The generalized claim is that all the physical constants are derived from four math constants. They did the work to explain this statement with 142 such constants. Over time, every calculation and conclusion will be reviewed. Currently, our study look at their four math constants:
Pi (Links go to Wikipedia when within the body of the subject)
On its face, these four math constants do not seem to have enough substance to define all the better known physical constants. Yet, the Waterman-Whitehead team may have the kernel of an idea that truly opens the way to define the first 67 notations in ways that inform isotropy, homogeneity, the nature of infinity, the nature of the finite, and the function of math constants to create the bridges to infinity and the physical constant to build bridges to physicality or space and time
Most of the world’s people live within what we might call, their OwnView. Even though subjective and often quite naïve, the elitists and the solipsistic and narcissistic among us, lift up that view as the best view, the only view, and/or the right view.
If and when we start to grow up, spread our wings and begin to explore beyond our horizons, we develop an objective view of the world. As we integrate more and more facets of our subjective and objective views, it begins to qualify as a WorldView (in the spirit of the old Weltanschauung).
In light of Boeke’s work, the next step for all of us is to bring whatever WorldView we have, and see how it fits and works within a view of the entire universe. Kees Boeke’s work is historically the very first UniverseView. Although Boeke only had 40 jumps and used base-10 exponential notation, it is still the first systematic view of the entire Universe.
2011: A Second Universe View Emerges From Another High School
A high school geometry class just up river from the French Quarter of New Orleans developed what appears to be the second systematic UniverseView. It is quite a bit more granular than Boeke’s work and it originated from the students’ work with simple embedded and nested geometries. Using base-2 exponential notation this group emerged with about 202+ doublings, layers, notations, or steps from the Planck Length to the Observable Universe. Eventually beside each length, the calculations from the Planck Time out to the Age of the Universe were added.
This fully-integrated UniverseView first emerged in December 2011 and was officially dubbed, “Big Board – little universe.” One of the initial boards was over eight feet high and the second and third generations were around 60 inches high. The entire universe, mathematically-and-geometrically related within 200 or so notations, seemed to bring the universe down to a manageable size!
Now, what do we do with it?
The first thought was that this UniverseView with its 200+ notations could be a good container for Science-Technology-Engineering-Mathematics (STEM) education. It puts everything in the known universe within a simple ordering system. Then, in January 2012, in the process of trying to find scholarly references to understand the foundations of their work, the students and their teachers discovered Kees Boeke. In so many ways, it was a vindication — “Somebody had been here before us.” Yet, even with all the fanfare around Boeke’s work, not too much was done to extract meaning from that model.
The base-2 model is quite different. It has simple geometries and a more granular mathematics. The students and teachers thought this ordering system might help to answer those historic queries by Immanuel Kant about (1) who we are, (2) why we are, (3) where we are going, and (4) the meaning and value of life.
Given this model has a starting point and an end point, the students and teachers opted to see the universe as finite. Always encouraging students to go deeper in their understanding of mathematics, their teacher, Bruce Camber, commented “To engage the Infinite it appears that we hold the objective and subjective in a creative balance and that balance is called geometry, calculus and algebra through which we can more fully discover relations.”
Boeke’s base-10 work has an important role in history. It gave the human family a starting point to see an ordered universe. The base-2 model takes the next step. Instead of just adding or subtracting zeroes, it adds 3.333 times more steps or doublings. It provides more data to explore the simplest continuities, relations and dynamics within and between each notation. Base-2 is the heart and spirit of cellular division, chemical bonding, complexification (1 & 2), and bifurcation.
Perhaps it is here that the academic community might begin to create a truly relational, integrated and functional UniverseView. Surely it is here that we find the rough-and-tumble within science.
So, although base-2 UniverseView is the second UniverseView, it seems to hold some promise. And though these are preliminary models, just a crack in the doorway, what a sweet and simple opening it is. Perhaps Kepler would be proud.
This high school group is now just starting to discover the work of real-and-graciously-open scholars. With the help of this larger academic community, our work just might somehow capture the spirit of one of the world great physicists throughout history, John Wheeler, when he said, “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? How could we have been so stupid for so long?”
Simple facts, figures, and logic in search of a theory. The chart with five columns for each of the base Planck units and 204 notations down, 1020 boxes, provides real data to examine the logic flows both across and up-and-down. The goal of this article is to examine no less than 10 boxes. Looking across would be 50 boxes; adding two or three boxes up and down would total over 150 boxes. Ten percent should be enough to learn if a simple logic actually flows through the numbers. If it does… well, perhaps we have a new science in the making.
201+ Notations Begin With The Planck Length. The simple mathematical progression that rendered the 201+ base-2 exponential notations was the result of following embedded geometries going smaller and smaller until in the range of the Planck Length. Going in the other direction, larger and larger, was achieved by multiplying by 2 until in the range of the Observable Universe. The total, just over 201 doublings, could not be found within the writings of the academic community. The base-10 work done back in the 1957 by Kees Boeke and his high school classes in Holland was abundantly indexed; there were no references to a base-2 progression from the Planck Length to the Observable Universe, especially as a result of following embedded geometries within the tetrahedron and the primary octahedron within that tetrahedron.
Planck Time to the Age of the Universe is applied. There is general scientific concurrence regarding the estimates of the age of the universe. That figure provides a better framework for the doublings of Planck Time, from the beginning of time to this moment, right now, our current time, which always defines the endpoint. Planck Time and Planck Length track together in informative ways. For example, the notation that defines one second is between 142nd notation (.6011 seconds) and the 143rd (1.2023 seconds). The doublings of the Planck Length are 180,212.316 kilometers at Notation 142 and 360,424.632 kilometers at Notation 143. As one might have expected, the speed of light is confirmed in between the two at 299,792,458 meters for one second. At this point in time the other three Planck base units have become quite large, larger than any common number within human experience.
This Quiet Expansion begins at the first doubling. Quite literally, there is no room for sound until out to the 108th doubling (the beginning of sound waves) and on out to the 119th (the full spectrum of sound ranges from Notation 108 to Notation 119). There is something quite helpful within a visceral sense of the number and parameter. Examining groups of numbers associated with a common human experience is more than helpful; it provides the infrastructure of logic. Yet, there is no point where simple logic flows across all five Planck units. Yet, as demonstrated, it is quite informative when even two such numbers correspond.
For example, one of the very smallest notations with an experiential human equation is Notation 93 where observable light begins to manifest. Notation 101 is within the range of the thickness of human hair. This, of course, is where a large group within Planck Length and Planck Time correspond. This is the human scale universe. And, within that group there is one place where length and temperature correspond.
Planck Length and Planck Temperature. First, it was a leap of faith to hold to our working premise, “Everything starts simply” and to place the extremely hot Planck Temperature at the top of the chart. That put a very common number between Notation 103 and 104 where the temperature has cooled to 98.6 degrees Fahrenheit. Here we find among many other common things, the human egg cell. At Notation 105 the temperature has risen to 894 Kelvin or a very hot 1149.53° Fahrenheit and at Notation 102 it has dropped to a very cool –58° Fahrenehit.
Planck Mass. The very smallest notation with a common figure is the 31st doubling (Notation 31) where we find 103 pounds (46.74 kilograms). For many people, it is a key weight threshold signifying our coming of age, quickly approaching being an adult. Within this doubling the other four figures are so small, it causes one to ponder. So much seems to be happening with each of these doubling, that 103 pounds encourages some speculation. How about this? Perhaps the 103 pounds is the sum total weight of this notation! At the top end of this column are the outrageously large numbers that come very close to estimates by some of the more speculative within the scientific community, especially if each number in this column is the sum total weight of that notation. In some peculiar ways, this just may be a measurable concept.
The Human Scale Universe. Within the human and large scale universe, there are many familiar things within the Planck Length notations, yet the other Planck figures remain largely remote.
Planck Time. Although we cannot meaningfully perceive much smaller than a tenth of a second (Notation 140), in 2010 machines at the Max Born Institute in Berlin measured down to 100 attoseconds (Notation 87). Perhaps each notation with the Planck Time column describes a range in which relations are defined. Some elements of that statement may be measurable.
What Is Is? If looked to discern any special logic, one’s conclusion might be that each notation, with its vast array of vertices and multiples of the Planck base units, define the terms and conditions by which that notation-qua-notation is. That is, these numbers define the “isness” of the notation.
So, let us look in depth at one second between Notation 143 and 144. The total mass ranges from 2.4268×1034 kilograms to 4.8537×1034 kg. It defines a range, “no greater than twice that amount, and not less than half that amount.” In a similar manner, the total energy has a range, 2.0913×1025 coulombs but not greater than twice this amount and no less than half that amount. The total of heat within the notation, a huge stretch of the imagination, is 2.4578×1014 K to 4.9156×1014 K. Though an unimaginable amount of heat to be spread out throughout this single Notation 142, it just may be a measurable concept.
Planck Charge. Let’s look at which notations Planck Charge becomes a common number. For example, a lightning bolt is typically around 15 C, large bolts up to 350 C. That is quite visceral, yet on the chart it is in the range of Notations 63 to 67, the run up to the transfer from the small-scale to the human scale universe. If it represents the sum total charges within each notation, it certainly provides us with something to ponder.
These five Planck base units create very large continuity equations. Though imputed, remember that this schema is also based on the simplest geometries. Taking the entire chart and the weight of its simple logic, it suggests that the symmetries of these imputed geometries and these continuity functions are infinite, and that length (space), time, mass, charge and temperature are finite. These 201+ notations seem to define a finite universe and each notation defines a range in which particular subjects and objects are bounded by their Planck base units doubling, thereby each notation has a certain functional uniformity which provides a range within which particular groups or sets of things work.
Questions are asked, “Is this model the abiding, on-going, current structure of things as they are? How?”
201+ notations, divided by three, renders a small-scale, human scale, and large scale universe. The application of scaling laws and dimensional analysis to the first 60 notations resulted in learning about the power of base-8 expansion. By the 20th notation there are plenty of vertices with which to build structures; that is 1,152,921,504,606,846,976 or 1.152 quintillion vertices. By the 60th notation, add 36 more places (zeros). That is a robust infrastructure with 1152921504606846976000000000000000000000000000000000000 vertices (perhaps point-free vertices).
There is what would appear to be an infinite number of possible constructions. Add in the 131 better-known dimensionless constants and the fundamental physical constants, there should be enough variables to accommodate the Standard Model in physics as well as the science that has resulted from the standard model in cosmology. Please note that at the 60th notation, the size of the Planck Length doubling is not yet large enough to accommodate a fermion. From the 1st doubling to at least the 60th doubling, all the “structure” may best be described as hypostatic, which means in this instance, the essence or underlying reality.
Humanity doesn’t physically appear within the Planck Time column until well into the 201st notation. There has been a dispersion of length (space) mass, charge and temperature throughout an ever-expanding universe. Obviously there is a lot of science to learn between Notation 101 to Notation 202, and it will all be in relation to the deeper dynamics between Notation 1 and 101.
Reflections and Projections. Our base-2 chart of the Planck Base Units was first published in February 2015. This is its first review. It is an introduction that requires many more years of work and analysis. It frames a detective story whose final chapter could be written in many different ways. To expand the grounds of the analysis will require going deeply inside the simple geometries within the first 60 notations to discern how these geometries extend undetected, but measurably present throughout the entire universe. The assumed universals — order and continuity, multiple grids of relations with symmetries as well as asymmetries, and dynamics that seem to conjure up transformative instants of harmony, degrees of perfection and the darkest forms of chaos within degrees of imperfection — will be studied in light of duality,finite andinfinite sets,group theory, and set theory. That study will focus on the correlations with advanced combinatorics,matroids,amplituhedrons, and the Buckingham pi theorem.
Much more editing and perhaps a little more writing to come.
Working notes: When this page is ready to be declared “a working first-draft,” I will post an index of related articles; and as a working first draft, this post will be the first in that list. -BEC
Editorial note: Our world seems increasingly crazy. This model just might help to open new insights that might mitigate some forms of that craziness. So though still quite rough, it’s being brought into the light of the public rather early. Also, by working on it in public, perhaps others will have comments and suggestions to shape its potential.
Perhaps a few more questions and comments would help.
What is it about a circle and sphere that pi is always-always- always true?
How does a number become a constant, irrational and transcendental all at the same time?
Let us compare pi to other unique numbers that have a special role among all numbers. These are e,0, 1, and I. They are all magical, but π stands out. So, let’s ask, “What are the shared qualities of these numbers?” Let’s study them to see if we can find any necessary relations.
We have the ratio between a circle and a line. Perhaps this is the fundamental transformation between the finite and infinite? Are circles and spheres always implicating or imputing the infinite?
That is a big question and enough to ponder for awhile.
Notwithstanding, there are many more questions to ask.
Some speculations: Pi may be the key to unlock the small-scale universe within the big Board-little universe
1. To get to the application of pi within the Planck Units, we’ll need to emerge from the singularity of the Planck Units. Is the radian a key to understanding this process? First, a radius is extended from the singularity. A radius extends into the preconditions for space and time, a now emergent small-scale universe. It makes that first arc equal to its own length. It does it again and again and again and again and again (six radians) and then makes that last leap, 2 pi, to complete the circle. Is this a reasonable scenario? Why? Why not?
2. We need to run through dozens of scenarios, often, and slowly and carefully. What scenarios are perfect and obvious?
For further discussion:
1. Is the Small-scale Universe the basis for the homogeneity and isotropy of space and time?
2. Does everything in the universe share some part of the Small-Scale universe?
3. How is Planck Temperature calculated? Does it begin with the other Planck Units and expand from that figure at the first notation?
Note: All of human history has occurred in the last doubling. Yet, all doublings remain active and current and dynamic. Continuity trumps time. Symmetries trump space.
What does sleep have to do with anything? If all time is current, within the moment, we particularize by the day and uniquely within a given waking day. Sleep seems to bring us into the infinite. Dreams seem to be the helter-skelter bridge between the finite and infinite. It seems that these naïve thoughts are worth exploring further.
Editor’s note: This page was first posted within Small Business School, a television series that aired for over 50 seasons on PBS-TV stations (1994-2012). It is the author’s business website, so many of the links go to that Small Business School website. Eventually all links will be redirected to pages within The Big Board – little universe Project.
Background: Our study of the Planck Length to the Observable Universe began formally on December 19, 2011. Though we thought about the matrix from the Planck Time to the Age of the Universe, it took until December 8, 2014 to add it the Planck Length chart. Logically, but non-intuitively, the two tracked well together. Based on that work, we started looking at our own foundations for understanding first principles, universals and constants.
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?
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.
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
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. We believe we can find most of her tilings within the models.
Within the Wikipedia article on Tessellation (link opens a new window), there is an image of the 22.214.171.124 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”).
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 with her. 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.
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
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.
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.
September 2014: If you think about it, most of the world’s people have never heard of the Big Bang theory (Reference 1 – the cosmological model, not the TV series). Of those who know something about it, a few of us are somewhat dubious, “How can the entire physical universe have originated from a single point about 13.8 billion years ago?” It seems incomplete, like there are major missing parts of the story.
To open a dialogue about this pivotal scientific theory is the reason for these reflections. And, if we are successful, all of us will have re-engaged our ninth grade geometry classes and we will begin to ask a series of “what if” questions about the origins of this universe.
We are doing a little fact check to see if the authors give those notations from Planck Time any causal qualities. It appears that they were not concerned about those base-10 notations until we pointed them out to them.
The first time period of interest to us is the first 20± base-10 notations which would be the first 67 base-2 notations. What happens between the Planck Units and the emergence of the elementary particles? These are real durations in time. A lot can happen.
We will be exploring this small-scale universe in much greater detail. By the 60th doubling there are quintillions-upon-quintillions of vertices with which to create many possible models. Also, in light of the work to justify the Big Bang theory, there is an abundance of information from all the years of research since the concept was first proposed in 1927 by Georges Lemaître.
A lot of pre-structuring of the universe could be quietly happening within such a duration (1/100th of a second). Using our most metaphorical, speculative thinking, one could imagine that the actual event within those first sixty notations was a gentle, symphonic unfolding, fully homogeneous and isotropic.* Although we should embrace all the key elements of today’s big bang theory, we should also be constantly asking, “What kinds of geometries would be required within each of the first 60 notations to render these effects?”
Perhaps the universe and our future belong to the geometers.
So, this article is to empower all of us to find the best geometers around the world to engage the Big Board-little universe model within what we call “the really-real small scale universe.” Of course, some of the work has already been done within the study of spheres, tilings, and combinatorial geometries.