“Behind it all is surely an idea so simple, so beautiful, that
when we grasp it — in a decade, a century, or a millennium —
we will all say to each other, how could it have been otherwise?”
– John Archibald Wheeler, 1911-2008, physicist
How Come the Quantum? from New Techniques and Ideas in Quantum Measurement Theory,
Annals of the New York Academy of Sciences, Vol. 480, Dec. 1986 (p.304–316), DOI:10.1111/j.1749-6632.1986.tb12434.x
Is this simple mathematical-geometrical view of the Universe meaningful? Can we open a dialogue about the question?
by Bruce Camber
Introduction: Initiated in July 2015 this article is now a Working Document. I realize that the basic concept is idiosyncratic. There are profound challenges, i.e. Planck Temperature and the order of dimensionless constants; and, there are more questions than answers. Although I do not want to waste your time, the reason for working in public is to get your insights, suggestions and comments. Links, footnotes, and endnotes are rough. This article builds upon earlier work that can readily be found in the Index Thank you. – BEC
In December 2011 we began our work on a very simple mathematical and geometric model of the universe; it was playfully dubbed, Big Board-little universe. We had started using the following parameters — base-2 exponential notation, the Planck base units, and the Platonic solids — in ways that created heretofore unobserved boundary conditions.
Our Three Initial Conditions: We started with a simple mathematical chart extending the Planck base units to the Observable Universe. We were studying basic geometries and nesting geometries. And, we used very simple logic.
1. A basic chart. There are just 202 base-2 exponential notations from the base Planck units of Length and Time to the Observable Universe and Age of the Universe respectively. In our chart these two base Planck units tracked together in informative ways and raised many questions. Here the operative function was multiplication by 2 while the two base Planck units were the known properties being multiplied. Notations took on a diversity of names depending on the functional qualities we were observing. A notation could be a cluster, domain, doubling, group, layer, set and/or step. The known universe was defined from about the 65th notation to the 201st-to-202nd doublings. A largely-undefined, very small-scale part of the universe was given a simple geometric and mathematical structure from the 1st to 65th doubling.
2. Geometries. We imputed a pervasive, simple geometry throughout the universe. This project started within our high school geometry classes by going inside the simple tetrahedron by dividing the edges by 2 and by connecting those six new vertices. We could see four half-sized tetrahedrons in each corner and an octahedron perfectly in the middle. We then went inside the octahedron; there we found six half-sized octahedrons in each corner and a tetrahedron within each face. Our geometry classes were exploring the question, “How far within could we go by continuously dividing by 2 each tetrahedral-octahedral layer?” Then we asked, “How far out can we go by continuously doubling what we had?” With just these two Platonic solids, we could tile and tessellate each layer and between layers or doublings throughout the entire model. We learned about the limits in both directions and we have begun learning about this progression called base-2 exponential notation.
Our initial structures were all three-dimensional. When we found many two-dimensional plates across all the notations, coherence throughout the universe seemed possible.
Although there is no evidence that these analogical constructions exist within every layer, we imputed, hypostatized, or hypothesized that in some manner of speaking, such analogues do exist, especially within the first 60 doublings. We could then ask the question, “Given this ubiquitous, four-dimensional web (continuum, matrix, grid), why does the universe work in the manner that it does?” In looking for answers, we have begun to see a means to attract, relate, bind, break or repel constructions within each, and between each, of the 201+ layers.
3. Logic. Our current chart redefines the continuity function to start with the infinitesimally small measurements, the base Planck units, and go out to their largest possible measurements using the Observable Universe and the Age of the Universe as the primary outer limits. Though imputed, this continuity function became our first principle for order in the universe yet it took a period of contemplation of the Big Board-little universe charts and images to begin to see the universe as a natural container for space and time.
As a container with a definitive beginning and current limits, the weight of logic seems to favor the conclusion that the universe is finite. That quickly raises questions about the infinite, such as, “If it is not defined by space and time, how is it defined?”
Within the tilings and tessellations of our pervasive-but-simple geometries and with our base-2 expansion from the base Planck units, we began finding an extraordinary diversity of possible symmetries and potential relations. We asked, “Could symmetry-making and symmetry-breaking through time be the basis for all dynamics? Could the illusive harmony be a perfection of those symmetries within a moment in time?” Unto itself, this logic seemed to become its own system of value and for valuations.” Perhaps the very nature of space and time is derivative; and order, relations, and dynamics and their three functional qualities — continuity, symmetry and harmony — somehow constitute the infinite and are infinite.”
This simple logic became an important building block to postulate our first principles. Our charts had become a model of the known and a largely-unknown, infinitesimally-small universe.
Who? What? Why? When? Where? How?
4. History. This highly-integrated view of the universe must now be tested within the history of logic, mathematics, philosophy and physics. If this embryonic model is to have a place within the work of scholars, it must be critically analyzed. And, we know it has a long way to go before it earns such a place within scholarship. It must address very basic related questions about duality, finite and infinite sets, group theory, set theory, then advanced mathematical concepts that seem to be necessarily related like advanced combinatorics, matroids, amplituhedrons, and the Buckingham pi theorem. Like breadcrumbs, these topics will be followed up in the near future.
We are still within a very young and naive stage in our development and there are many very-very basic questions to explore:
These are some of the subjects (or objects) that occupy our attention and focus our time. “Let’s go over the details just one more time to attempt to learn how this model provides new footings and foundations that could give rise to some of our current perceptions and accepted models and theories.”
5. Starting point or domain or … The key question is, “What is being measured by the doubling of each Planck base unit?” Something is being doubled within each notation of those five columns and 202 notations. Although the Planck base units are at a point some call a singularity and others, the Void, we now ask, “What happens when each is doubled? What is manifest that doubles?” …only natural units? These are always based solely on universal dimensionless physical constants. But, all of them? Some of them? If so, which come into play and when do they come into play and why do they come into play? There are many books and articles about these constants, however, our primary reference is the 2006 article by Tegmark, Aquirre, Rees, Wilczek (TARW), “Dimensionless constants, cosmology and other dark matters” where they identify 31 dimensionless physical constants (PDF). The Planck Length (space) and Planck Time are two of their 31.
Once we have begun to understand the TARW conceptual frame of reference, we will attempt to take on the other 104 dimensionless constants defined within Wikipedia.
Our short-term work is to begin to understand the published works of an expert with each of these constants. Perhaps we will begin to see how our two base units create a nondimensionalized plenum and vinculum so an “archetype” of mass(kg) and electric charge (q) begin to manifest and we begin to discern how the parameterizing functions of the Planck constant (h) including the speed of light in vacuum (c), the gravitational constant (G), the electric constant (ε0) and the elementary charge (e) as each comes into play. We assume somewhere along our progression of doublings, the fine-structure constant (α) will present itself as will all the other dimensionless constants.
What is manifest? First, we have the actual calculations by Max Planck for length, time, mass and electric charge. Questions abound. “How do these manifest? Though infinitesimal, is there a manifestation of something?”
Our first assumption is that the “somethings” could be either simple vertices or what are known as point-free vertices. Part of our on-going study, we are told by Freeman Dyson that we should be using dimensional analysis and scaling laws to count the vertices within base-2 exponential notation; thus, we should be multiplying the number of vertices by 8 with each doubling. If so, there could be eight vertices within the first or second doubling.
With the second doubling we have the simple calculations — multiplying by 2 — of base Planck units of length, time, mass and electric charge. Then we have the scaling number or 64 vertices.
The first twenty doublings open our analysis. The first eight vertices constitute the first chapter of the story. Theoretically or conceptually, here is the first abiding step to construct and sustain our little universe. Here we will start our analysis with the tools of causal set theory, cubic close packing, Pi, the dimensionless constants, and a perfected state with continuity, symmetry, and an infinitesimally short moment of harmony.
*Vertices or point-free vertices
With every one of the TARW 31 dimensionless constants, a guess will be made to see what happens to the number within each doubling. We will watch the simple logic of each doubling, especially between the 65th and the 70th doublings. When does that number punch out and become something that is reduced to practice? Or, in what notation does a dimensionless constant combine with anything that is manifest? When is there an apparent effect?
By the 20th notation, our vertex figure using dimensional analysis is up to an exabyte, the same number as 2-to-the-65th or 1.1529 quintillion vertices. We can see therefore that count continues out to 54 places (18 x 3) by the 60th notation. These numbers are so far beyond “large numbers” that it may seem meaningless. Certainly we all need to begin getting accustomed to very large and very small numbers! It seems that we could conclude that with so many vertices there is enough potential structure to contain every part of the Standard Model known to date.
Anything and everything seems possible.
6. Identity: Humanity at the center of this model of the universe. In December 2014, when we tracked the Planck Time next to the Planck Length, we found 201.264+ notations. Our very first chart in December of 2011 had 209 notations. We did not know where to stop. A NASA scientist helped us; he calculated 202.34 notations. Then a prominent French astrophysicist who did a calculation of 205 notations (See footnote 5).
From the 100th to 103rd notations we find sperm, hair, the thickness of today’s paper from a book or magazine, and the human egg, clearly a few of the basics that evolve to become humanity. And, of course, we recognize that there are many other objects within these four notations. Yet, within its simplicity, there was a quiet affirmation, “Perhaps we, the swarming sea of humanity, are not irrelevant. This model places us squarely in the middle of it all.”
7. The small-scale, human-scale, and large-scale Universe. In our chart of the Big Board – little universe there are 201.264+ notations. When divided by three, each scale would ideally have just over 67 notations. Following a longstanding convention within scholarship, we call these groups, the small-scale universe, the human-scale universe and the large-scale universe.
The small-scale universe ranges from the singularity of the Planck base units to notations 67 and 68. Within the 66th and 67th notations, protons, fermions and neutrons are indexed. Leptons, quarks may well be within the 64th and 65th domain. Some posit them at much smaller sizes. But, the measuring tape is mathematics and it is oblique mathematics to be sure. Common elements of the aluminum and helium atoms show up in the 68th notation.
This human-scale universe ranges from the 68th notation to the 135th notation. There have been times when we have been boldly speculative, perhaps just imaginative, thinking about the transition from the human scale to the large scale.
The large-scale universe ranges from the 135th notation to just over the 201st notation. Not just the domain for governments anymore, here the truly imaginative, speculative, and bold have gone where others would fear and tremble.
These three scales provide the second most-simple division of the universe and by studying the transitions between each, we will engage combinatorial mathematics, group theory and set theory in fundamentally new ways. The continuity conditions are redefined. Symmetry functions are expanded. And, there is a possibility of understanding something new about the harmony of the universe (see history of the Greats who used such terms, i.e. Pythagoras, Plato, Aristotle, Kepler, Newton and Leibniz).
We have begun to analyze other progressions or scales based on fourths, fifths sixths, and so on. In time, we may find something of interest.
8. Numbers and Operands (from Sequential Real Numbers, to Base-2 to Dimensional Analysis). We have observed how the simple mathematics of both base-2 exponential notation and dimensional analysis become unwieldy rather quickly by the 60th and 21st notations respectively. Virtually every day we say, “We need to go over this one more time. It seems that we are missing something.”
First, the notations (doublings or steps) are sequentially ordered, 1 to just over 201. What is that sequence? Is there any possibility that it could be related to the Fibonacci sequence? What is the very nature of addition?
Next, there is multiplication, division, and ratios. A former NIST scientist and mathematics professor at Brown, Philip Davis, cautioned that the circle and sphere are more simple than the tetrahedron. Of course, he is right. We are now learning more about cubic-close packing (ccp) and the world of pi. Within the first notation with its eight vertices, we now know that we have to understand ccp and anticipate that the entire small-scale universe is driven by ccp.
That will be an article in the near future.
At the top of this article is a quote from John Archibald Wheeler who was thinking about the standards for measurement within quantum mechanics. If Pi drives this small scale universe, we know Pi is an irrational number and transcendental number that never ends and never repeats. It gives each construction those qualities and those qualities reflect an essence of quantum mechanics; we know there is a lot to chase down here.
Also, one of the most simple ccp configurations will be the pentastar with seven vertices in the form of five tetrahedrons. There is a 7.38° (7° 21′) gap that we have called squishy geometry as well as quantum geometry; here are degrees of freedom that continue within the icosahedron (20 tetrahedral structure) and the pentagonal dodecahedron (60 tetrahedral structure). What is it all about? We are not sure, but we do know it is worth more study.
There are many notations as those Planck base units are being multiplied by 2, that raise questions. We say, “There are doctoral dissertations in there!” It is within our scope of work. Then it came time to ask, “What has over a quintillion units of something?” Today, we have answered, “Vertices or point-free vertices.” Are there any other possibilities?
What are the key operands? It seems that a vertex is a reasonable answer. It is a special kind of point defined by axioms, and these have no “…length, area, volume, or any other dimensional attributes.” Yet, within our logic these points give functional capabilities to continuity, symmetry and harmony. And, these points have within them the conditions for order, relations and dynamics.
We take the universe as a whole, just as it is given; however, we assume that it is all complete, integrated, where the historic is the current, the here and now.
2. The Platonic Solids: The simple geometries still hold new insights
3. A Simple Logic: Continuity, symmetry and harmony
4. History within Logic, Mathematics, Philosophy, and Physics:
5. Starting Points:
6. Identity: Humanity at the center of this model of the universe.
7. Three Scales of the Universe: Small Medium and Large (more to come)
8. Numbers and Operands (more to come)