Measuring an Expanding Universe Using Planck Units

Writing and editing on this summer day in the northern hemisphere, Friday, June 23, 2017

• Measuring an Expanding Universe Using Planck Units (work in progress)
• The Thrust of the Universe: What is it? (work in progress)
• Visualizing the Universe (work in progress)

Want to help? Drop me a note. Thanks. Bruce Camber

A novel approach to measure the universe is with the Planck base units and the power of 2 (doublings) also known as base-2 exponentiation. It renders very particularized results from the first moment of creation to the Age of the Universe. To open a discussion about this approach, a sampling of just six sets of numbers from the complete chart of 202 notations is provided. Each sampling was selected in part based on observing the emergence of a number that can be readily engaged from within our common experience. In the sequence of doublings, each is also a prime number, the basic building block of all numbers. The simple logic of these numbers is analyzed. Of course, we begin with the first set of numbers: Planck Length, Planck Time, Planck Mass and Planck Charge.

Six Derivative Sets of Numbers to Analyze

Notation: Planck Time (Seconds) Planck Length (Meters) Planck Mass (Kilograms) Planck Charge (Coulombs)
0 5.391 16(13)×10-44 1.616229(38)×10-35 2.176.470(51)×10-8 1.875545.956(41)×10-18
31 1.157794×10-34 s .470762×10-26m 46.79kg (103lbs) 4.0278116×10-9 C
67 7.95630×10-24 s 2.38509×10-15m 3.211962×1012 kg 276.78910 C
107 8.748×10-12 s 2.6224 mm (.10324 inches)
3.5315×1023 kg 3.0433×1014 C
149 38.47432 s 11,533,588.224 km 1.5532×1036 kg 1.3384×1027C
173 645,492,017.5 s 1.935015×1017 km 2.605×1043 kg 2.245×1034 C
199 4.331×1016 s 1.298×1024 km 1.748×1051 kg 1.506×1042C

Notation #0: Planck Numbers, Transformations Between the Finite And Infinite

Although Max Planck began developing these numbers in 1899 and first published them in 1906 (within his book, Theory of heat radiation), nobody paid much attention to what have become known as the Planck units. In 2001 Frank Wilczek (MIT) began publishing three articles, Scaling Mt. Planck (312, 321, 328) for Physics Today. Although others had done significant work using one or more of the Planck numbers, in our one-on-one conversations, Wilczek took some credit for lifting the Planck units up out of the category of numerology and to have opened the path for others to use and cite the Planck base units.

There were a few scholars who had even earlier intuitions about the significance of these numbers and dared to write about it. To understand that history, there are many stories that need to be reviewed. For example, in 1959 C. Alden Mead (UMinn) began his struggle to published his work about the Planck Length. Though finally published in 1964, the article, Possible Connection Between Gravitation and Fundamental Length Phys. Rev. 135, B849 ( 10 August 1964), was ignored by the scholarly community. At that time, the Planck Length commanded almost no respect as a fundamental unit of length.

Another Research Project: Even today, support for the Planck base units is far from unanimous. Among our many research projects, a study will focus on those who are not ready to recognize that these Planck units are in any way fundamental units for science.

Planck Time: Planck Time
Planck Length: Planck Length
Planck Charge: Planck Charge
Planck Mass: Planck Mass

Another Research Project. Studying and interpreting Max Planck’s 1899 logic to extract each of these base units is part of Big Board-little universe program. His logic to select each facet of reality, define every relation, every equivalency, and every ratio will be examined.

To interpret these numbers, we’ll need help:

Dimensionless constants. There are dimensionless constants that are never-ending and never-repeating numbers. Pi is the mother lode. Euler’s number, e, seems to be the next. All equations that use pi, like the fine-structure constant, qualify. All equations that use the reduced Planck constant may also qualify as bridges between the finite and the infinite and as a key to understand the simple.

Prime: Each of these six selected notations is also a prime number. Mathematics recognizes the special role of prime numbers. Our never-ending, never-repeating numbers set within a prime number may open the differentiation between groups within mathematics. Given our working assumption that everything starts simply, I would say that each group must start simply. We will try to discern the mechanisms for the simple-to-complex transformations (1) within a given notation, (2) then between successive notations, and (3) then with other groups of notations.

A new thought. Add the never-never numbers (all ratios). Another facet,  there appears to be four primary facets that define finite-infinite transformation dynamics as well as space-time dynamics within every aspect of mathematics, logic and science.

Notation #31: 46.79 kilograms or about 103 pounds

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.

Scaling Vertices: By the 31st notation the scaling vertices, line 9 with the chart, have dramatically expanded (1,237,940,039,285,380,274,899,124,224 or 1.23794×1027 vertices). In this model, called construction vertices, the first 67 notations are point free vertices as defined within the study of mereology or gunk theory, extended within the manifold studies of topology including systems theory and ontology.

One might guess that even “pointfree” vertices could somehow constitute 103 pounds! Though a somewhat flippant thought, at this notation the space per meter cubed is so infinitesimal, even 103 pounds feels like a stretch.

Prime numbers2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31. As noted, by this 31st notation, there could be as many as eleven different mathematical systems at work. Yes, we postulate that each prime that possibility. This notion was first introduced in an article from January 2016 about numbers. It is also the focus of line 11 within the horizontally-scrolled chart. At this time, this simple statement is introduced in a cursory manner and as a placeholder.

Planck time multiple: 1.157794×10-34 seconds.  This notation is in the range of the Big Bang Theory’s Grand Unification Epoch (10−36 10−34 seconds which also now includes Notations 25 to 34) and the Inflationary Epoch (10−33 seconds to 10−32 seconds, which now includes Notations 35 to 40).  This notation is postulated to continue to be the part of those keys that actually unlock both processes.  There is always a process whereby there is holding together and then a process whereby there is a separation and particularization of the so-called singularity.

Planck scales.  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.

The Planck charge multiple 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 thoroughly active because in this model the temperature is always approaching absolute zero and the mass is superconducting.

Grand Unification and Inflation: Although we currently accept the range given by the big bang theory for their Grand Unification Epoch and Inflationary Epoch, within the 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 and then finally a break out, an inflationary thrust, that continues throughout all notations.

And, to understand these numbers, we’ll need even more help:

Prime numbers between 2-to-202: 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 moment in time. The six prime numbers selected for this study are bold.

Every prime number has its own flavor and personality. There are over 101 different types identified by Wikipedia editors. Given their importance within this model of the universe, we will seek out mathematical experts to 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? Of course, among our current challenges, begin the process of answering these questions. In that light, we’ve made guesses regarding the ordering of the emergence of number, form and function within each prime:

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

– – – – – – – – – – – MORE TO COME – – – – – – – – – – WORKING DRAFT – – – – – – – – –

Notation #67: 2.38509×10-15m

Overview: The 67th notation is one-third of the total notations. In the earliest stages of developing this model, it was called the small-scale universe. The next 67 notation were referred to as the human-scale universe and notations 134 to 202 were considered the large-scale universe. Of course, those denotations will be subject to continuous review!

Planck Length: 2.38509×10-15 meters.  This multiple of the Planck Length is now in the area of measurement by CERN’s Large Hadron Collider.

Planck Mass: 3.211962×1012 kilograms. The mass for such a small space (3.211962×1012 kg) is beyond imagination. The earth is 6×1024 kilograms. In 1971 a team at Harvard, Baym, Pethick & Sutherland calculated densities could approach 4.3×1014kg/m3 before nuclear pasta and saturation densities are reached. Subsequently that figure has been further calculated to be ≃2.3×1017kg/m3 (nuclear saturation density). Although stretching beyond our comfort zones, we are eager to learn something as simple as meters-cubed (m3).

Question: In the progression of doublings, is it assumed to be m3 or is there another calculus we should be using within these mathematical transformations before we approach what is known as neutron drip densities, nuclear pasta and Coulomb crystals?

Planck Charge:  276.7891 Coulombs Equivalent to the discharge of a few average lightning bolts, in the International System of Units ( SI ), the coulomb (C) is the preferred unit of electric charge quantity.

Planck Time: 7.95630×10-24 seconds.  Within the big bang theory, the Quark Epoch begins at 10−31 and runs down to 10−12 seconds.  Within the Quiet Expansion model, this epoch would begin within Notation 41 and extend through Notation 107. At Notation 67, clearly halfway through the Quark Epoch, we propose that it be called the Quark Processes.  Within that scope there would be  multiple primes, all of which might initiate the special mathematics:  41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, and 107.  One might conclude there are many opportunities to develop the mathematics of the Standard Model of Physics. 

Common numbers can be difficult to interpret! We need help.

An Open Question: What is the logical sequence, from the most simple to the most complex, and which would be required before another begins? We will seek help from some of our best logicians and mathematicians, the scholars, who would be open to working with us, the neophytes.

Cycles, frequency and periodicity: The first prime numbers might tell a special story. Staying within the 202 notations, consider multiplying-by-2 the primes up to 101. Here may be the potential of many systems within systems within this model.

Planck Temperature:  3.25288×10-7K on a scale where the Planck Temperature is placed within the 203 notation and divided by 2 over and over and over again.

Notation #107

Overview: Although time is certainly within the measurable range of today’s devices, at 8.748×10-12 seconds, it is hardly perceptible by a human being. The length at 2.6224 millimeters is quite common. All things around the size of a small ant are included. Yet, the mass continues to be non-intuitive at 3.5315×1023 kilograms. Though still less than the weight of the earth, it is hard to imagine it as the size of an ant.

Planck Charge multiple: The coulombs scale has grown to a formidable 3.0433×1014 or 3.04 teracoulombs. We will try to assess the meaning of that number by also studying the faraday constant which equals 96,485.3399 coulombs.

We have substantial work to do within this analysis.

For example, would it be correct to think of 3.04 teracoulombs in terms of a lightening bolt?  Might it  be calculated as 304,000,000,000,000 or 304,000 trillion lightening bolts per second?  We need help!

Plank TemperaturePlanck Temperature: It is not one of the basic four Planck units.  Its derivation, however, resulted in an extremely hot number that became the basis of big bang cosmology.  These cosmologists had to ignore the four base units to make Planck temperature their focal point from which the theory would evolve.

With our inherent logic that everything starts simply and small, the Planck Temperature was placed at the top of the scale and it was divided by 2 over and over again. By the 107th notation, the Planck Temperature, 1.416.808(33)×1032 (K),  is now 3.5765×103 Kelvin.  Of course, with every new notation, it gets quite hot.

Kelvin is a very unusual temperature scale. At Notation 97 the number is 3.4927 K which is -453.38314 degrees Fahrenheit or -269.6573 degrees Celsius well within the range for high-temperature superconductors (-220 degrees Fahrenheit or-140 degrees Celsius at normal pressures, and -164 F and -109 C at high pressures). Consider these temperatures:

  • The human body temperature is 98.6 degrees Fahrenheit.  That equals 310.15 Kelvin and falls between notations 103 and 104.
  • The Sun’s temperature is 5778 K. In this working edition of the chart, it is between notations 107 and 108.
  • Quark-Gluon Plasma (QGP). Requiring an estimated temperature of 2×1012 Kelvin to create the Quark-Gluon Plasma (QGP), this process could begin as early as notations 136 and 137. It requires 175 MeV per particle. We’ll need help to figure that out!

The universe is less than one-hundredth of a second from its start. As we have observed, within this model, each notation is alive and well and still part of the sustaining infrastructure of the universe. Of course, there will be much more analysis forthcoming.

149:  Still basic information.  We have a substantial work to do within this analysis yet here we are gaining some confidence that these numbers are telling a plausible story that has a bit more logic than the stories told about inflation, super luminal speeds, and the inflatonMuch more to come…

Notation #149

The universe is now up to 38.47 seconds of activity. The Planck Length doublings result in a length of 11,533,588.224 kilometers.  That is almost 10 times larger than the diameter of our sun (1.392 million kilometers). The mass at 1.5532×1036 kilograms is now substantially greater than the sun (1.989×1030 kilograms). The coulombs number has grown to 1.3384×1027 or 1338 yottacoulombs.

We are asking professionals to what that coulombs figure might be equivalent. How many suns for example? 1,338,400,000,000,000,000,000,000,000 lightening bolts doesn’t really say much.

The temperature based on Planck units is now 3.146×1016 Kelvin. within the Big Bang theory, the projected temperature is 109 Kelvin.  One might assume that is is per meters cubed.  We will take the calculation for size, 11,533,588.224 kilometers, and figure out what the temperature rating would be.  Can you help? 

We have barely begun to work  with these numbers within our analysis.

Notation #173 Simple math problems

The universe is now 645,492,017.5 seconds old. That is 645 million seconds old. A quick conversion to years is somewhere between 16 and 32 years. We happen to know that a light year is between notations 168 and 169.

Calculation: 60 seconds × 60 minutes = 3600 × 24 = 86,400 seconds per day times 365.25636 SI days-to-the-year equals 31,558,149.504 seconds per year. 645,492,017.5 divided by 31,558,149.504 equals 20.4540515729 years.  This number is important to study. There is something very visceral about a second. Within these calculations we  the conversion number, seconds-to-year is 31,558,149.504 is used.

31,558,149.504 seconds to a year. This number has been computed as low as 31,536,000 using just 365 days per year, yet more often it is 31,557,600 using 365.25 days per year. The SI ( International System of Units) number is 365.25636 days per year. When talking about 13.81 billion years, that difference becomes a substantial variable. The Planck Time units from within a second for a year may be a more reliable figure to use in the future.

20.45 years old. A rather ideal age for a person, still young enough to be excused for our youthful ways, yet mature enough to be taken seriously. Within the greater model, 173rd notation is over 86% of the way through the total, yet it’s a long way to 13.892 billion years from 20.45 years. Though appearing more aggressive, the arc of the increase is still a simple doubling. From 173 to 202 is about 29 additional notations, divided by 3.33 would mean that we would add just under nine additional zeroes. That takes us from 20,45 years to 13.8 billion of years.

Placeholder information.  Yes, we are just beginning to work  with this analysis.

Notation #199

The universe is now 43,318,236,018,400,000 seconds (1.3727 billion years) old. That is just 43.318 quintillion seconds.

An analysis of the effect of gravitation on hypothetical experiments indicates that it is impossible to measure the position of a particle with error less than ΔxG=1.6×1033 cm, where G is the gravitational constant in natural units. A similar limitation applies to the precise synchronization of clocks. It is possible that this result may aid in the solution of the divergence problems of field theory. An equivalence is established between the postulate of a fundamental length and a postulate about gravitational field fluctuations, and it is suggested that the formulation of a fundamental length theory which does not involve gravitational effects in an important way may be impossible.

Cycles, frequency and periodicity: The first prime numbers might tell a special story. Staying within the 202 notations, consider multiplying-by-2 the following prime numbers: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, There are a few things we can identify within each progression. Our focus will be on those numbers in bold.

  • 2-4-8-16-32-64-128: This notation, 128, contains many common elements of life. The Planck Length multiple is 5.499 km or 3.42 miles. That would be the size of the universe at just .0000183 seconds, readily measurable with today’s instrumentation. The mass is an extraordinarily dense (7.406±×1029kg) considerably larger than the Earth’s mass (5.97×1024 kg) yet smaller than the sun (1.988±×1030 kg). The other measurements within notation 128 are, at this time, still being studied.
  • 3-6-12-24-48-96192:
  • 5-10-20-40-80160: Things the size of buckyballs or fullerenes are at notation 80. The first day, the distance light travels within 86400 seconds is between notations 160 and 161.
  • 7-14-28-56-112: Everything and anything that is around 8.3917 cm or 3.3 inches
  • 11-22-44-88176: Things the size of a virus are at 88.
  • 13-26-52-104: The size of this “.” A dot!
  • 23-46-92184: Nanowires (92) and 41,891 years, a young universe (184)
  • 29-58-116: All things around 1.34 meters or 53 inches.
  • 31-62-124:
  • 41-82164:
  • 43-86-172:
  • 47-94188:
  • 53-102:
  • 59-118:
  • 61-122:
  • 67- 202:
  • 71-142:
  • 73-146:
  • 79-158:
  • 83-166:
  • 89-178:
  • 97-194:
  • 101-202:

What could be happening as number-form-function aggregate? In how many different ways could these numbers be interpreted? First, there is the simple doubling of the Planck Units. Then, there is the scaling exponentiation of the construction vertices. And perhaps, there is this periodicity which just might also be doubled in the mix of processes.


(As a check, divide 645,492.5 by by 60 (seconds to minute) makes 10,758,200.2917 minutes. Divide by 60 makes 179,303.338195 hours. Divided by 24 makes 7470.97242479 days which divided by 365.25636 SI days-to-the-year is 20.454051573 years).  HITOSHI MURAYAMA, Dynamical Supersymmetry