CENTER FOR PERFECTION STUDIES: CONTINUITY•SYMMETRY•HARMONY • GOALS • December 2017 Homepages: Langlands I Langlands II|INFINITY|Inflation|KEYS|Original|REVIEW|Transformation
A never-ending blanket of the smallest possible spheres grappled-and-cinched by basic charge, light (space-time), mass, and special numbers.
by Bruce Camber First draft (revisions to come) Initiated: November 20, 2017 Updated: January 21, 2018
Austin, Texas: Since the very first discussions (December 19, 2011) about our chart of the 202 steps from the Planck scale to the Age of the Universe, the focus has been on the Planck Length / Planck Time and Planck Mass / Planck Charge. Each is multiplied by 2, and then each result multiplied by 2, over and over and over again, 202 times. There are over 1000 numbers that chart a quiet expansion and a map of our universe.
I have been unsure of the first ten notations (outlined on the left and partially displayed within the chart above). The first notation, the Planck Scale , is actually designated Notation #0 . To define Notation #1, I have turned to work of scholars from Plato to Langlands and beyond. This would be the first time both the historic and the current the work of scholars would be held up in the light of those 202 doublings.
I am rather sure that none of these scholars has researched and/or studied the doublings from the Planck scale to the Age of the Universe . Notwithstanding, I think this mathematical grid or matrix is the best possible place within which to test concepts and ideas no matter how idiosyncratic. It is a system, matrix or grid that encapsulates everything, everywhere, for all time. If it works within this scale (matrix, grid, or system), it deserves further scrutiny.
One of my early guesses was that the Langlands’ conjectures and programs will help to define these first groups of notations. Of course, Robert Langlands is a ground-breaking mathematician from the Institute for Advanced Studies in Princeton.
Given that my mathematics is limited, my progress has been slow. The obvious first challenge is to be able to understand Langlands’ work. The second challenge is to begin to grasp how to apply that work (and the work of a large cadre of Langlands programs scholars from around the world). In two prior posts  , I cracked opened that discussion. In the process of writing up these documents, I have been encouraged to visualize in rather new ways the potential and the initial thrust of the universe  just from within the generation of the numbers that define pi and Euler’s equation.  Of all non-ending, non-repeating numbers, pi and Euler’s equation give us an immediate visualization of a basic and perfect form.
So as I have said in several posts, I believe all those never-ending numbers by definition should be considered a bridge from the infinite to the finite and continuity, creating order and sequence, and symmetries, creating relations, define them.
|“Just A Second.”.Observations: Of the chart’s 202 notations the first moment of creation is Notation #0. But rather surprisingly, the very first second of the universe is a long way out there. It requires no less than 143 doublings. Remember the Wheat & Chessboard stories. That’s only 64 doublings. To get to that first second of this universe requires two complete doublings of the squares of the chessboards (128 doublings) plus 15 more! It falls between notation 143 and 144.  That is well over two-thirds of the way through the entire chart.
.That first second is a key. These first ten notations (a really real Big Ten) are on a scale that heretofore has had no meaning for scholars (or for me), and this Big Ten has never been studied per se by the academic community. I believe it is where we must start.
Observation: Note that it appears that time is being redefined. It looks less like an absolute duration (Newtonian time) and more like a processing time. Perhaps it is a bit more like exascale computing  (which would be painfully slow by comparison) than that which is measured and then fades into some whispery thing called the past.
.As a child we would play games and have to count to ten, 1001, 1002, 1003… each count approximating a second. Dub-Lub. Could we say that these first 144 notations are the initial pulse of the universe? Just that quickly, an exquisitely-deep complexity comes alive. Though we’ll attempt to make sense of each notation, notation by notation, it seems that all those notations, 1-145, or The First Three Seconds, should be taken as a whole. Of course, that title uses the work of Nobel physicist, Steven Weinberg, who wrote The First Three Minutes in 1977. I am sure Weinberg could not have imagined that there are over 150 doublings of the Planck units to get to his pivotal action!
.The first year, of course, is a light year, and it is between notations 168 and 169.  The universe is at about the size of our solar system. Perhaps the first year of one’s own life provides a good analogy. I speculate that we will all know what we have to know within those 169 notations. Thereafter, all the details that follow pick up within the current history of academic research and knowledge.
First we have the Planck scale with Planck Time (and light) built into all the equations. Then, the transcendental, irrational and incommensurable numbers have a thrust that is woven within the thrust of the Planck Charge, which is woven within the thrust within the four forces of nature — gravity, electromagnetism, the strong force, and the weak force.  Then, there are the dynamics of all the other non-ending, non-periodic numbers such as pi or Euler’s number. More of these were identified in a posting about numbers .
.Today’s visualization has notation #1 generating spheres since the first moment of time to this very day, hour, second, nanosecond, attosecond, yoctosecond, right to the infinitesimal interval known as Planck Time. I see this notational sheet actually layering on top of itself as imaged below. Start with Thomas Hales analysis of Kepler’s stacking of cannonballs  and it literally forces the emergence of Notation #2 and all subsequent notations. Please note: The processing time for each subsequent layer is 50% slower than prior layer. That could account for dynamics within close-packing of the spheres. Also, as the notations increase so do the circles and spheres from the earlier notation.
.What a wild-and-crazy image that is!
.To keep track of all the details, I have started a compilation for each notation,  Though it is very incomplete, it is a place to park ideas, comments, questions and insights, so I ask you to please be patient, however, your suggestions may prove especially powerful. Please know that you are always welcomed here.
|Notation #2Observations From Within the Mind’s Eye: The dynamic image of Kepler’s stacked cannonballs (referenced and linked just above and also below within the notes) shows the emergence of simple tetrahedrons from an alignment of the centerpoints of each sphere. This simple fact of geometry tell us much more than we can imagine. I will be coming back to this point often.
.At one time I was quite focused on the number of vertices. Then with Freeman Dyson’s suggestion , we began focusing on the scaling vertices. It begs the question, “Which constructions are the primary building blocks or the seed structures?”
.Because everything starts simply and Euclidean geometry appears to have the deepest intellectual foundation, I initially thought it would be a good place to start. I have a special affection for those five basic solids. Then I read that Euclidean geometry is a subset of projective geometry. That can’t be ignored. So let us start with projective geometry in this second notation and let it extend into every one of the 200 notations going forward.
.Now, that, too, is a special image that cannot be minimized.
.Within the next prime-number notation are the grounds for another dimension of mathematics and here I was once quite sure that Euclidean geometries would come next. Yet, at the Planck scale  there is so much going on, it is hard to guess what prioritizes the moment. We know the full complexity of all geometries is about to emerge. So, although we have studied a range of intellectual positions — Langlands, Barrows, Penrose, Rees, Wilczek (and so many others) — in this series of writings, I focus on Langlands. How would this group envision a logical construction path and its modalities?
Theories of Everything.These Langlands programs may actually benefit from the creative-and-adventuresome energies of those focused on defining a theory of everything. Often these people use simple geometries that have become complex. Here, for example, is an image of the Quasicrystalline Spin Network (QSN), what the Quantum Gravity Network  scientists consider to be the Planck scale substructure of spacetime. They may be right, however, my guess is that such an object could not manifest any earlier than the fourth notation where we’ve gone from spheres within the first doubling of the Planck scale, to projective geometry with the second doubling, and Euclidean geometry with third. The scaling vertices also become numerous — 8 for the first doubling (Notation #1), 64 for the second doubling (Notation #2), then 512 for the third doubling (Notation #3), 4096 for Notation #4, and 32,768 for Notation #5. To begin such complex constructions as the QSN, we’ll probably need most of the 4096 vertices and a good part of those 32,768 vertices within the fifth notation.
Where our first notation is primarily spheres orienting themselves to each other and the second notation is the beginning of projective geometry, I believe most of the third notation will be that tetrahedral generation pictured within that dynamic image of the stacked cannonballs. Within the tetrahedron is an octahedron. Within the octahedron are four hexagonal plates around the centerpoint, a half-sized octahedron in each of the six corners and a tetrahedron in each of the eight faces. I believe complexity begins here with this basic, basic structure. 
With all the other forces in play, there are many possibilities for bonding, cinching and grappling things together. So, now we propose a second layer of the fabric of the universe.
|Notation #3 A prime number, yet this notation is first a simple doubling of the second notation. We project Euclidean geometries begin and instantly become increasingly complex. As a subsystem of projective geometries, we now begin to attempt to figure out how these two systems work together and then how they both might work with Langlands programs..
Looking up any webpage that discusses Langlands and Euclidean and/or projective geometries opens a door, there are many sites and a lot of reading and thinking to do! It has becomes entirely clear that this will not be an easy assignment.
In a 2006 article by Edward Frenkel, he said, “We hope that in this way representation theory of affine Kac-Moody algebras may one day fulfill the dream of uncovering the mysteries of the geometric Langlands correspondence.”  (page 71) I suspect for all but a few among the professional mathematicians, this article will be a difficult read. For me, it is virtually impenetrable. But, I persevere, “What do affine Kac-Moody algebras look like? In what ways would they inform our understanding of this third notation.” I do not yet know the answer, but maybe you do. Please help!
What we have amounts to the third notation, with or without Langlands programs. This doubling has only 512 vertices to share so one might project that there is a thrust of perfection and the best of projective geometries are combining with the best of all the equations currently in position. We have numbers, geometries and equations all working within the same moment which almost instantly gives us the 4096 scaling vertices within the fourth notation. Let us logically attempt to create a range of possible interactions. All of them are possible, probable and doable at some point. Nothing here is static. Everything is pushing toward efficiency, elegance and beauty.
|Notation #4: Visualization.
The fourth notation is a simple doubling of the third notation. There may also be another kind of higher-order doubling of the second notation.
Projective and Euclidean geometries are striking out on their own yet there are only have 4096 vertices to share. With so many more possibilities for combinations of numbers, geometries and functions, the Langlands-Frenkel dream of a unity of mathematics is happening. What do we call it? Is it legitmately included within the Langlands programs at this point? It is difficult to know and surely we are anxious to hear from someone from among the Langlands experts to help us integrate what we have here with their work that started back in 1967.
The fifth notation, another prime, will open yet another possibility for a mathematical system to emerge. There will be 32,768 vertices with which to develop an integrated mathematical system. Within the sixth notation there will 262,144 vertics and within the seventh — it’s a prime — a new system could emerge and there will be 2,097,152 vertices with which to work.
- First principles and even more struggles with first principles
- Analysis and speculations: This is our third posting on a journey to learn from Edward Frenkel and his mentor, colleague and friend, Robert Langlands.  So, I will be looking through their writings for their statements about boundary conditions, spheres, pi, “never-ending and never-repeating numbers” to continue to develop this report on the fabric of the universe and notation 1, 2, 3, and 4. – BEC
Every notation may well be a layer of the fabric of the universe.
.It seems that the ever-changing, never-repeating, exquisitely-fine layer of notation 1 lifts up a projective layering where particular geometries begin to define notation 2, and it begins to push out both as a projective geometry and a derivative Euclidean geometry within Notation 3. Then, in much the same way Thomas Hales envisioned Kepler’s cannonballs, these additional basic structures begin to emerge. Yet with only 4096 vertices, there are limits of possibility.
The many factors for this initial push of inflation, especially coupled with Planck charge, makes for a constantly expanding universe. Yet, even as they emerge, each has the potential to be uniquely defined with one or more of the many inherent mathematical definitions within this nexus of transformation.  There is more to come and updates for what is already here!
.For access to the few of the pages just preceding this page:
- November 18: Defining Forms from Plato to Langlands
- November 15: Before we can understand the complex…We need to understand the simple things. An introduction to our study of the Langlands programs.
- November 12: Seven reasons to look more deeply at our chart (at the top). It is still a largely-unexplored model of the Universe
- November 9: Over 1000 Simple Calculations Chart A Highly-Integrated Universe
- November 8: Do we live in an exponential universe?
- Click on the number in the brackets to go back up into the article.
- Click on the description to go to a new page (tab) where this topic is further discussed.
 Within the academic community, the Planck scale has been called the Planck Epoch. It seems to be a misnomer. Epochs are usually set between less significant events. Within this model of the universe, it appears to be more like a process with a beginning but no apparent end. On one side is infinity and on the other side is the finite. The Planck processes appear to be a modulus or nexus of transformation between the two. All three together give us a working model of the universe.
 Our goal within this project is to have a detailed description of each notation. That will take a few years just to get it into a draft mode, and then several more years to tighten it all up. This is one of our very first links to an index of all notations and our working description of Notation 0, a nexus of transformation from the infinite to the finite.
 A chart that originates from the Planck units. This chart uses base-2 notation, and goes to the Age of the Universe, seems quite obvious. Yet, the obvious is not always quite obvious. Logic might tell us to start at one boundary, the smallest, and go to the other boundary, the largest, and use some simple type of connective tissue between them (base-2), but knowing the smallest and largest is not obvious. More...
 The first article in this series of four that introduce Langlands Programs: There are no pretensions about truly understanding the Langlands programs, but those of us outside that circle must start somewhere! To attempt to expedite this learning curve, I have commuinicated directly with Langlands and Frenkel.
 The thrust of the universe: Since the work of Alan Guth to define inflation in light of the big bang theory, thrust has taken a backseat to this rather novel understanding of inflation. Quite intentionally, this work will use thrust as a broader more inclusive category than big bang inflation. More…
 The study of pi and Euler’s equation: The academic studies of our scholars are voluminous, however, it seems that our understanding of pi is still rather basic. I have not yet seen any articles about the relation between infinity, pi and Euler’s number. If you know of any, please let me know! Thanks.
 One second. Heart beats range from 40 beats per minute to well over 100 beats per minute, but 60-to-100 is quite normal. There is something very basic about a second. There is no way to anticipate all the many ways of looking at the nature of a second. This opens our study.
 A Light Year: The distance light travels in about a year is about the size of our solar system. Light ideally travrels about 299,792,458 metres per second or 186,000 miles per second. So, that distance of a second is well within the orbits of our moon are 356,500 km (221,500 mi) at the perigee to 406,700 km (252,700 mi) at its apogee.
 The breakdown of big bang theory. This rather long article was first posted in June 2016. Although occasionally updated, this particular paragraph raises so many questions, it was here that the complexities of the Langlands programs seemed to embrace the many formulas, combinations, and possibilities suggested by this paragraph.
 Number & Numbers. If you were to google the words, “What generates number?” with the quotes, the first two references includes this page and an earlier report. It is a question that could also be asked, “How are numbers generated?” and there is only one reference of the first ten truly attempting to address the question. Add the word, “Langlands” and there are no references. It would appear that for Langlands programs that there is no particular interest in either question qua question. If so — and the issues about the very nature of number will continue to be explored — it speaks to the rather simple-naive approach taken here and the sophistication of the Langlands enterprise and family..
 Cubic closed-packing. This site uses a rather dynamic image from Wikipedia of cubic-closed-packing (ccp) helped me to begin to visualize the conversion of spheres to lines, triangles, tetrahedrons, and octahedrons. These images, however, come from current scholarship within university programs of structure in and around the 80th notation. This image is a key part of the study of atomic packing in crystals. Along with face-centered cubic packing (fcp) and Hexagonal-Closed Packing (hcp), these studies give us an academically-accepted foundation upon which to build-down to an even more fundamental seed structure. It seems to me that as the notations approach the first notation, though more primitive are possibly more perfect. My guess is that the thrust of continuity, symmetry and harmony from the infinite to the finite would hold this structure at a higher-order of perfection.
 Freeman Dyson is a scholars’ scholar. He has a sweet humility, but a strong presence. He doesn’t suffer fools gladly, but he is also very conservative about using labels. Some of the oldest and a few of the youngest scholars are open to concepts and ideas that fall outside the normal parameters of their work. Most scholars are too nervous to engage; they’re concerned that their name could be used to prop-up an idiosyncratic theory. Little do they know that by even making a reference to the big bang theory, that is exactly what they are doing. Those scholars that know the big bang is flawed are also reluctant to engage. Yet, until there is an alternative to the big bang theory, scholarship will continue swirl around in less than meaningful ways.
 The Planck Epoch is hardly an epoch. There is a fair amount of discussion about the Planck Epoch on the web. There is no discussion outside of our web domains about the Planck processes. If this model of the universe is any approximation of the real reality, then a major shift for us all will be to understand space and time in a very different light. Here space and time are the direct expression of light; and although part of that continuum is visible light, this light is pervasive, informative, and energizing.
 Quasicrystalline Spin Network (QSN), Quantum Gravity Network. We are all so sure of our positions however self-defining, incomplete and/or circular those positions may be. All of us are at one time or another a bit guilty of being overly enamored with our babies. Yet, within so much of this work is a kernel of truth and this reference is included to remind me to always be open to others even if their work seems self-defining, incomplete or circular. -BEC
 All the work of this website began when in 2011 many high school geometry students began enthusiastically creating these basic structures. To have discovered what could be an infinite regression of simple Platonic objects (tetrahedrons and octahedrons) raised the poignant question, “How afar within can we go? Is there a stopping point?” The answer was, “Yes” and the Planck Length was it!
 When I open any ArXiv document like this one from Frenkel, I always have two or three other tabs open just to look up words and concepts to see if I can begin to grasp the general essence of what is being said. Their work starts with analytic algebras. It is a mathematical language that must be learned from one’s undergraduate days and then engaged in depth throughout graduate school. Notwithstanding, most of us should at least understand the basic concepts. We will dive in more deeply in the next posting where we will also take on the next five or six notations. -BEC