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A Study of Notation #31:

The Numbers: A doubling of the Planck base units.

Notation	Planck Time	Planck Length	Planck Mass	Planck Charge	Scaling Vertices
31	1.1577×10-35 seconds	3.470762×10-26 meters	46.79kg  (103 lbs)	4.0278×10-9 coulombs	1.23794×1027

46.79 kilograms or about 103 pounds: Readily engaged from within our common experience
Overview: Our first group, after 31 doublings of the Planck base units, we discover that mass has become a rather common number. Length and time are still so small as to be non-intuitive. But the mass, a weight of 46.79 kilograms or about 103 pounds, is quite visceral for most of us. So, within this infinitesimal space within an equally infinitesimal duration of time, the universe is 103 pounds with a relatively modest charge.
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Scaling Vertices: By the 31st notation the scaling vertices, line 9 within our horizontally-scrolled chart, have dramatically expanded to 1,237,940,039,285,380,274,899,124,224 or 1.23794×10²⁷ vertices.
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One might guess that even “pointfree” vertices could somehow constitute 103 pounds! Yet, at this notation the space per meter cubed is so infinitesimal, even 103 pounds is a stretch. In this model, called construction vertices, these pointfree vertices are defined within the study of mereology or gunk theory, extended within the manifold studies of topology including systems theory and ontology.
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In prior articles we have considered the role of geometry and the evolution of ideal structures beginning with the circle, the sphere, cubic close packing, and the emergence of a line, triangle, then tetrahedron, its octahedron, and the four hexagonal plates that create structure within every octahedron. By the 31st notation, the potential complexity of structure is already overwhelming our imaginations.
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Eleven Prime numbers: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31. By the 31st notation, there could be as many as eleven different mathematical systems at work. We postulate that within each prime there is that possibility. This idea was first introduced in our article (January 2016) about numbers. It is also the implied focus of line 11 within the horizontally-scrolled chart.

31st doubling of Planck Mass: 46.79 kilograms or about 103 pounds begs the question, how can numbers have mass? What is mass? What is gravity? What is charge? How are the three interacting within this notation at that time and within our current time? How has that definition of mass changed over the years as all the notations have come to be?
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31st doubling of Planck Time: 1.157794×10^-34 seconds. This notation is within a range used by the big bang theory to define the Grand Unification Epoch, 10⁻³⁶ to 10⁻³⁴ seconds which includes Notations 25-to-34, and the Inflationary Epoch, 10⁻³³ seconds to 10⁻³² seconds which includes Notations 35-to-40. Each notation is postulated to continue to be the part of those keys that actually unlock both the unifying and the inflationary processes. There is always a process whereby there is holding together and a process of separation-and-particularization (of what those within big bang theory postulate as a singularity). Within this model, this singularity is more like a convention center because there is such a convergence of formulas, particularly ratios.

31st doubling of Planck Length: 3.470762×10^-26 meters We have been told many times that this length is too small to be meaningful. Our answer, “May be so, but may be not.”

31st doubling of Planck charge: 4.0278116×10^-9 Coulombs (nanocoulomb). Though small, it is known that the charge of one electron is about 1.6021766208(98)×10⁻¹⁹ Coulombs. This mass is so close to absolute zero, we can safely assume that all mass is superconducting.

The big bang theory’s Grand Unification and Inflation Epochs: Although we currently accept the range given by the big bang theory for their Grand Unification Epoch and Inflationary Epoch, within this Quiet Expansion model, this notation, and all those notations before and after, are part of an active, on-going definition of the universe. There are unification processes holding all those equations in place from the finite-infinite transformation point as well as inflationary thrusts, the doublings, that continue throughout all notations.

Planck scales. Within our current conception of time, this expansion appears to be instantaneous. It is not. It also appears to be silent so we named it a Quiet Expansion. It is the opposite of the big bang theory and what Alan Guth posits (Inflation, June 1993, National Academy of Sciences) as inflation. He uses scalar field theory to justify his concepts of inflation; of course, scalar field theory is not necessarily tied to Guth’s inflation and its principles may be used by other conceptual frameworks.

[Endnote: The “Inflaton” discussed in his book, The Inflationary Universe: The Quest for a New Theory of Cosmic Origins. Basic Books. pp. 233–234. ISBN 0201328402]

Within the Quiet Expansion model, every formula involved at this 31st notation along with the prime numbers involved will be thoroughly analyzed for the logic that generates the next level of activity within our charts.

Open Questions: Who are world’s experts in understanding prime numbers?

What do prime numbers do? Does each open the possibility to begin a new mathematical system, building on all others within the doubling complex, but introducing a new function heretofore not defined?

Speculative projections based on simple facts. Between 2-to-202 there are 46 prime numbers: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 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, 193, 197, and 199.

The next prime, 211, is beyond this definition and moment in time. Out of the 46 available prime numbers, the six numbers selected for this study are in bold type above.

Every prime number has its own flavor and personality. There are over 101 different types identified by Wikipedia editors. Are there mathematical experts within the studies of the functionality of prime who could help guide our thinking? Is it possible that each prime introduces a new mathematical set? Is this set properly described by set theory? Is each notation, within itself, defined by set theory and each transformation to the next notation defined by group theory?

To begin the process of answering these questions. We’ve made guesses regarding the ordering of the emergence of numbers, forms and functions within each prime. This listing comes from line 11 of the chart and an article about numbers.
2 • Euclidean geometries, starting with pi and cubic-close packing of equal spheres and lattice generation
3 • Bifurcation theory, including the Feigenbaum constant, and the various manifestations of the theory
5 • Golden ratio (Phi), the Fibonacci sequence and the nature of addition; five-fold symmetries, indeterminacy, the imperfect, fluctuation theory, and ratio analysis
7 • Computer automata theory with John Conway and Stephen Wolfram (this may be a special application of bifurcation theory)
11 • Group theory and projective geometry
13 • Algebraic geometries
17 • Langlands groups, Langlands correspondence, Langlands program, Langlands conjectures
19 • Zermelo–Fraenkel set theory (ZFC) and quantum gravity
23 • S-matrix theory, unitarity equations, Hermitian analyticity, connectedness
29 • Mandelbrot set, Julia set, Möbius transformations, Kleinian group