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A Study Of Notation #6  (still rough notes)

Extending pi-and-Euler’s equations, and projective, Euclidean, and icosahedral geometries

The sixth doubling
of Planck Time 
The sixth doubling
of Planck Length
The sixth doubling
of Planck Mass
The sixth doubling
of Planck Charge
3.450342×10−42 (s)
 32,768 to 262,144

Observation:  The number 6, often understood as three times two, is  the sixth doubling of the Planck numbers and within this system of numbers, geometries, and formulas.  The question is asked, “What happens within a notation that isn divisible by other notations? Does it have any special relation with Notation #2 and Notation #3?”

Planck unit doubles. Is it meaningful to say that 1.03×10-33 meters is six times larger than the Planck Length?  Does this notation help to stabilize all the different structures that have begun to emerge?  With an abundance of point-free vertices with which to make rather idealized constructions, what could possibly be going on at this juncture.

  • Spheres:
  • Projective geometries:
  • Euclidean geometries:
  • Simple doublings at Notation 4:
  • Pentastars, tetrahedral rings, tetrahedral systems & the icosahedral phase:
  • Simple doublings at Notation 6:

Here are a few of the pages preceding this page:


  1. November 15:  Before we can understand the complex…We need to understand the simple things. An introduction to our study of the Langlands programs.
  2. November 12:  Seven reasons to look more deeply at our chart (at the top). It is still a  largely-unexplored  model of the Universe
  3. November 9:  Over 1000 Simple Calculations Chart A Highly-Integrated Universe
  4. November 8: We live in an exponential universe.




Within the next prime-number notation are the grounds for another dimension of mathematics and here I was sure that Euclidean geometries would come next. Yet, at the Planck scale [2] 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 a series of four articles, I tried unsuccessfully to get into the Langlands programs, bring to answer the question, “How would this group envision a logical construction path and its modalities?

Observation: Within today’s scientific work, there are measurable units of charge and mass that are smaller than the Planck Charge and Planck Mass. It begs the question, “How can that be?

Hypothesis: Continuity, symmetry and harmony manifest within infinitesimal units of space and time, yet within parameters that measure charge and mass are measurable units that are smaller.

Questions abound:
•  If pre-Planck Length and Planck Time, are these charges within the transformation nexus between the finite and infinite?
•  Could these be earliest possible measurements of the radii and diameters of circlers and spheres?

The many measurements of mass that are smaller than Planck mass:  What are these numbers telling us about the nature of mass? From Wikipedia: “Unlike other physical quantities, mass-energy does not have an a priori expected minimal quantity, as is the case with time or length, or an observed basic quantum as in the case of electric charge. Planck’s law allows for the existence of photons with arbitrarily low energies. Consequently, there can only ever be an experimental lower bound on the mass of a supposedly massless particle; in the case of the photon, this confirmed lower bound is of the order of 3×10−27 eV = 10−62 kg.”

19 September 2017: Here is yet another fundamental challenge to the logic of the Big Board-little universe.  We will take one step at a time!

Although the Planck mass scale is somewhat disconcerting, the fact is at the top end of the scale is 1.399×1052kg within notation 202.  When that number is more carefully calculated for the current time, it will be closer to 4.4506×1052 kg, the mass of the observable universe as estimated by NASA.  The National Solar Observatory calculates it to be closer to 6×1052 kilograms.


Diophantine geometry

We use non-abelian fundamental groups to define a sequence of higher reciprocity maps on the adelic points of a variety over a number field satisfying certain conditions in Galois cohomology. The non-abelian reciprocity law then states that the global points are contained in the kernel of all the reciprocity maps.

Hasse-Minkowski theorem

recab(F×) = 0.