**Initiated**: December 1, 2014 **Very minor update**: Sunday, January 28, 2018

Tetrahedral-Octahedral-Tetrahedral (TOT) clusters and couplets tile and tessellate the universe^{1} (opens in new tab or window). In earlier writings, we have observed how the universe could be tiled in about 202 exponential notations or steps, layers, doublings, or domains.

**Please note**: Many links will open a new tab or window.

**The TOT Structure**^{2}** appears to be the “simplest, strongest, most perfect, interlocking three-dimensional tiling” within the Observable Universe**. The TOT can be used to make ball-like structures, clusters, lines, domains or layers. Here we can find, perfectly-nesting within every possible layer, a great chain of being seemingly suggesting that everything is related to everything throughout the universe.

**December 2011**: The Start of Our Research Using Base-2 Exponential Notation, Planck Length, And Plato’s Geometries.^{3} We used very simple math and got simple results yet also found hidden complexities. After doing a fair amount of analysis of our initial results, we continue to make new observations, conjectures and speculations about the forms and the functions within this universe. From all our data and study, it seems logically to follow that this tiling is the first extension of geometry and number (the sequence of notations) in a ratio.

**The most simple engaging the most simple**: Here may be the beginning of value structure.^{4} If so, it necessarily resides deep within the fabric of the universe, the very being of being. Could these very first doublings be the essential tension of creation?

**Here simplicity is based on a very simple logic, “ Everything starts simply.”**

^{ 5}

**NOTE**: The TOT as a tiling would be the largest-but 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).^{6} 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.

**The purpose of this article is to begin to introduce why we believe that this could be so**.

**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.

In earlier articles we observed how rapidly the vertices expand^{7} Yet, that expansion may be much greater once we understand the mathematics of doublings suggested by Prof. Dr. Freeman Dyson,^{8} Professor Emeritus, Mathematical Physics and Astrophysics of the Institute for Advanced Studies in Princeton, New Jersey. We are still working on that understanding.

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.

And, of course, we are just guessing though basing our conclusions on simple logic.

**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 tetrahedral-octahedral-tetrahedral 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.

**We Can Only Speculate**. We can only intuit the form-functions 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 million-trillion vertices (quintillion), using our simple math of multiplying by 2, we will see how that looks and begin to re-examine 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.^{8}

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 small-scale universe which for so many historic reasons has been ignored, considered much too small to matter.

Nevertheless, it seems to follow logically that this TOT tiling is in fact the reason the universe is isotropic and homogeneous.^{9}

**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.^{10} (A new image is under development with at least ten layers. A link will be inserted as soon as we have it.)

**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.

These same four plates (R2 tilings) can also be seen as triangle. There ares six plates of squares. One might assume that all these plates begin to extend from within the first ten notations from the Planck Length, and then, in theory, extend throughout our expanding universe.

Only by looking at our clear plastic models could we actually see these different R2 tilings.

We have just started this study and we are getting help from other school teachers.

We were challenged by Edkins work to see if we could find her plates within our octahedral-tetrahedral models and we were surprised to find most of her tilings within the model.

Within the Wikipedia article on *Tessellation* (link opens a new window), there is an image of the 3.4.6.4 semi-regular 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.

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.^{12}

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.

**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 three-dimensional tilings using just the tetrahedrons and octahedrons.^{13}

We are studying the Conway-Jiao-Tarquato (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 201+ base-2 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**

It takes on a new meaning within this domain of the very-very-very 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.

From where do these expressions of order derive? “*From the smallest scale universe…*” seems like a truism.

Perhaps this entire domain of science-mathematics-and-philosophy 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:**

**Endnotes, Footnotes, and other References**

1. This article is linked from many places throughout all the articles and documents. It is a working document and still subject to updates.

2. In 2006 I wrote to Dr. Francis Collins, once director of the National Genome Research Institute and now the National Institutes of Health. His publisher sent me a review copy of his book, *The Language of God*, and we spent several hours discussing it. The genome, the double helix and RNA/DNA have structure in common and it all looks a lot like a TOT line. Collins, a gracious and polite man, did not know what to say about the more basic construction.

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.

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.

*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.*

6. 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 201+ notations. Using the simplest math, multiplying by 2 (base-2), there is a rapid expansion of vertices. Yet, it can also be argued that vertices could also expand using base-4, base-6, and base-8. That possible dynamic is very much part of our current discussions and analysis. It is all quite speculative and possibly just an overactive imagination.

8. 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.

10. The two small images in the right column are of a very simple four-layer 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 octahedral-tetrahedral 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.

11. 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

12. 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 small-scale 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.

13. “*New family of tilings of three-dimensional 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

14 Our example of a TOT line was introduced on the web in 2006. In July 2014, this configuration was issued a patent (USPTO) (new window). That model is affectionately known within our studies as a TOT Line.