“Thus, in the last century only mechanical, chemical, thermal, electrical, luminous, and gravitational energy were known. Now, we know of nuclear energy, which constitute a much larger reservoir. But the infinite substructure of matter very probably contains energies that are as far beyond nuclear energies as nuclear energies are beyond chemical energies. Indeed, there is already some evidence in favour of this idea. Thus, if one computes the “zero point” energy due to quantum-mechanical fluctuations on even one cubic centimetre of space, one comes out with something of the order of 10^{38} ergs, which is equal to that which would be liberated by fission of about 10^{10} tons of uranium.”

Adam Becker, Author, “What Is Real?” by Basic Books, March 2018 Author and Astrophysicist http://freelanceastro.com RE: Nov. 15, 2018 Lecture, Harvard, “The Trouble With Quantum Physics And Why It Matters“

In 1971 I was a regular at the Boston Colloquium for Philosophy of Science (Cohen, Wartofsky, Shimony) and got caught by a lecture about the EPR paradox and John Bell’s work. In 1977, Viki Weisskopf (MIT) helped clear the way for me to visit with Bell at CERN and for a return a few years later. I was also able to visit with David Bohm at Birbeck College in London where with eight other graduate students we spent the better part of a day exploring points, lines, triangles and the tetrahedron. When I finally learned about Bohm’s death in 1992, I took down his book, Fragmentation and Wholeness, autographed and given to me in 1977. During what would be an honorary read, I declared out loud, “David Bohm! You never asked us what was most-simply inside the tetrahedron!” I quickly figured it out on my own.

In 1971, I was also part of the Philomorphs with Arthur Loeb at Harvard.

Forty years later, helping my nephew with his high school geometry classes, we chased that tetrahedron, going within, by dividing the edges in half, connecting those new vertices, to discover the four half-sized tetrahedrons in each corner and the octahedron in the middle. Doing the same with the octahedron, we found the half-sized octas in each of the six corners and the eight tetrahedrons, one in each face, all sharing a common centerpoint. Elegant.

Getting in the spirit of Zeno and people like Gian-Carlo Rota (Combinatorics, MIT), we chased the exponentially greater number of internal objects, back deeper and smaller within. In 45 steps, we were in the domain of particle physics. In another 67 steps within we were in the domain of the Planck scale. In 112 steps from the classroom, and we finally met Max Planck.

Such a finding it was but we had nowhere to go with it.

We shared our chart as a rather fun STEM tool but the first 64 notations were unsettling. We had numbers but nothing to match up with them, except perhaps such illusive things as strings, preons, gravitons, instantons, and dark matter and dark energy.

How stupid. How silly.

What would you do with these numbers? Shall we speculate that planckspheres manifest at the Planck scale, and have been filling the universe for the past 13.82 billion years, thus the expansion and, and, and….?

You know so much; you are playful. What do you do with such simple logic, simple math, but entirely idiosyncratic results. Shall we just flush it down and out of our systems? Thanks.

Most sincerely,

Bruce

PS. My last efforts within this arena was in 1980 in Paris where on one day at the Institut Henri Poincare, I studied with Costa de Beauregard and then the next day with J.P. Vigier. One day Vigier took me down to d’Orsay to meet Alain Aspect. Bernard d’Espagnat also joined us. Beyond just dropping the names of extraordinary scholars, at no time was the simple little sphere a point of any discussion. Isn’t that our collective problem?

In my first email to you, I asked, “Where are we going wrong?” Now, perhaps a bit more succinctly, I ask, “How can simple math and logic be so idiosyncratic? What are we doing wrong?” Should we attempt to grasp the first 64 doublings from the Planck-to-the-CERN scale? What is the deepest or smallest reach of CERN?

It is all quite idiosyncratic and often I feel like an idiot, but such is life!

Any and all critical comments are most welcomed. Thank you.

Of course, you are “…not too far removed…” from anything and your brilliance overflows. No self-deprecation accepted!

The first 67 or so notations are yet to be unmasked, but unmask them we will!

By the way, Viki opened the doors for me to visit with John Bell at CERN in 1977. On that trip I spent a day in London with Bohm and his aspiring PhD candidates. We talked about points, lines, triangles asking, “What are we missing?” When Bohm died in 1992, I took down his little book that he given to me, Fragmentation and Wholeness, and while reading, I finally asked, “What is perfectly enclosed within the tetrahedron?” I did not know. Even an old mentor, Bucky Fuller (he, too, was a “member” of Arthur Loeb’s Philomorphs) answered the question in a rather cavalier manner. There is more there than meets the eye. Base-2, applied to such simple geometries, opens profoundly-simple, eternally-complex systems. I so wish that I could have that discussion with Viki today.

Viki had a wonderful sense of the eternal. We became friends when I suggested to a Wall Street Journal friend and Boston-based writer that he do an A-hed article (the most-read column and considered a prize among the journalists) about Viki’s work within the Pontifical Academy.

We were at the faculty club over lunch. Too serious, my friend could not get Viki to speculate about a God particle. Of course, Higgs has total disdain for the expression. I say, “A particle it isn’t; a ratio it is.” Though that A-hed never shaped up for publication, it still needs to be written.

Months later in his home, we spent time looking through his wonderful collection of art and art books talking about eternal things. May I keep you informed of our progress? Simplicity is calling us.

When Lee Smolin and Anthony Zee attempted to combine Brans-Dicke gravitation with a Higgs-like spontaneous symmetry breaking potential, their preconditions of understanding were simply not simple enough. If space-time is derivative, finite, discrete and quantized… doesn’t it change everything?

Enough of my blather. I need to get back to your article and your ArXiv articles!

Congratulations on all your work posted in ArXiv and for your many books and articles. I am just now starting to wade into it all. I am trying to answer a question about base-2 notation from the Planck time to the age of the universe. In 2011 in a high school geometry class, we discovered just over 200 notations and we have been puzzled that nobody seems to find it at all interesting.

I guess that makes us idiosyncratic and probably a bit simple.

Now, quite a long time ago, I was a friend of Viki Weisskopf; and in the past few years, I have found the work of several MIT people to be most helpful, i.e. Wilczek on Max Planck (2001, Physics Today) and more recently, Guth on inflation and Tegmark on infinity.

Are bifurcation theory and base-2 related? Isn’t cellular production a base-2 phenomena? Where are we going wrong? (Our little history)

Could you help steer us in the right direction? Thank you.

PS. In and around 1975 I discovered and became fascinated with David Bohm’s work, Causality & Chance in Modern Physics (1957). On pages 163-164 he said:

“Thus, in the last century only mechanical, chemical, thermal, electrical, luminous, and gravitational energy were known. Now, we know of nuclear energy, which constitute a much larger reservoir. But the infinite substructure of matter very probably contains energies that are as far beyond nuclear energies as nuclear energies are beyond chemical energies. Indeed, there is already some evidence in favour of this idea. Thus, if one computes the “zero point” energy due to quantum-mechanical fluctuations on even one cubic centimetre of space, one comes out with something of the order of 10^{38} ergs, which is equal to that which would be liberated by fission of about 10^{10} tons of uranium.”

First email: Friday, September 03, 2010 @ 12:17 PM

Dear Prof. Dr. Leonard Mlodinow,

Not having the advantage of an advance copy, but following your Cambridge friend since the early ‘70s and enjoying his cooperation with our Schrödinger What is Life? tribute in 1979 at MIT, your collective work is somewhat known to me.

I have some simple-albeit-trick questions for you about basic structure and then a comment about metaphor.

The foundations, first principles of logic and life, begin with some ordering condition that creates continuity. We are able to measure. There is quantitative analysis. A science can emerge. A space is defined. Push the points and lines into three dimensions and an equals sign readily emerges. Relations are possible and can be quantitatively defined. Continuity equations now lead to symmetry equations.

Bear with me please.

Introduce duration and symmetries render all the diversity and complexity we know. Some of these dynamics are actually perfected in various manifestations of harmony. That is a simple overview of quantities.

With harmony, however, within the human condition, something rather special happens. We begin to have qualitative analysis. Numbers take on special meaning. There is interiority. Then, comes logic, linguistics, and a psychology and all the other expressions of humanity. Among those expressions are folks who think about universals. Put those in quantitative language and you have science, put them in qualitative language and you have personal dynamics, business, religion and the arts.

Too simple? I don’t think so.

Though not highly regarded by some, I enjoyed the friendship of David Bohm who encouraged us in a class to spend some time thinking about points, lines, triangles and tetrahedrons. When Bohm died in 1992, I took down his book, Fragmentation and Wholeness, that he had given to me as his guest in his class. After reading just a few pages, I thought, “Why did we not ask what is inside the tetrahedron?” I made a simple paper model.

For over a century, most discussions about simple geometries are not encouraged. As a result, most academics and practitioners of science could not answer the simple question, “What is perfectly enclosed within the tetrahedron?” School children should be able to quickly answer. At the center of the tetrahedron is the octahedron which poses an even more difficult challenge for most. Again the kids should know better. At Princeton a few years ago, I asked the surreal numbers man, John Conway (a good gaming guy), the question. He bought some time by asking me why I was so hung up on the interior structure of the octahedron. Then he said, “Let’s figure it out,” and of course, he did. Though a few of Bucky Fuller’s people could have answered the question, very few can.

And therein lies answers beyond our wildest dreams. The simplest interior structure of the octahedron is not simple at all. And, therein is not only the GUT, the TOE and a few TOES as well, there is more. This is what is. It is pure thought and for some reason, that exploration , if we prevail, is just about to explode. We’ll finally get to see the cohesiveness of knowledge, all of knowledge. Make it personal, twist a few metaphors within it, and you can get a lot of God talk. Keep it abstract and quantitative, and you have a science.

I know that you didn’t ask, but I thought you might enjoy a slightly different twist on our mutually-shared views of our universe.

On my first trip into Europe in 1977, I stopped in London to visit with David Bohm, then went up to Cambridge University, Clare College, to visit with Arthur Peacocke to talk about Newton-Leibniz debate, continuity-symmetry-harmony, and the future of science and religion. More to come…

And, because I have made so many references, I have started pages for you and Prof. Dr. Chuanming Zong. Your page is here: https://81018.com/2020/03/28/lagarias/ Of course, if you would like anything changed, deleted, or added, I will be glad to accommodate. Thank you.

Most sincerely,

Bruce

Second email: 27 March 2020

Dear Prof. Dr. Jeffrey Lagarias:

I thank you again and again for your scholarly work. I endorse your work! Yet, given our work is so idiosyncratic, you probably would prefer that I didn’t.

Notwithstanding, I am glad for “Mysteries in Packing Regular Tetrahedra(PDF).” Just about every day, I wonder what 1800 years of being wrong did to our scholarship.

^{10}Geometric gap: 0.12838822+ radians and 7.35610317245345+° degrees. Even today, March 2020, this gap is little studied and less known. Our first encounter with it was in 2016 upon writing the article, “Which numbers are the most important and why?” At that time, it seemed like Chrysler Corporation had branded that geometry as the pentastar. And though it is a five-tetrahedral representation, they never looked uniquely at the gap of 7.35610317245345+° also defined by 0.12838822+ radians. Two chemists (Frank & Kasper) came closest to opening the discussion in the 1950s. Two academics (Lagarias and Zong) did a preliminary analysis that was a tremendous help; the relation of this gap to the deeper geometries of life remains as a challenge. Our modest start is here: https://81018.com/number/#Pentastar

^{11}Aristotle’s failure is our failure. Perhaps the gravity and nature of this error is only now beginning to be understood. We all make mistakes. When we are challenged, we defend our concepts as best we can, and then adapt. We change or our associates change.

Some people become larger than life within their own time. Three examples are Aristotle, Newton and Hawking. All three were wrong about one key impression about the nature of life, yet their egos and their position and their person were so illuminated, it became increasingly difficult to challenge their assumptions.

Aristotle’s geometric gap, Newton’s absolute space and time, and Hawking’s infinitely hot big bang have each mislead scholarship and we all lost the scent and direction of the chase with its potentials for discovery and creativity. Throughout our ever-so youthful human history, such people can readily continue to mislead us. We have to be vigilant to review and re-review all the concepts we hold dear and begin to adjust them appropriately.

First email: Saturday, 31 August 2013 at 8:19:21 PM

Jeffrey C. Lagarias, Professor of Mathematics, University Chuanming Zong, Professor of Mathematics, Peking University

Just a terrific job. A wonderful read. Thank you.

Coming up on two years now, we still do not know what to do with a simple little construct: https://81018.com/planck-length-time/ I have a hunch that that object made of five tetrahedrons plays a key role.

Your work gives me a wider and deeper perspective.

Thanks.

Warmly,

Bruce ************* Bruce E. Camber

PS. Long ago I studied with David Bohm, Phil Morrison, and so many others like them, but to make a living, I became a television producer! We had the longest-running television series on PBS stations in the USA and the Voice of America around the world about best business practices. http://smallbusinessschool.org/page18.html

Here are some key points within my current thinking:

1. The universe is mathematically very small. Using base-2 exponential notation from the Planck Length to the Observable Universe, there are somewhere over 202.34 notations, steps or doublings. NASA’s Joe Kolecki helped us with the first calculation and JP Luminet (Paris Observatory) with the second. Our work began in our high school geometry classes when we started with a tetrahedron and divided the edges by 2 finding the octahedron in the middle and four tetrahedrons in each corner. Then dividing the octahedron we found the eight tetrahedron in each face and the six octahedron in each corner. We kept going inside until we found the Planck Length. We then multiplied by 2 out to the Observable Universe. Then it was easy to standardize the measurements by just multiplying the Planck Length by 2. In somewhere around 202 notations we go from the smallest to the largest possible measurements of a length.

2. The very small scale universe is an amazingly complex place. Assuming the Planck Length is a singularity of one vertex, we also noted the expansion of vertices. By the 60th notation, of course, there are over a quintillion vertices and at 61st notation well over 3 quintillion more vertices. Yet, it must start most simply and here we believe the work within cellular automaton and the principles of computational equivalence could have a great impact. The mathematics of the most simple is being done. We also believe A.N. Whitehead’s point-free geometries should have applicability.

3. This little universe is readily tiled by the simplest structures. The universe can be simply and readily tiled with the four hexagonal plates within the octahedron and by the tetrahedral-octahedral-tetrahedral chains.

4. And, the universe is delightfully imperfect. In 1959, Frank/Kaspers discerned the 7.38 degree gap with a simple construction of five tetrahedrons (seven vertices) looking a lot like the Chrysler logo. We have several icosahedron models with its 20 tetrahedrons and call squishy geometry. We also call it quantum geometry (in our high school). Perhaps here is the opening to randomness.

5. The Planck Length as the next big thing. Within computational automata we might just find the early rules that generate the infrastructures for things. The fermion and proton do not show up until the 66th notation or doubling.

I could go on, but let’s see if these statements are interesting to you in any sense of the word. -BEC

_______________________________________ Determinant becomes undecidable, uncomputable and unpredictable _______________________________________

Bruce E. Camber
Big Board-little universe Project: http://81018.com
500 East Fourth Street #484, Austin, TX 78701
camber@81018.com

April 22, 2020

Abstract

Apply base-2 exponentiation to the Planck base units and the universe is parsed within 202 notations or doublings. These initial Planck units are derivative and finite. All the values by which each is defined opens questions about the nature of the finite-infinite relation. Within this emerging model, infinity is: 1) continuity creating a finite order and time, 2) symmetry creating finite relations and space, and 3) harmony creating finite dynamics and a space-time moment. No other definition of infinity or the infinite is engaged. Within this construction there is a small range of notations, the dynamics of which are determinant and are also understood to inculcate the following: decidability, computability, and predictability. Then comes a range of domains, the dynamics of which transition to the indeterminate and a transmogrification to undecidability, uncomputability, and unpredictability. There is a domain of perfection with no quantum fluctuations and a much larger domain of imperfection where quantum fluctuations have become dominant.

The Focus. We are projecting that within the first 64-notations, just below thresholds of physical measurements, there is a dynamic range of perfectly-defined domains, and then, an even more dynamic range that transitions between determinacy and indeterminacy. These first 64 notations, we believe, are the grounds for the decidable [1], the computable [2], and the predictable [3]. An irony is that although logically determinant, these values are too small to be measured. Once measurable, the measurements become indeterminate. It is a fundamental transmogrification to undecidability [4], uncomputability [5], and unpredictability [6].

Planck base units. We begin with the Planck base units of time, length, mass and charge. We take these face values as a given and ask the question, “What would the universe look like if the very first moment begins with the instantiation of those four values?” Of course, these values are the result of equations with additional values used by Max Planck to render his four basic numbers.

Consider the four equations and their numbers for space (length), time, mass and charge (Illustration 1).

Background. In our high school geometry classes the question was raised, “If the Planck Length and Planck Time are the smallest possible units of length and time, does it follow that these are also the very first units of length and time? [7] Does it follow that these equations, with all their dimensionless constants, come together to become the very first moment of physicality?” We were unwittingly opening the “CDM of the universe” and wondered if Steven Weinberg would call it a “grand reductionism” [8].

Our postulate is that these Planck’s units are really-real physical entities, not zero-dimensional point particles, but an actual entity defined by those base unit values. So our next question was, “What would this entity look like?”

Every equation is in part defined by the speed of light, pi (π) and the Planck Constant.

Because our students were studying basic geometric structures, they had a few answers. Yet, after some discussion, the students of pi, circles, and spheres prevailed. We then assumed not one sphere, but an impossibly-fast, steady stream of primordial spheres emerge. We then wondered what the next dynamic could be.

We decided to invoke Kepler (1611) and his sphere-stacking exercise of that year. (Illustration 2). So analogically, like Kepler, we now have this infinitesimal, raw stacking of spheres. We consider the first ten notations. Within Notation-10 there could be as many as 256 spheres. However, if we follow the advice of Freeman Dyson regarding scaling vertices and dimensional analysis, there could be as many as 67,108,864. Go to the (Chart, column 10, lines 8 & 9). We decided that at some point we would learn a deeper logic and we would be able to decide.

The dynamics still beg the question, “What happens next?” Our students had a quick answer, “The spheres come alive.”

First, there are the dynamics within cubic-close packing of equal spheres. The radii “discover” radii (see Illustration 3), and triangulation begins aka, triangulated coordination shells [9]. The discovery process continues and a tetrahedral layering begins enclosing octahedral cavities.

Spherical perfections. Within this thrust to create perfect continuity, perfect symmetry, and perfect harmony, this infinitesimal universe takes shape. The plancksphere [13] dominates. In theory, there is nothing that is undecided, uncomputed, and unpredictable. It is all a quiet emergence within a simple perfection. It’s creating an isotropic and homogeneous universe. Infinitesimal and way too small to measure, these are domains reached only by logic and mathematics. Actual physical measurements of a length begin around the 67th base-2 notation; and, the first unit of time, the attosecond, is not measured until the 84th notation. Thus, there is an extraordinary amount of intellectual space to tie logic, numbers and geometries together. Of course, it will be a challenge, but it just may be relatively straightforward for some scholars.

It is a study of perfected states in space-and-time.

The Expansion: Geometric, arithmetic, and exponential. Scholars within Langlands programs and string theory have done major work to define this space and its automorphic forms [14]. Here is a discovery process whereby every equation within Langlands programs has a place within a highly-structured environment. Every radius of each sphere (a string) opens Witten’s equations of state and the Seiberg–Witten invariants [15].

This simple base-2 ordering system quickly becomes complex. Each of the nineteen subsequent prime-number notations — 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, and 61 — introduce even more complex mathematics. The remaining prime numbers — 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 197 and 199 — open new physical potentials.

Also, base-2 is just one dynamic of this expansion. This universe appears to be opportunistic so may well use the other prime number bases — base-3, base-5, base-7, base-11, and base-13 (right up on up to base-101 — to introduce yet even more complex functions [15]. Of course, a majority of notations are included within base-3 (67), base-5 (40) and base-7 (28).

Within this model all notations are always active. They build off of each other.

Finite-infinite. This system is its own self-enclosed system, a working finite model theory, with its own rules and axioms that are grounded in a problematic statement — the origins of this perfection are not finite. Opening the finite-infinite relation is an age-old enterprise so we close that door rather quickly with this simple definition of the infinite: It is the qualitative expression of continuity, symmetry and harmony whereby continuity begets order, symmetry begets relations, and harmony begets dynamics. The finite is the quantitative expression. That’s it. Nothing more.

The Results. Within a little over one second, the base-2 expansion is out to the 144th notation. Planck Length is 360,424.632 kilometers. Planck mass is a hefty 4.8537×1034 kilograms. Just to put that in context, the mean average distance between the earth and the moon is 384,402 km (238,856 mi). It is 356,500 km at the perigee and 406,700 km at the apogee. The sun’s weight is around 1.989 × 1030 kg so at this notation, the density of the universe is like a neutron star.

The universe as we know it begins to take shape between Notation-196 and 197. Here the universe is at 10,829,559,004,640,000 seconds or about 343.15 million years.

Just within these 202 notations are a few highlights of this base-2 model:

The first 1000 years is between Notation-178 and 179.

The first million years is between Notations-188 and 189.

And, the first billion years is between Notations-198 and 199.

This model is primarily about the very early universe. Within the process, while it is being filled with planckspheres and with the emergence of geometries from the simplest to the increasingly complex, a five-tetrahedral cluster will manifest.

Although Aristotle thought it was a perfect configuration [16], and that understanding had been perpetrated by scholars for over 1800 years, the truth became apparent in the 1500s. Yet, even at that point, the real realities remained under-reported and less understood. As recently as the 1950s, chemists who recognized Aristotle’s mistake, calculated that size of that gap. It is an important gap.

Pentagonal, icosahedral and Pentakis-dodecahedral structures have such a gap or the surfaces are stretched and the internal angles are not exactly 60 degrees. In the 1960s the first concepts around aperiodic tilings were introduced. In 1976 Roger Penrose introduced his unique tilings and Alan Mackay followed up experimentally to show how a two-dimensional Fourier transform (with rather sharp Dirac delta peaks) manifests a fivefold symmetry. In 1982 Daniel Shechtman began his public-struggle to open the exploration of quasicrystals. This struggle to understand these geometries are current and on-going.

Pentagonal faces introduce new dynamics. The most fundamental dynamic would be the beginning of quantum fluctuations and its aftermath, undecidability (subject), uncomputability (relation), and unpredictability (object).

We are in search of answers to the question, “When and where do these fluctuations manifest?” We’ve beg for help. These are all new studies for us. Our simple history begins in 2011. Our critical history didn’t really begin until 2016. Notwithstanding, we are speculative people and believe the fluctuations actually and measurably begin to fluctuate, first between notations somewhere above Notation-64, and then between sets of notations, and within groups of notations.

Our Fuzzy Universe. In 1945 John Wheeler (Princeton) and Richard Feynman (Caltech) proposed quantum field theory or QFT, and, it has increasingly become a bedrock of physics [17]. Very well-defined, it can be argued that QFT has the deep roots to the unpredictable and indeterminate within the sciences, mathematics, and even logic, linguistics, philosophy, and consciousness. However, those within QFT studies have not yet considered the 202 base-2 notations and the implications of Planck base units, an exponential universe, a dynamic finite-infinite relation, and the dynamics of a structural evolution from the sphere and cubic-close packing of equal spheres. Within this model, the old epochs of big bang cosmology get readily absorbed by an all-natural inflation. There are at least eighteen very special claims that provide a possible foundation for QFT that deepens its roots and broadens its reach.

Gödel’s constructions.[18] Gödel gives Newton’s absolute time a place within General Relativity. And, given Einstein’s special relations with Max Planck, we are still in search of any references where Einstein or Gödel engage the Planck base units. Though Gödel was a teacher-professor (1940-1978) at the Institute for Advanced Studies and Einstein and Gödel had long walks, the infinitely-hot big bang probably got in the way. Even with Gödel work on numbering and base-2, he did not clearly demarcate a beginning of the universe.

Conclusion. We will continue asking scholars about our simple configuration as we learn more about those topics required to make our universe work. Thank you. –BEC

Endnotes, Footnotes, References, and Resources

[1] Decidability (Subject). A key to this theory and construct (Wikipedia) is the coherence of its logic. Are Planck Time and Planck Length the smallest possible units of space and time? Some scholars say it is. By starting at those Planck base units, does that base-2 progression necessarily include everything, everywhere, for all time? Max Planck never applied base-2, so he was not able to declare the base-2 progression of doublings to be a logical system. We do. It is totally predictive. One validation point is between the 143rd and 144th notation; it logically confirms the distance light travels in one second. That number is 299,792 km and it is within .1% of the speed of light confirmed within the laboratory. That requires a certain coherence of our four most-basic natural units, those Planck units, and their dimensionless constants. That would include c with special relativity, G with general relativity, and ħ with quantum mechanics. Also, there is ε0 (vacuum permittivity) with electromagnetism and kB (Boltzmann constant) with temperature/energy). More…

[2] Computability (Relation). The question is asked, “Is it possible that computable functions, including Turing degrees, are not necessarily set within just Notation-202? In this model where space and time are derivative, computability theory logically begins within Notations 0-to-1 and builds logically to include Notation-202. The algorithms of computational logic, like dimensionless constants, do not necessarily reside within machines. Questions about the mind and consciousness are stretching us to start a theory of computation that logically starts within Notations 0-1 and grows to Notation-202. More…

[3] Predictability (Object). Because the only notation that has a past and future is Notation-202 —all others are complete, yet fully dynamic and fully symmetric — there is still change. That thrust for change is a dynamic that effects every notation. To begin to enter this intellectual space, some of the more recent work regarding predictability is being engaged:

Within this model, time is finite and derivative. More…

Transmogrification: Starting with Notation-1 and going as high as Notation-67, logic prevails, yet our universe remains opportunistic. There is a thrust for more continuity, symmetry and harmony, reaching for a higher perfection. The first ten notations have been generally associated with Plato’s forms which today are associated with the automorphic forms of Langlands programs and string theories. Essentially this is the first-order of the planckspheres. We hypothesize that the next set of notations are a second-order for structures and a third order of planckspheres is for substances in the spirit of Aristotle’s Ousia. The emergent face of forms, structure and substance is qualities, the fourth face of the planckspheres. Perhaps as early as Notation-40 the first group of five-tetrahedrons, twenty tetrahedrons (icosahedron), or sixty tetrahedrons (Pentakis-dodecahedron) are manifest. When there is an ordering within the system, that gap, a squishy geometry, creates the indeterminant. We are projecting that transitions between Notation-50 and Notation-67. That’s a guess. More…

[5] Uncomputability (Relation). The historic brain-mind discussion will be part of these discussions. The Mind, that is, all minds, are currently projected to emerge within Notations 50 to 60 while consciousness as we understand it today would always be within the current time within Notation-202. To begin to grasp the boundaries conditions between computability and uncomputability we are engaging many documents including:

Perhaps the most illusive and difficult of all our studies, of course, there will be more…

[6] Unpredictability (Object). In 1970 I became enamored with the work of John Bell at CERN labs in Geneva. He had a new way of looking at the EPR Paradox. In time I went to meet him stopping along the way to visit with David Bohm and Carl Friedrich von Weizsäcker. My goal was to better understand how the relations could be the primarily real within the subject-relation-object formula. It was only by making space and time derivative did the simple formulation of continuity-symmetry-harmony, a moment of perfection begin to open up. We wanted a container universe but did not discover the container until 2011. Now, all these articles are beginning to make some sense. Unpredictability is built into the very essence of geometries of the universe, yet the universe itself can be profoundly knowable. Freedom, creativity and moments of perfections are all built into the structure of unpredictability. Though it is not the fundamental structure, it intersects with the most fundamental. So, to help better understand this nascent model, we continue to explore a few related articles:

These are new studies for us and we have “miles to go” before it all coheres. More…

[7] The story behind this story. There are links throughout this website to the December 19, 2011 story of our high school geometry students and their teachers who chased tetrahedrons and octahedrons down into the Planck scale and then out to the age and size of the universe. Here are other key links to tell a bit more of that story:

When we could find no place within our grid for Plato’s Eidos and forms, Aristotle’s Ousia, binary operations, axiomatic set theory, pointfree geometries, Langlands programs and string theory (please see line 11 of our horizontal chart), we decided, “That’ll all be within the first 67 notations.” We knew then there would be an endless amount of work to do within this model.

The advantage of youth and naïveté resulted in this early story about our work together.

Naïveté is often a curse; but in this instance, it allowed our most simple concepts to emerge. More…

[8] Grand Reductionism. We started as everything does — simple. We take very little steps and ask simple questions. We try to respect the scholarship that has gone on before us. When we become confused, we step back to something more simple. So, it was with deep respect that we engaged the CDM approach to the universe and Steven Weinberg and his book, Facing Up. Yet, we could not imagine a larger “grand reductionism” so, we did wonder!

We continue wrestling with his work and with these other scholars:

Beyond the Dynamical Universe: Unifying Block Universe Physics and Time as Experienced, by Michael Silberstein, W. M. Stuckey, Timothy McDevitt, Oxford (2018) Also: https://www.relationalblockworld.com/

In 1979 I first met Steven Weinberg at his office in Jefferson Laboratory at Harvard. He had not received his Nobel prize, but The First Three Minutes was out. Everybody fights for their legacy.

[9] Triangulations. Our initial studies of the work of F. C. Frank (H. H. Wills Physics Laboratory, University of Bristol, England) and J. S. Kasper (General Electric Research Laboratory, Schenectady, N.Y.) opened the concepts within cubic-close packing of equal spheres, the triangulated coordination shells, and the emergence of the tetrahedron from just four spheres. That all opened the way to engage The Physics of Quasicrystals, World Scientific, 1987 edited by P J Steinhardt and S Ostlund. We struggle to grasp the work of scholars within this area:

Objections to set theory as a foundation for mathematics

Jonathan P. K. Doye and his group are very helpful

[10] Fourier. Our first introduction to the Fourier transform was through Steven Strogatz on Pi day in 2015. His article for The New Yorker Magazine resonated at that time and it still does today. Now we are attempting to really dig into the Fourier work. Of course, we have a long way to go. Here are some of the scholars to who we are currently turning for help:

A huge study, we have barely scratched the surface. More…

[11] Lorentz. Linear transformations are part of the dynamics within a notation. Yet, there is a homogeneity with all the contiguous planckspheres so geometries may readily extend within notations and across notations. It seems that the dynamics of all geometric models of the universe may hold insightful keys. With that mindset, we are open to all studies of space and time symmetry:

Lorentz Transformation

Rovelli: Reconcile Planck-scale discreteness and the Lorentz-Fitzgerald contraction

Planck scale space time fluctuations on Lorentz invariance at extreme speeds

[12] Poincaré. In 1980 I worked with Jean-Pierre Vigier and Olivier Costa de Beauregard at the Institut Henri Poincaré. Our focus was solely on the EPR paradox, Bell’s theorem, and the experimental work of Alain Aspect at the SupOptique or “IOGS” in d’Orsay (just outside of Paris). Never did we look back at the work of Henri Poincaré. Today, a focus is on the Poincaré sphere and its underlying Lorentzian symmetry as a geometrical representation of Lorentz transformations. More work with these scholars:

The Poincare Conjecture: In Search of the Shape of the Universe, Donal O’Shea (2007)

Sphere in Various Branches of Physics, Tiberiu Tudor (February 2018)

Harmonic Analysis on Symmetric Spaces—Euclidean Space, the Sphere, and the

[13] The Infinitesimal Sphere. Also called the Planck sphere, it is a key, core concept and we will continue to research it until we find the best possible resources that go back as early as possible. To date, we start with John Wheeler’s work with quantum foam believing that it could hold a key. Others are:

Discrete Model of Electron, April 2019, DOI: 10.13140/RG.2.2.28408.49920, Discrete Universe Project, Jose Garrigues-Baixauli, Universitat Politècnica de València, Spain PDF

Physical Significance of Planck Length, Thanu Padmanabhan, Ann. Phys. 1985 165(1) 38-58

We continue wrestling with the work of scholars within this domain, so there will always be more…

[14] Automorphic forms. We turned to the scholars within the Langlands programs and string theory to learn about automorphic forms. We are learning about Loop Quantum Gravity n . They have done sustained work since the 1970s and have done a major amount of work to define its automorphic forms:

Automorphic forms (Wikipedia) “One of Poincaré’s first discoveries in mathematics, dating to the 1880s, was automorphic forms.”

Langlands program (Wikipedia)

Is there an analytic theory of automorphic functions for complex algebraic curves?, Edward Frenkel (ArXiv – December 2018)

We continue wrestling with the work of scholars within this domain, so there will always be more…

[15] String theory. One of the world’s leading scholars within string theory is Ed Witten. He is also a gentleman. Because the majority of his career has been in the shadow of big bang cosmology, his work has had an impossible starting point with which to contend. There is no easy migration to a theory that pushes time-space-and-light together at the Planck scale, and then with mass-and-charge at the next level (c2). It will be fascinating to see if they will do better within a cold start that redefines the historic æther, and gives their discipline the radius of the plancksphere within Notation-1 and every plancksphere through Notation-67, and at least the first 67 notations of each subsequent notation.

The logarithmic equation of state for superconducting cosmic strings, Betti Hartmann, Brandon Carter, November 2008 arXiv:0803.0266

Seiberg–Witten invariants

We continue wrestling with the work of scholars within this domain, so there will always be more…

Of course, base-2 is the first exponential expansion of this model such that no point within the universe, right from the first instant, is more than 202 notational steps away. Yet, I believe our opportunistic universe will also test base-3 which would aggregate a 67-step shortcut through to Notation-201. Base-5 would provide a 40-step shortcut through to Notation-200. Base-7 would provide a 28-steps to Notation-196, base-11 just 18 steps to Notation-198, and base-13 just fifteen steps through to Notation-195. These clusters of notations possibly can introduce even more complex functions.

The largest square of a prime, 13^{13} is 169, obviously under 202; and, 17^{17} is over (289). More…

[16] Geometric gap. One of history’s greatest thinkers made a most fundamental mistake that was repeated for about 1800 years. That is a tragedy of epic proportions. We are all taught to have such great respect for scholars, sometimes it holds us back. Aristotle (384–322 BCE), one of the greatest Greek philosophers and a polymath obviously had imperfect models of the tetrahedron, otherwise he would have seen and felt this geometric gap. Five tetrahedrons all sharing one common edge opens that gap.

1800 years. The greats that followed him repeated his mistake and we failed to grasp a most-essential quality of simple geometry. One of our primary source article is “Mysteries in Packing Regular Tetrahedra (PDF)” by Jeffrey C. Lagarias and Chuanming Zong. They relied heavily on the Dutch article by D. J. Struik, Het Probleem ‘De impletione loci’, Nieuw Archief voor Wiskunde, Series 2, 15 (1925–1928), no. 3, 121–137. Two chemists, F.C. Frank and J.S. Kasper with their article, Complex Alloy Structures Regarded as Sphere Packings, took it further and calculated that gap. An key part of it all is the work of Daniel Shechtman, I. Blech, D. Gratias, and J. W. Cahn, Phys. Rev. Lett. 53, 1951 (1984). https://doi.org/10.1103/PhysRevLett.53.1951, Google Scholar Crossref

We are among a very few who claim that this gap is the basis for quantum indeterminacy, imperfections, free will, unpredictability, undecidability and uncomputability, so there will be much more to come…

[17] Fuzzy Universe. The concept of a warm and fuzzy universe certainly flies in the face of current cosmology and even with the physics that has grown out of the work of people like John Wheeler and Richard Feynman. Yet, perhaps we have the makings for a mathematics of a hyperconnected universe. Let’s open it up for discussion. Let’s see if our universe is as hyperconnected as the internet and our brain. More…

Aristotle’s mistake was so ingrained and so pivotal, quantum fluctuations have remained a mystery to this day. Here the belief that physics is geometry bears out in a most simple way to explain a most difficult concept.

Newton’s absolute space and time held back scholars with a deep understanding of continuity, symmetry, harmony, and yes, even beauty. The infinite became a place where only fools dared to tread.

Hawking was trapped by his disease and by becoming a Lucasian professor. He couldn’t deny Newton,. Had he, perhaps he would have been moved to redefine the infinite as continuity (order), symmetry (relations) and harmony (dynamics) whereby space and time would have become derivative and relational.

Even with his work on numbering and base-2, Gödel never applied his logic to a base-2 model of the universe from Planck’s units to the current time. It is our loss. More…

I started my Bohm trek in 1971 with Causality & Chance in Modern Physics, 1957, pages 163-164, That little section is here: https://81018.com/nobel-laureates/#RR1 A few years later I had my first opportunity to visit with David Bohm at Birbeck College (and then yet another all-day session a few years later). Bohm inspired me to go inside the tetrahedron and octahedron: https://81018.com/tot/ And, just in the past week I have sent my first notes to Aharonov and Hiley: https://81018.com/alphabetical/ We’re all getting old and most have died. Yet, there is a possibility we are getting closer to the real core. Maybe our idiosyncratic ideation might hold a clue.

Long, long ago, I started following David Bohm upon reading Causality & Chance in Modern Physics (Routledge, 1957). Ted Bastin’s Quantum Theory and Beyond (Cambridge, 1971) was a wonderful introduction to everyone, including you! I don’t know why it has taken me 50 years to write to you to just say, “Thank you.”

Ted Bastin and I became friends. He was a guest in my home for over a week back in the 1970s. I visited with Bohm in London and Bell in Geneva in 1974 and 1977. Bohm gave me a copy of his little treatise, Fragmentation and Wholeness. I was at BU, part of Robert Cohen and Abner Shimony’s group. Through Ted, I also became an irregular with H. Pierre Noyes’ Alternative Natural Philosophy group at Stanford. In 1980 I studied with Vigier at the Institut Henri Poincaré. In 1981, feeling a bit like a whirling dervish, I dropped out and returned to a business that I had started in 1971.

Something was off with our models.
Renormalization wasn’t the answer.
Newton’s absolute space-and-time… not the answer.
And, everybody ignored Planck’s base units (and still do). I didn’t.

It took awhile, but in 2011, helping a nephew with his high school geometry classes, we went deeper and deeper inside the tetrahedron (dividing the edges by 2 and connecting the new vertices). In 45 steps, we were within the CERN scale of particle physics; and in 67 additional steps, we were within the Planck scale. There we decided to start with the Planck units, used base-2, and we were back within the classroom scale in 112 steps. We then went out to the age and size of the universe in another 90 steps. Just 202 notations outlined the universe. Although Kees Boeke used base-10 in 1957, it was just a scale of the universe and not the start of a working model. Very early in our discovery process we had an intuition that base-2, simple doublings, was a primary functional activity within physics. Yes, something as simple as cubic close-packing of equal spheres and sphere stacking just might be the start of a possible rapprochement to the old concepts of the æther.

I have spent too much time chasing this simple model of 202 steps. I need expert advice. What do you think? Does it have merit? Can it add anything to the discussion?

PS. Once I start to study a scholar’s work, I also start my own online notes of the references and resources around that person. Here is the page that I have started for you: If ever you’d like me to add or subtract from that page, I would be delighted to do so. -BEC

References within this website:
1. Musser G. (2017) Spacetime Is Doomed. In: Space, Time and the Limits of Human Understanding (Springer, 2017) edited by Shyam Wuppuluri and Giancarlo Ghirardi Foreword by John Stachel, and afterword by Noam Chomsky

Most recent communication: January 28, 2020, 7:37 PM

Dear George:

A couple of years ago we exchanged notes. Communicating our work has never been easy. Very early, back in 2012, John Baez told me that our model is idiosyncratic, but at that time we did not know how out of the mainstream it was. Yet, I continued to work to identify its weaknesses and strengths. Have you ever looked at the chart of raw numbers? https://81018.com/chart/

Our work unwittingly continues the work of David Bohm, one of my mentors from the 1970s.

The current homepage for the site is http://81018.com
I am still asking the questions: Is it meaningful? Is it worth pursuing?

Thanks.

Warmly,

Bruce

^{†} Musser G. (2017) Spacetime Is Doomed. In: Space, Time and the Limits of Human Understanding (Springer, 2017) edited by Shyam Wuppuluri and Giancarlo Ghirardi Foreword by John Stachel, and afterword by Noam Chomsky Particularly see the references.

Third email: Dec 12, 2018, 8:11 PM

Hi George –

I am still at it. You may remember an earlier email where, in a high school geometry class, we created a model of the universe by doubling the Planck base units, then doubling the results over and over again, until in 202 doublings (base-2 notation), we are at the size and age of the universe. The chart is here: https://81018.com/chart

The current homepage for the site is http://81018.com
I am still asking the questions: Is it meaningful? Is it worth pursing?

Your insights would be helpful. Thank you.

Warm regards,
Bruce

PS. I have aggregated some links to your work. Is there anything you would add or delete? Thanks. -BEC

Second communication: Feb 20, 2018, 5:24 PM

Thank you, George, for being on top of this note to you.

Of course, our work started out as a simple ordering tool.
We could set everything within a length that was a multiple
of the Planck Length. We were very surprised that it only took
202 doublings to get to the approximate size of the universe.
Reference: http//81018.com/home

We thought it was most peculiar that there were no references
to this simple mathematical progression on the web. Of course,
did find Kees Boeke’s base-10. One of my old professor
friends was Phil Morrison who with his wife Phylis created
the book, Powers of Ten based on Boeke’s work.
References: https://81018.com/big-board/ https://81018.com/why-now/

When we added the other Planck base units of time, mass and charge,
and observed the logic of that natural inflation without a big bang,
we wondered, “What is this simple math doing?” https://81018.com/chart/

When it came to us that this was e, Euler’s equation, and
it defined an exponential universe, we thought we had something quite
special. But, nobody else did. https://81018.com/exponential-universe/

We began to wonder about our commonsense and logic. Then, the
more we thought about our simple model, it appeared that every
notation was necessarily an active part of the current definition of
the universe and that time was some kind of an illusion or was always
entirely local so we went back to the other Lucasian professor,
Sir Isaac, and delved into his debate with Leibniz.
Reference: https://81018.com/pursuit/#2

Now, this is getting very weighty and most of the kids went off
to play football. Some of them hung in and are as curious as I am:
What’s wrong with our simple model besides “everything”?
Where does the logic break down or does it? Is it possible that
the universe is an exponential notation machine?
Reference: https://81018.com/finite-to-infinite/

Personally, I think we fell into something much larger than we are.
So, advice? We are all ears! Thanks.
-Bruce

First communication: 20 Feb 2018, @ 4:40pm

RE: Models of the universe

Dear George:

Our work originates from a high school geometry class where we followed the tetrahedron-octahedron, going within, by dividing the edges in half, deeper and deeper, 112 steps to the Planck scale. We then went out just 90 more steps by multiplying by 2, to get in the range of the Observable Universe. We thought it was a good STEM tool. On further consideration the first 67 notations to the CERN-scale began to intrigue us. Then, we added Planck Time to our Planck Length chart, then we added Planck Charge and Planck Mass. We began to think that we lived in an exponential universe and thought Euler would be pleased. Certainly the Hawking-Guth team would not be. Within the chart, there is a natural inflation that does not defy all logic. Then we began looking for alternatives to absolute space and time…

Will it ever stop? Please somebody, help us! We are drowning in issues that are way-way over our heads. But, our simple model? What do we do with it? Why is it wrong?

I am confident you can get us back on the straight and narrow! Maybe you can become an advisor!?!