UPDATED: SUNDAY, MARCH 20, 2016 Commentary/Reflections (new tab)
An Early Draft
“I have learned that many of the Greeks believe Pythagoras said all things are generated from number. The very assertion poses a difficulty: How can things which do not exist even be conceived to generate? But he did not say that all things come to be from number; rather, in accordance with number – on the grounds that order in the primary sense is in number and it is by participation in order that a first and a second and the rest sequentially are assigned to things which are counted.”
– Theano, On Piety (as reported by Thesleff, Stobaeus, and Heeren)
Using the model of the universe generated through the Big Board – little universe Project where there are just over 201 base-2 notations from the singularity of the Planck base units (particularly from Planck Time) to the Age of the Universe, the question to be addressed is, “Which numbers come first and why?” Mathematical logic calls out the most simple-yet-powerful numbers that can be used to build and sustain a highly-integrated universe. Our other assumptions are here. Each of these key numbers and number groups are introduced; each will then become the focus of additional study, further analysis, and the basis for a more-in-depth report about each number. Our initial numbers are:
(1) 3.1415926535897932384626433+ or π or Pi
(2) 74.04804896930610411693134983% or the Kepler Conjecture
(3) 0, 1 where the numbers are: zero and one
(4) 7.356103172453456846229996699812° called the Pentastar gap
(5) 1:1.618033988749894848204 or the Phi ratio
(6) 4.6692016091029906718532 which is a ratio called the Feigenbaum constant
(7) 110 of Stephen Wolfram’s rules
(8) 6.6260709×10−34 J·s or Planck constant plus all related numbers
(9) Groups of dimensionless constants, all known mathematical and physical constants
(10) 13.799±0.021 billion years, the Age of the Universe
Please note: Links inside the body of the article most often open a new tab or window within a Wikipedia page. For those occasional inks that do not open new windows, please use the back-arrow key to return to the referring page. All links within the Endnotes will eventually go to source materials if posted on the web.
Page 1 of 10 https://bblu.org/2016/01/08/number/
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Most of us know the universe is infused with numbers. It seems nobody really knows how all these numbers are organized to make things and hold it all together.
In our work with high school students there is a constant demand that our numbers be intellectually accessible. Simplicity is required. So, it is rather surprising that we ended up engaging the Planck Length (and the other Planck base units) very early in our study of the platonic solids. We also started to learn about base-2 notation and combinatorics. We had to do it. We had divided our little tetrahedron in half so many times, we knew we were in the range of that limit of a length, and we wanted to find a place to stop. Eventually, to get more accurate, we started with the Planck Length, used base-2 exponential notation, and multiplied our way out to the Observable Universe.1 It took just over 201 doublings. What? Huh?
That little fact is as unknown as it is incredible (even as of January 2016 when this article was first posted).
In December 2011 we could find no references to the 201+ notations in books or on the web. We did find Kees Boeke’s 1957 work with base-10 notation. It was a step in the right direction, but it had no lower and upper boundary, no Planck numbers, and no geometry. It had just 40 steps amounting to adding zeroes.
We were looking for anything that could justify our “little” continuum. We didn’t know it at the time, and we later learned that we were looking for those deep relations and systems that give us homogeneity and isotropy, a cosmological constant, and an equation of state. Though we already had put everything, everywhere throughout all time in an ordered relation, we had no theoria, just the praxis of numbers. We tried to set a course to go in the direction of a theory that might bind it all together.
The first 60+ doublings constitute a range that scholars have been inclined to dismiss over the years as being too small; some say, “…meaninglessly small.” Yet, being naive, it seemed to us that the very simple and very small should be embraced, so we started thinking about the character of the first ten (10) doublings. Trying to understand how to “Keep It Simply Simple,” we were pleasantly surprised to discover that there was so much work actively being pursued by many, many others throughout academia and within many different disciplines to develop the logic of the most simple and the most small.
Within the studies of combinatorics, cellular automaton, cubic close packing, bifurcation theory (with Mitchell Feigenbaum’s constants), the Langlands program, mereotopology and point-free geometry (A.N. Whitehead, Harvard, 1929), the 80-known binary operations, and scalar field theory, we found people working on theories and the construction of the simple. Yet, here the concepts were anything but simple.2
It is from within this struggle to understand how all these numbers relate, we began our rank ordering of all possible numbers. This exercise helps to focus our attention.
Planck Length and Planck Time. One might assume that we would put the five Planck base units among the most important numbers to construct the universe. As important as each is, it appears at this time that none of them will be among the Top 5. Although very special, the Planck numbers are determined by even more basic and more important concepts and numbers. At the very least, all those numbers will come first.
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First Principles. The work to find the Top Numbers was preceded by an end-of-year report after four years of studying and using the Big Board-little universe charts. That report titled, Top Ten Reasons to give up little worldviews for a much bigger and more inclusive UniverseView 3, was done with comedian David Letterman in mind. He often had a Top Ten on his show.
“#10” for us it is, “Continuity contains everything, everywhere, for all time, then goes beyond.” One of the key qualities to select our most important numbers is the condition of continuity and discontinuity starting with the simplest logic and simplest parts.
A Quick Review of the Top Ten Numbers in the Universe.
Because many scholars have addressed the question, we did a little survey.
- Pi was somewhere on most everybody’s list among our sample.
- 0 and 1 are obvious keys for most scholars.
- Numbers like e and Phi (and the Fibonacci sequence) are on most of these lists.
- There is this one equation by Euler — ei*Pi + 1 = 0 — where five key numbers are used in what Richard Feynman considered to be the crown jewel of mathematical equations about identity.
- Though infinity is not a number but a concept, it was on many lists.
- The numbers for the speed of light, c, and the gravitational constant, G, are also on many lists. These numbers are keys used by Max Planck for his calculations of the Planck base units back in 1899.
- The Planck Constant is quite rightly on many lists.
- The Feigenbaum constants, Buckingham Pi theorem, the fine-structure constant, and other dimensionless quantities and physical constants were cited less often.
- We added two numbers not cited at all: mathematician Thomas Hales‘ number from his proof of the Kepler Conjecture and what we call the Pentastar 7.38 degree gap.
Scholars and thought leaders. Our limited survey began with leading thinkers in the academic-scientific community and then thoughtful people from other disciplines:
- Astronomer, Martin Rees, author, Just Six Numbers: The Deep Forces That Shape The Universe (See Section 3, “Well-known subsets,” and click on 3.2, “Martin Rees’ Six Numbers)” or go to our summary page for Lord Martin.
- Cosmologist, physicist, mathematician, John D. Barrow, author, The Constants of Nature: The Numbers that Encode the Deepest Secrets of the Universe. 2003.
- Philosopher/scientist, Roger Penrose, Oxford, The Penrose Number
- Physicist-mathematician, James D. Stein, author, Cosmic Numbers (begin on page vii) and a summary, Popular Mechanics (September 2011).
- Clifford Pickover, Wonders of Numbers summary
- Journalist David Pegg has a Top 25 (May 9, 2013).
- Business analysts Ben Duronio and Walter Hickey have a top 10 (July 8, 2012)
- Geometer, Steve Waterman, and chemistry professor, Michael Anthony Whitehead have just four math constants that they used in the derivation of 142 physical constants. Because of their conclusions, their numbers are being scrutinized.
Base-2 notation. Yes, our work with base-2 notation originated from within a high school. We have no published scholarly articles and there has been no critical review of our emerging model. Nevertheless, we forge ahead with our analysis of numbers and systems.
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Goals. Our singular goal is to try to construct our universe using mathematical logic. We begin with the magic of the sphere. Our #1 number is Pi (π).
#1 = π
Numerical constant, transcendental and irrational all rolled into one
For us Pi (π) seems to be a very good starting point. Non-ending or continuous, it is also non-repeating which is discontinuous. This most simple construction in the universe requires just two vertices to make the sphere. How does it work? It appears to give form and structure to everything. Using dimensional analysis and scaling laws, this progression of the first 20 notations shows the depth of possibilities for constructions when multiplying by 8. Our open question: In what ways do the Feigenbaum constants within (bifurcation theory) apply?4
|B2||Vertices||Scaling Vertices (units:zeroes)||Bifurcation*||Ratio*|
|12||4096||68,719,476,736||*This bifurcation and|
|13||8192||549,755,813,888||ratio columns come|
|14||16,384||4,398,046,511,104 (trillion:12)||from a Wikipedia article|
(discussion begins on the next page)
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Discussion. Pi still holds many mysteries waiting to be unlocked. Among all numbers, it is the most used, the most common, and the most simple but complex. We assume, that along with the other mathematical constants, pi (π) is a bridge or gateway to infinity. We assume it is never-repeating and never-ending. It is “diverse continuity.” There are enough scaling vertices within ten doublings to construct virtually anything. So, to analyze a possible logical flow, any and all tools that have something to do with pi (π) will be engaged. Again, among these tools are combinatorics, cellular automaton, cubic close packing, bifurcation theory (with Mitchell Feigenbaum’s constants), the Langlands program, mereotopology and point-free geometry (A.N. Whitehead, Harvard, 1929), the 80-known binary operations, and scalar field theory. Perhaps we may discover additional ways to see how pi gives definition — mathematical and geometric structure — to our first 60-to-67 notations. What are the most-simple initial conditions?
More Questions. What can we learn from a sphere? … by adding one more sphere? When does a tetrahedral-octahedral couplet emerge? When do the tessellations emerge? At the third notation with a potential 512 scaling vertices, surely dodecahedral and icosahedral forms could emerge. Within the first ten notations with over one billion potential vertices, could our focus shift to dynamical systems within the ring of the symmetric functions?
#2 = Kepler’s Conjecture
Not a very popular topic, one might ask, “How could it possibly be your second choice?” Even among the many histories of Kepler’s voluminous work, his conjecture is not prominent. To solve a practical problem — stack the most cannon balls on the deck of a ship — he calculated that the greatest percentage of the packing density to be about 74.04%. In 1998 Professor Thomas Hales (Carnegie Mellon) proved that conjecture to be true. By stacking cannon balls, all the scholarship that surrounds cubic close packing (ccp) enters the equation. The conjecture (and Hales 1998 proof) opens to a huge body of current academic work.5 There we found this animated illustration on the right within Wikipedia that demonstrates how the sphere becomes lines (lattice), triangles, and then a tetrahedron. With that second layer of green spheres emerges the tetrahedral-octahedral couplet.
This image file (right) is licensed under the Creative Commons Share-Alike 2.5 Generic license.
Revisions. As we find experts to guide us within those disciplines where pi has a fundamental role, undoubtedly sections of the article will be substantially re-written and expanded. Our goal has been to find the most logical path by which all of space and time becomes tiled and tessellated. Perhaps there is a new science of the extremely small and the interstitial that will begin to emerge. These just might be foundations of foundations, the hypostatic, the exquisitely small, the ideal.6 We plan to use all the research from Kepler to today, particularly the current ccp (hcp and fcp) research from within our universities, in hopes that we truly begin to understand the evolution of the most-simple structures.
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#3 = 0, 1
The numbers, zero (0) and one (1) begin the mapping of pi to Cartesian coordinates. Beginning with a circle, each sphere is mapped to two-or-three dimensional Cartesian coordinates. It is the beginning of translating pi to sequences and values. The first iterative mapping is a line, then a triangle, then a tetrahedron, then an octahedron. When we focus solely on this subject, with experts to guide us, perhaps we can engage the study of manifolds that are homeomorphic to the Euclidean space.6
#4 = Pentastar gap =
The little known 7.356103 degree gap is our fourth most important number, the possible basis for diversity, creativity, openness, indeterminism, uniqueness and chaos.7 That Aristotle had it wrong gives the number some initial notoriety; however, it is easily observed with five regular tetrahedrons which would have eight vertices. It appears to be transcendental, non-repeating, and never ending. Where the tetrahedron with four vertices and the octahedron with six have been been whole, ordered, rational, and perfect, tessellating and tiling the entire universe, the potential for the indeterminate which has the potential to become the chaotic resides somewhere deep within the system. We believe that place just may be right here.
Within this infinitesimal space may well be the potential for creativity, free will, the unpredictable, and the chaotic. Here may well be the basis for broken symmetries. Of course, for many readers, this will be quite a stretch. That’s okay. For more, we’ll study chaotic maps and the classification of discontinuities.
#5 = phi = φ = The Golden Ratio
= φ = 1:1.618033988749894848204586
Of all the many articles and websites about the golden ratio and sacred geometry, our focus is on its emergence within pi and within the platonic solids. Phi is a perfection. It is a mathematical constant, a bridge to infinity. We are still looking to see if and how phi could unfold within the tetrahedral-octahedral simplex. Could that answer be within Petrie polygons? The magic of the golden ratio does unfold with the dodecahedron, the icosahedron, and the regular pentagon. Within this listing, phi has bounced back and forth with the Pentastar gap. Which manifests first? Is it manifest if it is inherent?
Starting with this article, we have begun an active study of Phi and its relations to pi and the Platonic solids. Although there are many, many papers about phi, none are from our special perspective of 201+ notations.
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#6 = Feigenbaum constants
δ = 4.669 201 609 102 990 671 853 203 821 578
We are the first to admit that we are way beyond our comfort zone, yet to analyze and interpret the processes involved within each of the doublings, each an exponential notation, requires tools. This Feigenbaum constant gives us a limiting ratio from each bifurcation interval to the next…. between every period doubling, of a one-parameter map. We are not yet sure how to apply it, but that is part of our challenge.
It gives us a number. It tells us something about how the universe is ordered. And, given its pi connection, we need to grasp its full dimensions as profoundly as we can. We have a long way to go.
#7 = Rule 110 cellular automaton
There are 255 rules within the study of elementary cellular automaton. Rule 110 was selected because it seems to define a boundary condition between stability and chaos. All 255 rules will be studied in light of the first ten notations to see in what ways each could be applied. Any of these rules could break out and move up or down within this ranking. Steve Wolfram’s legacy work, New Kind of Science (NKS) is online and here he lays the foundations for our continued studies of these most basic processes within our universe.8
#8 = Max Planck numbers
We have been working on our little model since December 2011. Over the years we have engaged a few of the world’s finest scientists and mathematicians to help us discern the deeper meaning of the Planck Base Units, including the Planck Constant. We have studied constants from which the Planck numbers were derived, i.e. the gravitational constant (G), the reduced Planck constant (ħ), the speed of light in a vacuum (c), the Coulomb constant, (4πε0)−1 (sometimes ke or k) and the Boltzmann constant (kB sometimes k). This engagement continues. We have made a very special study of the Planck Base Units, particularly how these numbers work using base-2 exponential notation and with the Platonic solids. We had started with the Planck Length, then engaged Planck Time. Finally in February 2015, we did the extension of Planck Mass, Charge, and, with a major adjustment to accommodate simple logic, Temperature. We have a long, long way to go within this exploration. Essentially we have just started.9
Notwithstanding, there is a substantial amount of work that has been done within the academic and scientific communities with all the Planck numbers and those base numbers that were used to create the five Planck base units. Perhaps chemistry professor, C. Alden Mead of the University of Minnesota began the process in 1959 when he first tried publishing a paper using the Planck units with serious scientific intent. Physics professor Frank Wilczek of MIT was the first to write popular articles about the Planck units in 2001 in Physics Today (312, 321, 328). From that year, the number of articles began to increase dramatically and experimental work that make use of these numbers has increased as a result. https://bblu.org/2016/01/08/number/#7
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#9 = Mathematical & physical constants
Given we started with pi (π), it should not be surprising that we are naturally attracted to any real data that shows pi at work such as the Buckingham π theorem and the Schwarzschild radius.
We will also bring in Lord Martin Rees “Six Numbers” as well as the current work within the Langlands programs, 80 categories of binary operations, scalar field theory, and more (such as the third law of thermodynamics and zero degrees Kelvin).
In studying the functionality of these many numbers, especially those among the dimensionless constants, we believe this list will evolve and its ordering will change often. In searching the web for more information about about dimensionless constants, we came upon the curious work of Steve Waterman and an emeritus chemistry professor at McGill University in Montreal, Michael Anthony (Tony) Whitehead. I showed their work to a former NIST specialist and now emeritus mathematics professor at Brown University, Philip Davis. He said, “There are always people who wish to sum up or create the world using a few principles. But it turns out that the world is more complicated. At least that’s my opinion. P.J.Davis” Of course, he is right; Einstein did a good job with e=mc2. Because claiming to find all the physical constants derived by using pi, the isoperimetric quotient, close cubic packing and number density is not trivial10, we’ll be taking a second look. Perhaps they are onto something! We have brought their work out in the open to be re-examined and in so doing we will re-examine over 140 physical and mathematical constants. This work is also ongoing.
#10 = The Age of the Universe
13.799±0.021 billion years
This number is important because it creates a boundary condition that is generally recognized for its accuracy throughout the scientific and academic communities. Though it may seem like an impossibly large number of years, it becomes quite approachable using base-2 exponential notation. Without it, there is no necessary order of the notations.
Although there are many different measurements of the age of the universe, for our discussions we will use 13.799±0.021 billion years. The highest estimate based on current research is around 13.82±0.021 years. Also, within this study there are some simple logic problems. In 2013, astrophysicists estimated the age of the oldest known star to be 14.46±0.8 billion years.
Notwithstanding, using base-2 exponential notation all these measurements come within the 201st notation. At the 143rd notation, time is just over one second. Within the next 57 doublings, we are out to the Age of the Universe. So, with the Planck Time as a starting point and the Age of the Universe (and our current time) as the upper boundary, we have a container within which to look for every possible kind of doubling, branching and bifurcation. We can study hierarchies of every kind, every set, group or system. Eventually we can engage holomorphic functions within our larger, ordered context, i.e. the seen-and-unseen universe.11 https://bblu.org/2016/01/08/number/#8
This article was started in December 2015. It’s still in process. Your comments are invited.
Page 9 of 10 EARLY DRAFT.
Endnotes about our open questions, plus a few references:
1 Our Initial exploration of the types of continuity and discontinuity: Continuous-discrete, continuous-quantized, continuous-discontinuous, continuous-derivative… there are many faces of the relations between (1) that which has a simple perfection defined in the most general terms as continuity yet may best understood as the basis of order and (2) that which is discrete, quantized, imperfect, chaotic, disordered or otherwise other than continuous. These are the key relations that open the gateways between the finite and infinite.
Questions: What is a continuum? What is a discrete continuum?
2 We are simple, often naive, mathematicians. We have backed into a rather unique model of the universe. To proceed further we will need to understand much more deeply a diverse array of relatively new concepts to us; we are up for the challenge. We have introduced just a few of those many concepts that attempts to define the very-very small and/or the transformations between the determinant and the indeterminant. There will be more!
3 Of the Top Ten Reasons, the first three given are our first principles. We know it is an unusual view of life and our universe. The sixth reason advocates for a Quiet Expansion of our universe whereby all notations are as active right now as they were in the very earliest moments of the universe. When space and time become derivative, our focus radically changes. It opens a possible place for the Mind down within the small-scale universe. Our current guess is between the 50th and 60th notations. The archetypes of the constituents of our beingness are between notations 67 (fermions) to notation 101 (hair) to notation 116 (the size of a normal adult). Then, we live and have our sensibility within notation 201, the current time, today, the Now. So, this unusual view of the universe has each of us actively involved within all three sections of the universe: small scale, human scale, and large scale. To say that it challenges the imagination is a bit of understatement.
4 Open Questions. There are many open questions throughout this document. It is in process and will surely be for the remainder of my life. All documents associated with this project may be updated at anytime. There should always be the initial date the document was made public and the most recent date it was significantly updated. Although the Feigenbaum constants are our seventh number selected (and there are more links and a little analysis there), we will attempt to find experts who can guide us in the best possible use of these two constants within our studies. Bifurcation, it seems, has an analogous construct to cellular division, to chemical-and-particle bonding, to cellular automaton (especially Rule 110,) and to the 80 categories of binary operations.
5 Wikipedia, ccp, and genius. Jimmy Wales is the founder and CEO of Wikipedia. His goal is to make the world’s knowledge accessible to the world’s people. He has a noble vision within precarious times. In order to be published within Wikipedia, the material has to have its primary sources of information from peer-reviewed publications. As a result, Wikipedia is not where “breakthrough” ideas will first be presented. Blogging areas like WordPress are a more natural spot and Google quickly indexes all those blogging areas. It took only a day before they found this article. So with a little ingenuity one can quickly find many new references to new ideas and then go to Wikipedia to find the experts on that subject. Prior to this research, we had barely scratched the surface of ccp. We did not know about the Feigenbaum constants or Kepler’s conjecture. For sure, we had never seen the cannonball stacking illustration that helped us to visualize the process by which a sphere becomes a lattice, becomes a triangle, and then becomes a tetrahedron. We are quite confident that our first four numbers are the right selections possibly even within the right order. If you believe otherwise, of course, we would love to hear from you.
Page 10 of 10 IN PROCESS
6 A hypostatic science. Our small-scale universe, defined as the first 1/3 of the total notations, ranges from notation 1 to just over 67. It is established only through simple logic and simple mathematics. Because it cannot be measured with standard measuring tools or processes, validating its reality requires a different approach. Because it cannot be measured with standard measuring tools or processes, validating its reality requires a different approach. Our first indication that it may be a reality is found between notations 143 and 144 at exactly one second where the speed of light “can be made” to correspond with the experimental measurement of the distance light travels in a second. Currently it appears to be one notation off which could be as brief as just one Planck Time unit.
One of our next tasks is to carry that out to a maximum number of decimal places for Planck Time and Planck Length, and then to study the correspondence to a Planck second, a Planck hour, a Planck Day-Week-Month, a Planck Light Year, and finally to the Age of the Universe and the Observable Universe.
Our goal is to determine if this is the foundational domain for the human scale and large-scale universe. We are calling this study a hypostatic science because it is a study of the foundations of foundations.
7 From SUSY to Symmetry Breaking and Everything In Between. One of the great hopes of the Standard Model and many of the CERN physicists is that supersymmetries will be affirmed and multiverses will wait. Within the Big Board-little universe model, their wish comes true. Plus, they gain a reason for quantum indeterminacy and embark on a challenge to apply all their hard-earned data acquired to embrace the Standard Model to the most-simple, base-2 model.
Here are four of our references through which we learned about the heretofore unnamed pentastar gap. The Lagarias-Zong article (#4) is where I learned about Aristotle’s mistake.
1. Frank, F. C.; Kasper, J. S. (1958), “Complex alloy structures regarded as sphere packings. I. Definitions and basic principles”, Acta Crystall. 11. and Frank, F. C.; Kasper, J. S. (1959), and “Complex alloy structures regarded as sphere packings. II. Analysis and classification of representative structures”, Acta Crystall. 12.
2. “A model metal potential exhibiting polytetrahedral clusters” by Jonathan P. K. Doye, University Chemical Laboratory, Lensfield Road, Cambridge CB2 1EW, United Kingdom, J. Chem. Phys. 119, 1136 (2003) Compete article, ArXiv.org as a PDF: http://arxiv.org/pdf/cond-mat/0301374
3. “Polyclusters” by the India Institute of Science in Bangalore, illustrations and explanations of crystal structure. PDF: http://met.iisc.ernet.in/~lord/webfiles/clusters/polyclusters.pdf
4. “Mysteries in Packing Regular Tetrahedra” Jeffrey C. Lagarias and Chuanming Zong.
8 Cellular Automaton. Although the discipline is intimately part of computer science, its logic and functions are entirely analogous to mathematical logic, functions, and binary operations. We have just started our studies here with great expectations that some of this work uniquely applies to the first ten notations.
9 The Planck Platform. All the numbers associated with the generation of the Planck Constant and the five Planck base units, plus the Planck units unto themselves are grouped together until we can begin to discern reasons to separate any one number to a notation other than notation 1.
10 The Magic Numbers. Mathematical constants, dimensionless constants and physical constants are studied in relation to the isoperimetric quotient, close-cubic packing, number density and to bifurcation theory and to the 80 categories of binary operations. We will working with the processes developed by geometer, Steve Waterman, and chemistry professor, Michael Anthony Whitehead and the generation of the 142 physical constants.
11 The first 67 notations. Given the work of CERN and our orbiting telescopes, we can see and define most everything within notations 67 to just over 201. The truly unseen-unseen universe, defined only by mathematics and simple logic, are: (1) the dimensionless constants, (2) that which we define as infinite, and (3) the first 60-to-67 notations. It is here we believe isotropy and homogeneity are defined and have their being. It is here we find the explanation for the most basic cosmological constant. It is here the Human Mind takes its place on this grid which claims to include “everything-everywhere-for-all-time.”
Please note: We are still working on this article. A running commentary is being developed within the LinkedIn blogging area for Bruce Camber. Besides editing the overall document, we’ll are still working on the end notes using some of these reference materials.
11 thoughts on “On Constructing the Universe From Scratch”
It is a brave soul who seeks to build the universe from scratch. The logic of mathematics is surely an excellent tool, but hey, we are building from scratch, so like any medieval journeyman we first have to build our toolset: what do we want our math to do, how should we craft it, we must set to and learn to use the tool we have crafted – only then can we attend to the universe itself. Oh, my, but how do we start to think about building mathematics if we have no mathematics to think with yet? Philosophers at the back of the room smile inwardly. Mathematicians pride themselves on mathematical logic and its applications to other areas of thought. Philosophers, who brought those mathematicians into the world thousands of years ago, know better. Philosophical logic is the foundation of all rational thought. Only when we apply it in all its glory to numbers or to the shapes of things does it become mathematical.
So now, we must start deeper in, dig down to the primal mysteries. Strip away the numbers, strip away the shapes of things, what are we left with? When George Boole discovered the logic which now underpins the digital age, he wrote it up in a book called “The Laws of Thought”. Even today, the computer designer’s diagram of digital states is called a truth table. But Boolean algebra is not deep enough in. There are other algebras, and beneath them lie other logics, other modes of rational thought.
Einstein always said that his insights came from asking the simple questions that any child might ask. So here goes: Why is two always bigger than one? If a thing is true today, will it always be true? Might one be bigger than two tomorrow? When I re-read this, why is it the same as when I wrote it? In fact, how do I even know I am re-reading the unchanged text that I wrote yesterday? What is change anyway?
In answering such riddles we find unanswerable rules of being. Here are a few (in which a “thing” is an abstract thing or idea, as we have not got to a physical world yet):
– If two things have exactly the same properties in all respects then they are the same thing.
– If a thing is itself today, it will always be the same thing.
– If two things have different properties, they are different things.
– If an argument is true today then it will always be true.
– If a thing exists then not-that-thing also exists.
For anything of the slightest degree of complexity, these rules must all apply. Break any one of them and the thing collapses in chaos, any hope of meaning is lost. We talk of principles such as identity, consistency, negation and so on. While we seem to understand each principle as something distinct from the others, in practice none can stand without the others. Identity means nothing if it is not identity the next time we consider it. Consistency is impossible if a statement which was true is now suddenly false. There is no halfway house, no single “this is how the process of rational construction is itself built on reason”. It has to begin somewhere, and this is where it begins. We talk of identity, consistency, negation and so on, but at the foundational level each is just a face of the others. From them, all else arises. This is the central mystery of being.
Now let us take a step up, it is time to build mathematics. That seems easy, we might start by observing that Boole’s True and False might also represent 1 and 0 respectively, and bingo! we have all the mathematics that a digital computer can handle. While not every number can be reduced to 0s and 1s, let us gloss over the complications and consider some numbers.
Some, such as zero 0, one 1, the natural number e of logarithms and exponentials, the square root of minus one i, the golden ratio τ upon which the Fibonacci sequence 1,1,2,3,5,8,13… converges and many others, prove of enormous importance, appearing all over the place in our mathematical machinations. Pi π emerges as a function of e, i, 1 and 0: π = (ln(-1))/i, more famously expressed by Leonhard Euler as e^iπ + 1 = 0. When we come to build our universe, these numbers may be regarded as foundational. Nevertheless, we must not forget that they in their turn are founded on the mystery of being and not the other way round.
In passing, we may note that the rational numbers are discontinuous – for continuity you need the real numbers, those infinte seas of the unnamed and unwritten which lie between each and every pair of rationals. Computers with their 0s and 1s can work only with the discontinuous rationals, yet they often contrive to manipulate continuous things. If nothing else, continuity is an elusive idea.
So, now that we have our mathematics, perhaps at last we can begin to build our universe. In passing it is worth noticing how it takes an arbitrary selection from all of mathematics and embodies that selection in something we call the laws of physics. Why this particular selection, why not some other equally rich and satisfying selection from the pile? And so the second great mystery of being is passed by with barely a thought, we press on. And here, there are two significant things we latch on to. First, the big one. The universe appears as a vast fabric of space and time, measured by the travelling of light and warped by the presence of matter. We notice that we can make and use a ruler, marking it with 0 and 1 and their brethren. We start discovering that set of laws, pulled from the pile, which we call geometry. We rediscover π, e and τ.
Our friends have been busy. They latched on to the very small. They found the Planck units, quanta and the standard model of particle physics, and in there numbers like Planck’s constant h and the fine structure constant α. It is just as well that the principle of continuity was kept out of the foundational logic because at this level everything breaks down into quantum foam.
How do these two scales relate? Ah, wouldn’t every physicist love to know!
We humble bystanders can offer only a handful of insights based on our rulers. Here are a few samplers. The universe grew from a tiny beginning at finite speed for a finite time. It must therefore be finite in size. That defines the largest possible wave which can resonate in it, which in turn defines the smallest possible energy quantum which can exist. This is called the Planck unit of energy. Then, this finite universe has a finite density and finite energy. The sum total of positive mass and energy is the maximal conceivable quantum which can exist. This defines the smallest wavelength and highest frequency which can exist, giving us the Planck length and time respectively. A black hole looms large, we are falling uncomfortably close as we pass by. Our geometers set out to measure what they see but quickly become deeply puzzled. One, having recently been thinking a lot about general relativity and the shape of space, cries out, “Hey! pi in a circle isn’t pi any more, it has become a variable dependent on distance from the black hole. Look, here’s a place where the gap I call pentastar has closed and five regular tetrahedra meet exactly around their common edge.” Another begins, “Hey! …” but what they cry is yet to be known. Will it be a physicist with a brand new Grand Unified Theory who shouts it out, or will it be one of us?
Now is a good moment to take time out (Or, save it to the end for I have little left to say here) and read a study in how to approach numbers – old style: http://www.steelpillow.com/blocki/philosophy/antikythera.html
So many numbers in the building of our universe! Which ones matter, where and why and how? It is easy to be lazy, to pick an arbitrary favourite and run with it. One thing we can be confident of – the important numbers in geometry and physics, along with at least some aspects of continuity, are less foundational than the important numbers in pure mathematics or the principles of rational logic. Now that you have read this little essay, you know which numbers come from where and matter the most, and have perhaps even the haziest inkling of why and how.
Now, let’s take another look at that Top Ten list.
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Guy, How does one thank anybody for taking one’s work seriously enough to write such a thoughtful response. I am most gratified. Though I am in the process of commenting on that article within my LinkedIn page — http://linkedin.com/pulse/constructing-universe-from-scratch-bruce-camber — I will respond in greater detail when those reflections are complete. Thanks, Guy. -Bruce
For me the first task in “building from scratch” is to find the most powerful, simple number that has the greatest possibility of being there at the beginning. That number will begin to define the mathematics to engage. That was the three-year discussion with Philip Davis about the simplest three-dimensional shape. I said the tetrahedron. He said the circle. It took me awhile, but I adopted his point of view but didn’t know what to do with it. When Steve Waterman introduced me to cubic-close packing (ccp), and then observing that animated gif within Wikipedia, I finally concluded, “Let’s pivot off pi.”
Yet, to suggest that pi has a dependency on the other constants — “Pi π emerges as a function of e, i, 1 and 0” — doesn’t appear to me as a necessary condition. Can you help me out with your statement, “Pi π emerges as a function of e, i, 1 and 0.” What must I read to see the deeper logic? Thanks. -Bruce
You will find the gist of it here: https://en.wikipedia.org/wiki/Euler%27s_identity
including the quotation that “it is absolutely paradoxical; we cannot understand it, and we don’t know what it means, but we have proved it, and therefore we know it must be the truth”. So the logic does not go a lot deeper, it just comes to a dead stop.
The key point here is that e, i, 0 and 1 are pure abstract numbers defined in other realms of mathematics. Pi has no such abstract mathematical significance elsewhere, it must wait until after these numbers have been placed on the table because in purely abstract terms it derives its significance from them.
So what does this mean for geometry? Well, there are many geometries, some Einsteinian, some not. Some such as our familiar “Euclidean” geometry give pi a special significance, while others do not. For some (hyperbolic), the circumference of a circle is not 2.Pi times the radius but is larger, for others (elliptic) it is smaller, but in both cases the ratio of circumference to radius is heavily scale-dependent. Only in “flat” Euclidean geometry is it independent of scale – but then, if there is one thing we can learn from Einstein, it is that the universe is not flat and Euclidean but curved. So if we wish to build scale-independence into our physics, pi is actually a bad place to start.
Must dash off now. Hope this helps.
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Fun. I had just re-read that reference to Euler’s identity earlier today. Now, I’ll have to go back and sleep on it! Thanks for the challenge, Guy. Just brilliant. -Bruce
Guy, I particularly enjoyed your comment, “But Boolean algebra is not deep enough in. There are other algebras, and beneath them lie other logics, other modes of rational thought.” Certainly at the assumed singularity of the Planck base numbers, we are deep enough in. Yet, nobody knows what is going on at that very mysterious place. I suspect, quite contrary to others, that the answer might be, “Not very much.” There is an infinitesimal amount of room to move. Surely we can discount both the quanta and fluctuations for awhile.
The length is so small and the duration so short, perhaps all that happens in the first 20 or so notations is the expansion of point-free geometries that are optimized by the perfections of the infinite’s continuity, symmetry and harmony. Perhaps all than can happen is pure and simple geometry. Perhaps Euclid is a good place to start. Though I make no pretensions of understanding A.N. Whitehead and mereotopology, the study of the logic of perfection may be the best possible path to deem the nature of this infinitesimal space and time. Perhaps all that we have is pi’s simple definition whereby our best metaphor was given to us by Kepler and his stacking cannonballs exercise. Perhaps here is the convergence from the infinities of pi to the finiteness of the stack. Perhaps like a Turing machine, this perfection just “tests” combinations of point-free vertices. Given at this stage in our exploration, still trying to move off the singularity, perhaps the first doubled numbers will tell us something, give us some clues: https://planckbaseunits.wordpress.com/#Planck
Those values do not tell me much except, of course, the Planck Temperature. Here we find a very small value. The logic of this Big Board -little universe project is that “everything starts simply” and the Planck Temperature is assumed to be the largest possible temperature given the wavelength constraints of the Planck Length, so it was started at the top of the column. Of course, we could discover within this logic of perfection that it is actually starts in notation 202 or higher. More than global warming, perhaps there is such a thing as universe warming. Perhaps the 2.725° K cosmic microwave background radiation (CMBR) and the measurements of the FIRAS experiment have another explanation that can be found within the the 201+ notations.
Silliness perhaps, but this is where we find ourselves today. There is a simple logic that we are following even if it takes us to what initially seem to be illogical conclusions. We will live with them for awhile to see if we can build upon it or discover where it can be adjusted to fit the evidence of experimental data.
The cleanest and most simple geometry is called projective geometry. Where Euclid singles out a special case of “parallel” lines which never meet, projective geometry makes no such complicated special case – all lines meet somewhere. Hyperbolic and elliptic geometries provide their own complications – only projective geometry is clean and simple.
In its purest form projective geometry has no idea of measurement, whether of length or angle: a circle, an ellipse, a parabola or a hyperbola (the conic sections) are indistinguishable from each other. These is no such thing as a square and it is indistinguishable from any and every perspective drawing of a square. There is no Pi in here, no circle to construct the circular functions of sine and cosine.
But there is something deeper and more mysterious, too subtle to explain right now – there are projective measures and there are cross-ratios. From these, some idea of number emerges. But the numbers are not fixed in the space like a ruler glued to the floor, they are mutable and can be constructed anywhere, and once constructed can be moved elsewhere under a projective transformation. Some number lines cycle around a point like the spokes of a wheel, converging in infinite series on some particular line that one has singled out. Others zigzag between the two ends of a ruler, extending from zero in both directions – the odd numbers on one side, the even numbers the other.
It is only when we grab a familiar-looking ruler and declare it both evenly-spaced and universal that we can build Euclidean geometry. And here, we get a little surprise. It makes no difference whether we grab a straight ruler or a circular ruler – a protractor if you like – we can construct each from the other. And at last we discover Pi.
There is a penalty: the straight ruler pushes one particular line in the plane out to infinity – the horizon – and in space it pushes a whole plane out there. Ordinary lines which meet in this infinite place are said to be parallel – this is essentially Euclid’s complication: savagely cut away the prisoner at infinity so that it no longer exists – leaving infinity like the smile of the Cheshire Cat without the cat but, like the end of the rainbow always beyond reach – and you finally reach Euclid. But what a journey! Surely the beginnings of things can’t be that complicated and arbitrary?
I would recommend Lawrence Edwards; Projective Geometry, 2nd edition, Floris Books, 2003. He takes us on a graphical journey through these mysteries and even beyond.
The great HSM Coxeter’s book on the subject is more of a text book, but it also introduces an algebra which can be used to do projective geometry without ever resorting to your ruler, be it real or metaphorical. This reveals geometry at its heart to be pure logic, becoming clothed as geometry only when it gives form to some fabric of space and existence.
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Great. Projective geometry. Lead the way. Euclid’s fifth, that parallel postulate, has caused many to stop and wonder. I’ll get the textbooks suggested (and those related), and we’ll put the article on ice until we can apply new insights from both. In my initial looking into projective geometry, I also looked in on other geometries: Spherical, Non-Euclidean (Elliptic Hyperbolic), Synthetic, Analytic, Algebraic, Riemannian, Differential, Symplectic and Finite. I am always overwhelmed with my lack of depth and so appreciate any and all guidance you can give me and our little team!
Questions: Reading online about projective geometry, what does it look like within the first notation out from the assumed singularity of the Planck Base Units? What is the most simple result of that doubling?
Pure projective geometry has no idea of distance and hence no idea of scale – it is scale-invariant. The simplest projective construction of any significance is a set of seven points and seven lines called the Fano plane. Abstractly, one does not even need to define what a “point” and a “line” are, the logic works just fine: for example the Fano plane can be treated an example of a “discrete” geometry in which a line is just a set of points with nothing in between, not even ordered in any particular sequence. At the Planck level we may wish to treat a point as a fuzzy blob of spacetime or perhaps a tiny loop of spacetime, and a line as a connection or relationship between a sequence of such blobs. But the scale down there is not inherent in the geometry, it is – perversely – created by the upper-scale limits to the Universe as I explained earlier.
In fact, I suspect that the number of doublings from one end of the scale to the other is in fact a measure of the age and expansion rate of our Universe: in the beginning it was tiny and no low-energy waves could exist. The Planck length and time would have been quite large. In fact at the earliest moment science can say anything about it at all, the Planck length would have been the size of the Universe itself, the positive energy of its narrowband quantum radiation balanced out by the negative gravitational energy of its tightly-curved spacetime fabric. One might call the subsequent moment the Age of the First Doubling. Two further doublings would have to have taken place before even the Fano plane could be constructed.
Remember that continuity is such a slippery thing? Well, it seems to emerge as further doublings take place, allowing more complex geometries to create between-ness and hence ordering.
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Great to see your very quick response, Guy. I will be the first to admit, however, that I am going to need time with this response. I have already been through your posting twice and now need to start delving into all the secondary resources that I can find so I understand it better! If and when I do, I’ll respond hopefully in a way that we can begin constructing our universe, or at the very least, I can revise my rank order of those ten numbers and number groups! -Bruce