Structure of spacetime at the Planck Scale: An infinitesimal sphere

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The First Particle Is Not A Particle Per Se.
It’s pure geometry and mathematics.
by Bruce E. Camber

Abstract (as a series of hypotheses from a high school thought experiment)
The first particle has all the structural and dynamic elements of a most simple sphere. It is defined by the four Planck base units of time, length, mass and charge. It is an infinitesimal, archetypal, primordial sphere that defines the first moment of space-time. The Planck scale is also defined by dimensionless constants so here we propose several mechanisms to begin to bridge the Planck scale with the electroweak scale. Our model using base-2 and the Planck base units originated in 2011 and it begs for more analysis. For example, if Planck Time also defines a rate of expansion by taking as a given that there is one “Planck particle” per unit of Planck Time and Planck Length, it’s a different model of our universe. Seemingly logical, please suspend your harshest judgments in order to explore whatever mechanisms we can imagine in light of the Standard Model for Cosmology (ΛCDM or Lambda cold dark matter) and the Standard Model for Particle Physics with all their successes and problems. [*].BEC

Key words: Structure of spacetime at the Planck Scale, Planck particle, physics at the Planck scale, shell particle, plancksphere, infinitesimal sphere, archetypal sphere, sphere dynamics

There are many concepts about the nature of the first particle. This article postulates that it is defined by the most basic structural and dynamic values of a most-simple sphere and that is further defined by the four Planck base units of time, length, mass, and charge. Of course, it is many orders of magnitude smaller than a fermion (neutrino) and is too small to show up on any measuring device. It would be considered categorically as we do dark matter and dark energy.

Naming: Planck Particle or Planck Sphere or infinitesimal sphere.
Some will want to call it a Planck Particle. That term has been used by others where there seems to be a consensus that this hypothetical particle is a tiny black hole whose Compton wavelength is equal to its Schwarzschild radius.[1]

For our current considerations, the 1916 calculations by Karl Schwarzschild for Einstein’s field equations “…for the gravitational field outside a non-rotating, spherically symmetric body with mass” is placed on a bit of a hold to determine how it fits in with our emerging model. Although quickly celebrated, Schwarzschild’s original work within our postulated model evokes Alfred Whitehead’s (Process and Reality, 1929) concept, a fallacy of misplaced concreteness. The concept of “a tiny black hole” is too quickly used as a primary definition upon which to draw too many conclusions about the role and place of its exacting numbers. It should also be noted that Compton’s wavelength is also placed on hold and John Wheeler’s concept of a blackhole did not emerge until 1952, and effective use, not until the 1960s.

All these definitions have a place and role. Compton’s wavelengths should play a more exacting role as the studies of each hypothetical particle becomes a key piece of this larger puzzle. Where the current physics community sees a black hole, we see a space-time particle with a vast array of variables. Within their linear view of space-time, there is no conceptual space within which to breathe, imagine and postulate anything up to the electroweak scale. Within our base-2 exponential view, there are no less than 64 doublings to consider.

Any postulation about the first instant, the first sphere, and the first moment of space-time ought to be part of the most-complete picture. Yet, to my scant knowledge, most of these postulations have not been defined within the first 64 of 202 base-2 doublings of those four Planck units. That grid of 202 doublings starts at the “first particle” and first moment of space-time and goes to the current expansion of the universe and current time right now. Our chart follows the numbers.

The term, Planck Sphere, has been used by several scholars; the work of Victor J. Stenger [2] stands out. In his book, The Comprehensible Cosmos: From where do the laws of physics from?, he says, “The energy density within that sphere, for this brief moment, will be on the order of (1038)/(10-33)3 = 10121 electron volts/cm3 which we call the Planck density. This density would be equivalent of a cosmological constant.” And later he says, “…a Planck sphere is akin to a black hole whose entropy is maximal for an object of the same radius.” It appears that the general consensus of the academic community is with Stenger, a Planck sphere is akin to a black hole. [3]

Thus, naming the first particle is not straightforward. For now, the simple term, the first particle, will be used along with “infinitesimal sphere.” The hadron was the last confirmed particle in 2012. Again, this first particle is highly speculative and many orders of magnitude smaller than any hadron. Nevertheless, someday this first particle may be considered for the Wikipedia topic, Timeline of Particle discoveries? [4]

Parameters as mechanisms
Our definition of a sphere starts with continuity-symmetry-harmony defined within pi.). The very nature of those dynamics establishes the basis for homogeneity and isotopy. Also, stated in prior articles, there is a natural inflation. Quantum mechanics and its indeterminacy only begin to emerge within geometries of indeterminacy. [5] Although not a topic that attracts much attention, the emergence of very specific geometries through which quantum fluctuations could become a major component of emergence studies.[6] The implications of those geometries have not been fully considered and, even less so, that prior to those notations where indeterminacies (quantum fluctuations) actually manifest, states of perfection can be logically inferred. Highly speculative, our goal is to use our most simple formulations within logic to engage those parameters (principles) and mechanisms (functions) that give rise to mathematics and physics (and eventually all the other sciences).

Within this model, as developed in several prior articles, the infinite is profoundly within the finite. It is not finite, but actively and constantly imparts qualities to the finite. So, yes, those scholars who follow David Hilbert are asked to stay open. Pi’s three primary facets of the infinite are real realities of every circle and sphere. These qualities condition the finite.

Everything-everywhere-for all time, is in accordance with numbers, geometries, and equations; and, it all is a manifestation of the infinite qualities of continuity, symmetry and harmony.

Extending the discussions about Theano on Pythagoras, that statement above has become the byword of this website. To carry it a step further, a priority is given to defend this statement, “Everything-everywhere-for all time is also necessarily an expression of exponentiation, Euler’s number, e, and the Buckingham pi theorem.” The challenge to unpack such postulations is further complexified by the dimensionless constants of the Planck base units and the distinct types of harmonic functions necessarily part of every sphere. We postulate that this first particle is the archetypal form and function of all particles, strings, and automorphic forms.

Shell Particle
The first particle is anything but simple. It is the encapsulation of many facets of the infinite (continuity-symmetry-harmony) and the bridge between the finite-and-infinite. It is projected to be a shell for hypothetical particles as well as all known particles. Within the first 64 base-2 notations, the configurations within that shell are virtually infinite. With a very-small measure of confidence, I project that Langland programs, strings and M-theory, and SUSY can all be readily worked into the dynamics of those 64 progressions.

From earlier discussions about Infinitesimal spheres stacking and packing
, and given one particle per Planck unit of time and length, the rate of expansion using Planck Time computes to over 539 tredecillion particles per second. Dimensional analysis and our much earlier interactions with Freeman Dyson may cause us to adjust that statement. This rather different cosmological constant, 539 tredecillion particles per second, unfolds our base-2 chart of just 202 notations. Between Notations 143-144, Planck Time is one second and Planck Length computes to be 299,792,422.79 meters. That calculation, Planck Length divided by Planck Time, is within .001% of the NIST/ISO value for the speed of light set in 2019. The extrapolations for each of the 202 notations are within a range of .001% to .1% of the speed of light. Analyses of those figures are ongoing. Analyses to discern other well-known values throughout the chart are also ongoing.

Quantum openness
With these studies, quantum mechanics is considered an artifice of geometry that only manifests with a five-tetrahedral structure (and as of May 2023, five-octahedral structures as well) sharing an edge that creates a 7.356103+ degree gap. This natural gap is also within every expression of dodecahedral and icosahedral structures; here spatial dynamics are currently generally classified as quantum fluctuations. Such fluctuations are not an inherent part of the first particle and possibly other classes of particles (See Wikipedia).

The 202 Notational Grid
To add to the many variables already cited, in many earlier articles, it has been stated that all notations are always active. Every notation within the first 64-to-67 notations are profoundly active and dynamically changing. Such a concept redefines space-time and it redefines “the Now.”

That first particle has no less than four variables of the Planck units, the three primary variables of the the three facets of pi (π), the dimensionless constants within the Planck base units, and the many expressions of harmonic functions generated within every sphere. Cubic-close packing of equal spheres further complexifies as it generates new forms and functions within the tetrahedral-octahedral structures. With every prime number notation even more possibilities unfold. Then every prime number base progression adds another possibility. Where does it stop? It’s anybody’s guess! And, yes, there have been a lot of facts and guesses since the days of Einstein and Planck.

Our challenge now is to examine our emerging model in light of the bleeding edges of physics. We all know the answers; we simply have not properly defined the playing field and its most basic forms and functions. That is the purpose and goal of these studies: Langlands programs, String theory and M–theory, SUSY (including work Beyond the Standard Model), Causal sets and causal set theory, Loop Quantum Gravity, Scalar Field Theory, Spectral Standard Model and Causal Dynamical Triangulation. [7]

Within our simple model, the universe is just beginning and already it has the potential for far more complexity than within existing models as currently understood. It doesn’t discount any prior work; our work is to context it so these studies open even more potentials for diversity than so far imagined.

So, of course, your comments are most welcomed. Thank you. -BEC


Footnotes & Endnotes

* Our models and our scholars. It seems that the search is for three master keys: (1) a Finite-Infinite key, (2) the “Integrative Systems that Structure the Universe” key, and (3) the “Redefinitions of Space-Time (Mass-Charge)” master key. There are many prior homepages that touch on these three topics and there are many ways to access those pages. One of my favorite ways is to click on the left arrow at the top of each page. That will take you back, homepage by homepage, to the beginning of this website in August 2016. Work on these concepts, particularly the base-2 progressions, started in a high school in 2011. Here are my presuppositions that fall outside the mainstream: The definitions of a perfected state, particularly the continuity-symmetry-harmony expressions began back around 1971.

[1] Schwarzschild radius. Michel M. Deza; Elena Deza. Encyclopedia of Distances. Springer; 1 June 2009. ISBN 978-3-642-00233-5, p. 433. (PDF) Also: See Wikipedia: Karl Schwarzschild and The Planck Particle today:

[2] Victor J. Stenger, The Comprehensible Cosmos: From where do the laws of physics from? (2006), Prometheus Books, pp. 134, 295, and 298. Also see: A Scenario for a Natural Origin of Our Universe, (PDF), arXiv:0710.3137 [gr-qc]

[3] Victor J. Stenger, Defending The Fallacy of Fine-Tuning (PDF), p. 7, 2012, arXiv:1202.4359

[4] Particle discovery. Wikipedia’s Timeline of Particle discoveries, retrieved January 10, 2022

[5] Geometries of indeterminacy. Wikipedia, Quantum indeterminacy, retrieved January 10, 2022 Also, see my earlier work, Determinant becomes undecidable, uncomputable and unpredictable.

[6] Indeterminacy. Mysteries in Packing Regular Tetrahedra (PDF), Jeffrey C. Lagarias, Chuanming Zong, 2012

[7] Logic and Puzzles. Wikipedia topics: (1) Langlands programs, (2).String theory and Mtheory, and (3) SUSY (including work Beyond the Standard Model), (4).Causal sets and causal set theory, (5) Loop Quantum Gravity, (6) Scalar Field Theory, (7).Spectral Standard Model and (8) Causal Dynamical Triangulation.


Editor’s notes: Perhaps it is not evident, but I try to write as if I had a group of high school students and other teachers all collaborating with me, reading every word. Perhaps eventually we’ll get out of the weeds and closer to first principles and the first particle. To that end, the most dynamic part of this page follows. These are the evolving references, emails, and instant messages, yet be forewarned, sometimes these people are quite deep in the weeds! -BEC


References & Resources ________Prior / Next

Go to back to prior references from within this website. Also, review these earlier documents:
1. This work began in 1971 within the study of the 1935 EPR paradox.
2. It was part of a conference at MIT in 1979 in search of first principles.
3. There are many pages that consider the first instants of the universe.
4. There are also these presuppositions and assumptions.
5. Equally speculative is the concept that these foundations give rise to our ethics and values.
6. My struggle page. It’s among the current pages being updated, yet quite incomplete.



  1. Christoph Schiller: A classically-trained physicist at Universität Stuttgart (Germany) and received his Ph.D. in physics at the Université Libre de Bruxelles (Belgium), and a most-creative spirit and talent for writing about technical concepts for the average person. I believe his six volumes on the foundations of physics can help fill in some of the gaps in my education.
  2. Pierluigi Poggiolini: One of the world’s leading research scientists, a specialist within optics, light, and nonlinearity, his work challenges us to figure out a way to test our concepts.
  3. Elena Deza: Author of Dictionary of Distances (with Michel Deza, Elsevier, 2006), Encyclopedia of Distances (with Michel Deza, Springer, 2009; 4th ed., 2016) and others, she is a professor of mathematics at Moscow State Pedagogical University.
  4. Chiara Marletto: A Research Fellow within the University of Oxford’s Physics Department of University of Oxford, she is an active member of Wolfson’s Quantum Cluster and of the New Frontiers Quantum Hub. Her work on constructor-theoretic concepts caught our attention.
  5. Karl Schwarzschild: A Letter to A Legend (first draft)
  6. Sophie Gibb: Her team brought to life The Routledge Handbook of Emergence (2019). With Robin Findlay Hendry and Tom Lancaster, that book is filled with different types of analysis of emergence. Searching for some corroborative work, the concept of an actual geometry for indeterminacy was not found. The concept is still quite naive and young.
  7. Karen Crowther: Author of Appearing Out of Nowhere: The Emergence of Spacetime in Quantum Gravity(PDF), 2015, she is a professor of the foundations of physics at University of Oslo.
  8. Moataz H. Eman: Author of Covariant Physics: From Classical Mechanics to General Relativity and Beyond, Oxford University Press
  9. Molly Ball: Time Magazine, National Political Correspondent. 
  10. Arthur Holly Compton: A letter to a legend, Nobel laureate, 1927, whose work opened the way to quantum indeterminacy.



@EtheHerring @Exconsul @DurhamIAS @Routledge_Phil The Routledge Handbook of Emergence opens key concepts. How did it all begin? What is the most basic building block? Here is a very different model — – a mathematically integrated universe.

Sent to many others: If there is ever going to be a little harmony in this world, we’ll need to break out of our little worldviews for an integrated view of the universe. Here’s a simple start:

Here’s a variation on that theme:

@TIME @mollyesque (Molly Ball, Time Magazine) We’re all tied up within narrow worldviews. Thanks Isaac Newton! We need a mathematically-integrated view of the universe. We started in December 2011 in a New Orleans high school — It works. And it places today in proper context. It’s simple; not easy!

@SenSchumer @LeaderMcConnell @SpeakerPelosi @GOPLeader Tweet: Why are we so discombobulated as a people, a country, and a world? We live within narrow worldviews; a highly integrated view of the universe is needed to understand each other and even basic ethics.

@TheEconomist The world is not enough. We are discombobulated people because we live within narrow worldviews; a highly integrated view of the universe is needed to understand each other and even basic ethics. and

@BehrouzGhezel We all need to be lifted out of our little worldviews to see the entire universe so we can think-and-write-and-context in light of everything-everywhere-for-all-time. The beginnings of such a view are here: Also see:


Invitations and Collaborations

With whom do we collaborate? Of the hundreds of people who visit this site every month, who among them might want to extend a right hand and say, “Let’s work together.” Our only thrust is that the foundations of this universe and life itself be seen in light of infinity and the continuity-symmetry-harmony that the infinite engenders. Please, talk to us. Thank you. -Bruce


Key dates for this document, particle.


+ About the two dates at the top of the home page Close to 6 AM (TZ-19 or USA CST) each day, the days listed at the top of this page get advanced by one digit. It should be a relatively easy program to write, yet I rationalize that I do it manually just to remind me of our granular (sun-to-earth) sense of time. TZ-19 is time zone #19 assuming that the International Date Line is #1 and Greenwich Mean Time falls within Time Zone #13 (TZ-13). Notwithstanding, we all learning that the only time is Now.