic and homogeneous are working assumptions about the deep nature of the universe. Homogeneous means it has a uniform structure throughout and isotropic means that there is no directional bias. This work about tilings provides a foundation for both assumptions.
9. The two small images in the right column are of a very simple fourlayer tetrahedron. The Planck Length is the vertex in the center. The first doubling creates a dynamic line that can also be seen as a circle and sphere. The next doubling creates the first tetrahedron and the third doubling, and octahedron and another tetrahedron, the first octahedraltetrahedral cluster also known as an octet. The fourth doubling may be sixteen vertices; it may be many more. When we are able to understand and engage the Freeman Dyson logic, the number of vertices may expand much more rapidly. Again considering the two images of a tetrahedron in the right column and its four layers, today we would believe that it amounts to three doublings of the Planck Length. When we begin to grasp a more firm logic for this early expansion, we will introducing an image with ten layers to see what can be discerned.
10. I went searching on the web for images of tetrahedrons and tessellations or tilings of hexagons. Among the thousands of possibilities were these very clean images from Jo Edkins for teachers. Jo is from the original Cambridge in England and loves geometry. She has encouraged us in our work and, of course, we thank her and her family's wonderful creativity and generosity of spirit. http://gwydir.demon.co.uk/jo/tess/bighex.htm
http://gwydir.demon.co.uk/jo/tess/grids.htm#hexagon
11. Virtually every mathematical formula that appeared to be an abstraction without application may well now be found within this Universe Table, especially within the very smallscale universe. We will begin our analysis of set theory, category theory, exponential objects, topos theory, and Lie theory to show how this may well be so.
12. "New family of tilings of threedimensional Euclidean space by tetrahedra and octahedral" Article URL: http://www.pnas.org/content/108/27/11009.full
Authors: John H. Conway, Yang Jiao, and Salvatore Torquato
