Pi, Circles, Primordial Spheres at a Nexus of Transformations

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The First Instant of the Universe

by Bruce E. Camber Related: WorldThe ThreeHistoryExpansionConsciousness

Background:1 In 1899 Max Planck began calculating what are now known as the Planck base units. In 2001 Frank Wilczek (MIT) wrote three related articles for Physics Today, all titled, Scaling Mt. Planck (I, II, III). Those articles opened a wider discussion about these units; and today, Planck Length and Planck Time are generally accepted as the smallest possible units of space and time. In 2011 our high school geometry class applied base-2 notation (doublings) to the Planck units. Our chart goes from the two smallest to the largest possible units of space and time (the most current approximate size-and-age of the universe). That scale has just 202 notations. Rather quickly it became apparent that this scale compared favorably with the “epoch scale” suggested by big bang cosmology. In our 2011 model there is a heretofore unexplored infinitesimal scale of at least 64 notations, so small that these are only accessed through math and logic. That scale opens many more foundational questions about the very nature of space and time.

Purpose: That infinitesimal scale represents a new parameterization of space, time, mass and charge; it is readily accessible through a very simple logic, base-2 notation, the Planck base units, and dimensionless constants. The infinitely-hot start of big bang cosmology is not necessary. Redefinitions of space, time, and infinity are queued. No singularities, this nexus of transformations is a keyhole to make direct correlations with the Langlands programs and string theory.

Methods: Functional optimization, logic and simplicity are the operational requirements, yet always within the framework of the 202 base-2 notations. Particularly, the primary focus of this article is on several natural groups of notations. The first twenty-five notations come before big bang hypotheses are possible. The first 64 notations of length and time are too small for our measuring devices. The first 143 notations out of the 202 capture the first second of the life of this universe. Here we have a very different experimental environment to test new concepts.

Key working results to date:
(1) Of the seventeen big bang epochs, this base-2 scale encapsulates fourteen of its major epochs and marginalizes the first three.
(2) Base-2 exponentiation becomes the foundation for a natural inflation which then becomes the foundations for homogeneity and isotropy.
(3) Dark matter and dark energy are defined by primordial spheres, the first emerging with Notation-1 and then 1044 spheres per second literally flowing through and creating all 201 dynamic and fully-symmetric notations. Currently those same spheres are creating Notation-202 at the same rate, 1044 spheres/second. Isn’t that 539 tredecillion planckspheres per second? For more, go here.
(4) The spheres constitute what was once considered the aether, a working concept used to interconnect all things, everywhere, for all time. As this model is in some way validated or re-textured, many more results will be considered.

Conclusion: These 202 notations, particularly the first 64, open a new model of the universe that is worthy of our time to explore.


Calculations and Navigation: There are known problems with our calculations. We are constantly working on them and welcome your insights. Also, any link that pointed to pages outside of this website have been moved into our Footnotes-Endnotes section. Links pointing within the website open a new tab or window.

The Story: In 2011 we outlined a model of the universe with just 202 base-2 notations starting at the Planck Scale and evolving to the current time and age of the the universe. It became a STEM tool, then a science fair project. Though quite idiosyncratic and highly-speculative, it had enough simple logic to continue to pursue it. In 2016, we developed a horizontally-scrolled chart to analyze the flow of numbers more carefully, both their relations within any one notation and along the entire expansion from Notation-1-to-202. Special attention is given to the first 64 notations. Given that these had never been part of any model, each notation will continue to be analyzed with appeals for the help from among the leading, living scholars around the world.


A Different Orientation. Since 2011 the 202 base-2 notations have had a few names. The first was Big Board-little universe. For the purpose of this article this theoretical framework will be referred to as the 202 base-2 notations. It is a mathematically-integrated view of the Universe. Because all notations are always active, it requires much more thought about the derivative nature of space and time and the role of the finite-infinite relation.

Not an infinitely-hot beginning. Hawking’s big bang has only been partially successful. Partial is partial. Several key scholars say, “It’s wrong.” Among them is one of Hawking’s many co-authors, Neil Turok. We agree with Turok that the universe is perpetually starting. In our emerging model, Notation-1 is like all notations and it is always active and it resides deep within everything, everywhere, for all time. The 202 base-2 notations all start cold and have a natural inflation. Yet, the numbers actually compare well with most of the epochs of the Hawking’s infinitely-hot start. Ultimately this model absorbs the big bang. Yet, the question still remains, “How does that first notation of these 202 notations come to be?”

The first instant is the first sphere and the first moment of space and time. Dimensionless constants define the Planck base units. Among them, the oldest, best-known, and most-used concept is pi with its circles and spheres. So, let us examine pi, circles and the sphere in light of the 202 base-2 notations. This theoretical framework, subject of many preliminary analyses, has much more to reveal.


The first sphere is primordial, constant, ever-changing, always the same. Pi, a dimensionless constant, doesn’t end and doesn’t repeat. It doesn’t meet the definition of finite. If it is not finite, what is it? Can we take as a given, that it is the beginning of a definition of the infinite?

If we take it as a given that these words begin to define the infinite, how might those words-numbers-results be characterized? It has a random orderliness that is constantly defining a circle or a cycle that never ends, constantly changes, but it is always the same. Perhaps that is the best possible definition of continuity and order.

In earlier articles we concluded one plancksphere is rendered every plancksecond. That results in approximately 539 tredecillion planckspheres being generated every second, a baseline expansion of the universe (Planck Time: 5.39116(13)×10−44 seconds).

Sphere stacking is taken as a given; each sphere is creating its own space-time-density. Within each notation spheres stack uniquely. Looking at the numbers, perhaps that density is best examined around the one second mark. Notation-144 is 1.2023 seconds. The mass is about 4.8537×1034 kilograms. The anticipated diameter is 299,792.458 kilometers. That mass is well beyond the mass of the sun at 1.989×1030 kilograms (with its much larger diameter of about 865,690 kilometers).

Assuredly, we are asking questions about blackholes and we are reviewing the mathematics around the concept going right back to Einstein. Is there a better metaphor? Is it analogous with, and possibly profoundly related to, dark energy and dark matter? We are beginning to think so.


Perspectives.2 In 1611 Johannes Kepler wrote ‘On the six-cornered snowflake‘ and part of it included reflections about stacking cannonballs on the deck of a ship. As recently as 2002, Thomas Hales proved Kepler’s Conjecture to be true. That sphere-stacking legacy opened the path to close packing of equal spheres. And, thanks to Cornell mathematician, Steven Strogatz, we are beginning to understand how Fourier’s Transform is also involved. In 1822 a French mathematician and physicist, Joseph Fourier, introduced functions that evolved from work with sine-cosine-tangent and the nature of periodicity. Here the interior of all spheres are interconnected. We speculate that such functions — harmonic functions — are profoundly inherent in everything-everywhere-for-all-time. A circle is stretched out as a cycle and these cycles readily binding all 202 notations from the Planck scale. Yes, highly speculative, we probably should affirm that these spheres are not empty but are defined by the Planck base units and an unknown number of dimensionless constants; and the smaller the spheres get, the more intense that activity appears. So, it is perhaps a bit ironic that our study of spheres continues to build upon Kepler’s work with cannonballs, and then within the atomic scale atomic-packing factors (APF), and now within the Planck base units.

At the Planck scale, long before big bang cosmology engages, our model has already created twenty-five highly-integrated and calibrated notations. In this emerging model, inflation begins immediately. Gravity and electromagnetism do as well. Within the sphere, it appears that there are internal dynamics that pull within and there are external dynamics that push out. My simple logic tells me that electromagnetism pushes out and gravity pulls in. Perhaps at the Planck scale, the Fourier dynamics may logically reveal that electromagnetism and gravity are the Janus face of each other. It is certainly not a new thought. Even more speculatively, the density numbers in the earliest part of each notations suggest either analogies to blackholes or perhaps the inverse definition of one (Notations 1-67). Being rather whimsically speculative, perhaps there are just the 135 (Notations 67-to-202) for the creation and 135 (Notations 202-to-67) for recycling… And yes, yes indeed, to be sure, that is whimsical speculation.

There are major discussions within the current dialogues of the physics community that focus on the open issues; however, until these discussions begin within the Planck scale within the 202- base-2 notations, it seems that those discussions will not be simple or basic enough.

Technicalities and today’s challenges. Although Max Planck was calculating infinitesimal space and time in 1899, it was too small to explore. The National Institute for Standards & Technology (NIST)3 of the USA works with most standards organizations. None have a formal name for units as small as 10−27, 10−30, 10−33, 10−36, 10−39, and 10−42 seconds. Within our model those numbers have purpose and meaning. The yoctosecond at 10−24 seconds is the smallest number with a name. As a result, we have formally requested CODATA,3 which is an agency of NIST, BIPM, and other international standards organizations, to name at least five, possibly six, groups of these pivotal numbers. In the grand scheme of things, each may be infinitesimal, but here each is a key foundation of the universe.

Grasping Density. Our four prior articles were an exercise to begin to become more familiar with basic questions about this scheme, then with basic concepts that define it, and then with our most-simple formula, one Plancksphere per Plancksecond, as an expansion rate of the universe. This article and the prior four articles will be updated as this infinitesimal scale is further opened up by today’s scholars as they begin to see relations well beyond our scope and capacities.

Planck Time is calculated to be 5.39116(13)×10−44 seconds. If “one plancksphere per plancksecond” is correct, and there are five-hundred, thirty-nine tredecillion planckspheres per second, it is a relatively easy calculation to determine the number of planckspheres for each notation.

Back in 2012 Freeman Dyson suggested to us that dimensional analysis should be applied to this expansion. Now, reflecting on his comments in light of this article, I would say to him, “That gives too much primacy to space and time. These spheres are creating space and time so it seems to me that the numbers of units would be a base-2 expansion, not base-8.”

For more, see the base-2 expansion (line 8) and the base-8 expansion on line 9 within the horizontally-scrolled chart. And because this is all so speculative, your insights and comments will always be most welcomed.

Base-2 expansion and the sphere count. By Notation-6, there would be only 16 planckspheres. To begin to visualize the dynamics, let us return to the five images (below) that are often used within this website. You’ll notice that in the first panel (left-to-right), the image displays eighteen spheres. By Notation-7 there would be 32 spheres. By Notation-12 there will be 1024 spheres. By Notation-25 there are 8,388,608 spheres. By Notation-64 there are 1.8446744×1019 spheres; and by Notation-144, there would be 5.57518×1042 spheres. There are many open questions. Can we reconcile the sphere count at 5.57518×1042 with the rate count of 5.39116(13)×10−44 seconds?

Sphere Dynamics: There are external dynamics between spheres as evidenced within the stacking within the first box below on the left. We are also examining projected internal dynamics that are generated through the many faces of the Fourier Transform. The next five graphical images (below) represent cyclical data that have been labeled Fourier-A to Fourier-E. Although the Fourier Transform is scale independent, it appears that little work has been done with it at the Planck scale. Understanding the possible applications of all five representations of cyclical functions is part of our work on this very special model of the universe. We are seeking experts who have deep working knowledge of the ways that Fourier-A to Fourier-E may open new pathways through each notation, especially those first 64 notations which hold possibilities of unimagined diversity and complexity.

Sphere to tetrahedron-octahedron couplet
Sphere Stacking-Packing
File:Circle cos sin.gif
Lagrangian point


Throughout our ever-so-limited human history, there have been diverse speculations about the nature of infinity. However well-intentioned and earnest these have been, in this work, such speculations are set aside for a more mathematical approach.

Pi, circles and spheres continue to inform us. This work began back in 1972 when a simple perfection of the infinite was characterized as “continuity rendering order, symmetry rending relations, and harmony rendering dynamics.” Today, it is recognized that those simple perfections are all manifest within pi, circles and spheres. Although each is well understood, very limited work has been done to date at the Planck scale and then into a gateway or nexus of transformation between the finite and infinite. As we have seen from the chart of base-2 notations, these infinitesimal notations are so dense, there is no room for fluctuations. Those three simple perfections — continuity, symmetry, harmony — will keep things in line. Even within the first second of the universe, right on up to Notation-144, I can well imagine that everything is still “in line.” That’s just a guess.

From perfection to imperfection. When and where that first gap manifests is an open question. There is a “thrust for perfection” that conceivably could go on for a year (Notation-169), perhaps 1000 years (Notation-179). We don’t know, however, at some point five tetrahedrons will share a common edge, a simple fluctuation begins, systems engage and fluctuations become part of our universe. Our guess, perhaps just whimsical speculation, is that the first manifestation of some kind of fluctuation is where the homeostasis with the internal-and-external activities within the sphere become out-of-balance. Once established as a system within space-time, conceivably it moves back down into the earlier notations until the thrust for perfection doesn’t allow room for a fluctuation. At this point in our thinking, it seems that the domain of perfection would go from Notation-1 to about Notation-50 where we hypothesize the beginning of Systems and the archetypes for the Mind. Here we finally begin to see an infinite diversity of possibility within those five dynamics pictured above. Here, probabilities can be exploring the easiest possibilities.

With every new second within each notation, new layers of inter-connectivities emerge. Instead of just continuity, symmetry and harmony, the formulas are extended:
1. The infinite is continuity and it begets order, begets numbers which begets time.
2. The infinite is symmetry and it begets relations which beget geometries which beget space.
3. The infinite is harmony and it begets dynamics begets Systems begets space-time.

The Dynamics of the Infinitesimal:4 According to some scholars, some types of measurements can be inferred right down within Notation-50 in the range of 1.819679×10-20 meters. To date, we find no quark-related sizes that are smaller. Typically quarks are no smaller than 10-18 meters which ranges from Notation-56 to Notation 59. To find language to describe possible phenomena, we go back to Plato’s Forms, Aristotle’s Ousia, Langlands’ Automorphic forms, Witten’s strings, and so many other basic-basic structures that have no connection to the Standard Model for Particle Physics or the Lambda Cold Dark Matter model.

Scale Invariance and the Fourier Transform.5 Every image of a circle and a sphere needs to be considered and reconsidered for applications within the Planck scale. Because this model is logically scale invariant, all such images should apply within the finite-infinite transformation and within the Notations 1-to 201. Because Notation-202 is still in-process, directional and asymmetric, scaling within Notation-202 could be limited and/or adaptive. We hypothesize the primordial sphere as in Lemaître, opens sphere stacking and sphere packing. We uncovered that dynamic image online in 2014. It gave us a visualization of the transformation from spheres to interior functions, straight lines, and finally the Euclidean-Platonic geometries. So we are guessing. The functions of Fourier A to Fourier D may actually be fairly good representations of waves and standing wave patterns. Fourier E is even more speculative. It comes from celestial mechanics; and to the best of my knowledge, it has never been suggested to represent any activity on the human-scale, small-scale or infinitesimal-scale of our universe. The Lagrange point applied to the Planck scale just might open a pathway to automorphic forms and strings and it has some semblance to the visualization of the standard model of particle physics suggest by Chris Quigg in the October 22, 2020 issue of Quanta Magazine.5

There is so much more to discern. There are so many open issues… Foremost among them is the promise of even more uniqueness and diversity introduced by prime numbers. Also, this model starts with base-2. Others have argued that base-3, base-4, base-5 would do equally well. Now, I believe that is too quick a judgment. The prime number bases provide unique access to unique starting points; and within this model, all such starting points are held together within the base-2 structure. Mix into these considerations, all the other most-used dimensionless constants, this universe has a palette of possibilities far greater than the panoply of colors, sounds, smells, textures, and odors that we currently identify. Thank you.


Footnotes and Endnotes

Some of our concepts and language need clarification, especially if you are a first-time visitor. Any link that currently points to a page outside of this website is initially redirected here before going outside of this site. All links pointing to pages inside the website will open in a new tab or window. Also, these footnotes-and-endnotes are still being developed.

[1] Background. This introduction within this box (with its light-yellow background) is the most recent attempt to summarize work done since 2011. This box and its key links remain here especially for our first-time visitors.

[2] Perspectives. Old concepts have new meanings when looking at the universe as a highly-integrated, scale-invariant whole. As we return to 1611 to the work of Johannes Kepler, we begin to realize how very young our sciences are. Here Kepler is just beginning to formalize the concept of sphere stacking with the advent of the cannons and cannonballs just 150 years earlier. The irony of it being included within an article entitled, ‘On the six-cornered snowflake,’ should not be lost. As well, not until 2002 did Thomas Hales prove Kepler’s Conjecture to be true. We are all such simple and arrogant people. Thus, our next venue within these studies must be to be to understand the human mind and its rather natural capacity for deceit and self-importance.

[3] CODATA, An Agency of NIST, BIPM, and other SI-related Organizations: Because names are so important to shape the personality of a word or domain, we have appealed to CODATA to officially intervene. This body assigns official names to measurements. Even Wikipedia recognizes the importance of these agencies.

[4] The Dynamics of the Infinitesimal: The entire universe appears to share the first 64 notations. Individuation possibly begins to appear with Notation-50 to about Notation-54. Plato’s Forms, Aristotle’s Ousia, Langlands’ Automorphic forms, Witten’s strings. Other basic-basic structures that have no connection to the Standard Model for Particle Physics or the Lambda Cold Dark Matter model.

[5] Scale Invariance and the Fourier Transform. It all starts with a sphere and then sphere stacking which opens to sphere packing. Uncovered online in 2014, this image gave us the visualization of the transformation from spheres to interior functions, straight lines, and finally the Euclidean-Platonic geometries. All the descriptions and links go directly to Wikipedia.
Fourier A. “This wave pattern occurs often in nature, including wind waves, sound waves, and light waves. It is often said that the cosine function leads the sine function or the sine lags the cosine.”
Fourier B. “The sine is a trigonometric function of an angle. The sine of an acute angle is defined in the context of a right triangle: for the specified angle, it is the ratio of the length of the side that is opposite that angle, to the length of the longest side of the triangle (the hypotenuse).”
Fourier C. “A circularly polarized wave as a sum of two linearly polarized components 90° out of phase.” Also, see the Poincaré sphere.
Fourier D. “Animation showing four different polarization states and two orthogonal projections.” Also, see polarization states and circular polarization.
Fourier E. Most speculatively, because the Planck numbers appear to be scale invariant, let us see if that logic applies if we take a basic definition within celestial mechanics, the Lagrange point, and try to apply it to the Planck scale. Within 24-hours of speculatively hypothesizing that the Lagrange points describe activities within the infinitesimals of the Planck scale, a visualization came to my attention that has a semblance of it. An article by Natalie Wolchover in Quanta Magazine (October 22, 2020) introduces a visualization based on the work of FermiLab’s emeritus theoretical physicist, Chris Quigg. The Quanta team took it further. Their visualization helps enliven Quigg’s work and opens metaphorical and analogous thinking.


References and Resources

While an article is being written, there will be many references opened, mostly to ArXiv and Physical Review articles. Some are included here. Most references are reminders to me to return to the work of these scholars and to think more deeply about their constructs of reality.

  1. China: Some Chinese scholars are very important consensus builders. For every 100 views of this website, close to 10% are from China and another 10% are from the United Kingdom. Typically within a given day, there will be visitors from as many as 15 different countries. To facilitate explorations by our Chinese friends, Baidu automatically translates the entire site. Here is a link to our current homepage in Mandarin: 这是当前主页 (translation can be slow).
  2. Tensor and pairing interactions within the QMC energy density functional, Martinez, Thomas, Guichon, Stone (April 2020)
  3. A Higher Dimensional Cosmological Model for the Search of Dark Energy Source, PS Singh, KP Singh, November 2020
  4. Quark-meson coupling model. What if the Mass-Length-Time [M0 L0 T1] of the quark is just too gross or large? You know that quarks are not directly measured per se but confirmed only by theoretical predictions. Best guesses puts them in the range of 1.81×−20 meters to 7.27×10-20 meters which puts them in our range of Notation-50 to Notation-52. Within this model the current speculation is that these notations are the edge of informational fluctuations, cellular automaton, and the beginning of Systems.
  5. Assume big bang cosmology is wrong, so our current worldviews are necessarily wobbly, incomplete, and too small. We need an integrated view of our universe that starts right at the very first instant and comes to this moment, the now, the current time and current expansion of our universe. Mathematically, in its most simple form, it requires 202 base-2 notations from the first instant to the current expansion today.



Most emails are to scholars who know something about the issues involved. Their problem is like our problem; they, too, get set in their ways. They have their belief systems and they tend to stick to them. As politely as possible, these emails invite their help to look at our data in new ways.

Dec 2, 2020, 11:17 PM Ewan Palmer, a London-based reporter for @Newsweek.
Dear Ewan, It’s been awhile since you’ve been on Twitter. I just sent you this highlighted paragraph below. My current work is here: https://81018.com/instant/ Best wishes, Bruce
@EwanPalmer@joelockhart We are all so flawed. The primary flaw is our limited worldviews. On both sides of every equation! If we migrate to an integrated, mathematical view of the universe — https://81018.com/chart/ — it opens a new perspective and responsibility for integrity. See: https://81018.com/ethics/ Editor’s note: Joe Lockhart is a CNN political analyst.

Saturday, December 5, 1:02 PM, About Densities directed to Edward Kolb, Dean, professor, Kavli Institute for Cosmological Physics, Chicago, Illinois Key Question: “How about return to Lemaître’s primordial sphere? If such a sphere manifests, defined by the Planck base units, there would be about 539 tredecillion spheres per second. Too idiosyncratic to pursue?



Link to tweets that challenge people to stretch and to attempt to see the universe and not just the world that is in front of them.

9:51 AM · Dec 16, 2020 @justinbieber Be careful, Justin. Our little worldviews get in the way of having an integrated view of the entire universe – everything, everywhere, for all time. It’d inspire new lyrics and a gracious, loving heart. A start: http://81018.com I’d be glad to help explain.

12:57 PM · Dec 4, 2020 @erikvance Deep within is an intuitional network which sees the discontinuities such as a son with only his socks. You knew; you just thought it was too special to interrupt that flow. And it is! …especially in Mexico City… but not in Baltimore! https://81018.com/instant/ is a start (to define that intuitional network). Note: Erik is a staff editor at @NYTParenting and author of Suggestible You.

10:25 PM · Dec 3, 2020 @NRoberts_atHome The problem with religions, Christianity included, they are so intimidated by the sciences, they really haven’t had the guts to grasp infinity anew to be able to explain it to the intellectual, scholars and academics. For more: https://81018.com/infinity/

12:37 PM · Dec 4, 2020 @alwaleed_philan @Alwaleed_Talal http://www.kingdom.com.sa Until we’re freed of Newton’s absolute space and time, we have a handicap; our understanding of the universal and infinity is tied down. We need an integrated view of the universe: http://81018.com is a start!

4:39 PM · Nov 29, 2020· @loru22o and @NassimHaramein …our first 64 notations give us all the breathing room we need: https://81018.com/chart/ There’s much more here: https://81018.com -Bruce

10:38 AM · Nov 29, 2020 @atrupar@voxdotcom What if Newton was wrong in 1687 (when he published Principia)? What if space and time are derivative, not absolute? What if our worldviews are too small? What if an integrated-view of the universe is needed to right the wrongs? https://81018.com/chart is a start.



Afterthoughts. Within the perspectives of this website, time’s arrow is entirely limited to Notation-202. All other 201 notations are symmetric, space and time, like mass and energy, are derivative, yet profoundly related to each other and light. It is a radically idiosyncratic view of our universe that emphasizes at least 25 ways to engage the details of the universe.


Key Dates for this file, instant

Stacked magnetic spheres (four, four, then one), Kandinsky spheres (background), is a dynamic.

The sphere prior to quantum fluctuations creates a domain of perfection within our physical universe.

Names of Very-Large and Infinitesimal Numbers

Largest-named to smallest-named
  —Numbers-names from large to small —
quadrillionth0.000 000 000 000 001femtof
quintillionth0.000 000 000 000 000 001attoa
sextillionth0.000 000 000 000 000 000 001zeptoz
septillionth0.000 000 000 000 000 000 000 001yoctoy

NamePower of 10Decimal
Quadrillion1015etc …
Vigintillion106363 zeros

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