CENTER FOR PERFECTION STUDIES: CONTINUITY•SYMMETRY•HARMONY GOALS.March.2022
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An unusual STEM tool from a 2011 high school geometry class is based on simple geometries and mathematics. This study began when students and teachers divided the edges of a tetrahedron in half and connected the new vertices. A little like Zeno’s paradox of dividing by 2, going within, there are 112 steps to get to the Planck Length. Going out, there are just 90 steps (doublings) to the Age and approximate size of the Universe today. That chart of just 202 base-2 notations maps the universe from the first moment until this day. Logically and by definition, this map or model starts with something that is archetypal and primordial at the very first moment of time and then goes out to encapsulate everything, everywhere, for all time. So comprehensive, it has taken years of thought and analysis to identify key components of the model and those key components are now introduced for further analysis. What started as a simple exploration that created a STEM tool is now an avenue for lifelong learning.
Key words: STEM, STEM education, physics education, base-2 notation, Platonic solids, close cubic packing of equal spheres, sphere dynamics, pi, finite-infinite, topology, combinatorics
Plato’s five solids, the first concepts to define space, are often covered within a single session of our Geometry 101 studies. By asking several key questions, that one session became a STEM tool for lifelong learning. The first key question was, “How far within the tetrahedron can one go by dividing its edges by 2 and connecting those new vertices?” The class had constructed a model of the tetrahedron with three layers of measurements. The first layer is the simple tetrahedron.
The first division yields four tetrahedrons, one in each corner, and an octahedron in the middle. Dividing the edges of the octahedron in half, there are smaller octahedrons in the six corners and eight tetrahedrons, one in each face. We continued building many different configurations. See close-up image.
On paper, the class made forty-two (42) additional calculations and the edge of the tetrahedron was now within the CERN-scale of particle physics. In sixty-seven (67) additional calculations, the edge of the tetrahedron was within the Planck scale. The results of dividing or multiplying by 2, we learned, is called base-2 notation. 
At some notations we identified an application for a natural doubling. We have come to believe that for most notations there will be a discernible application, a dynamic for doubling that goes beyond our simple mathematics from stacking. The prime number notations deserve special study.
The class then used the ISO calculation for the Planck Length for the first measurement of a length. In 112 doublings that progression returned the students to the classroom. They continued to double that multiple of the Planck Length. Within just 90 additional doublings, the length was out to the estimated size of the universe. When Planck Time was added, that 202nd notation had also encapsulated the estimated age of the universe at 13.81 billion years.
The two other Planck base units for mass and charge were added. The logic of that expansion opened many new questions about continuity equations, symmetry relations, and sphere dynamics. As these studies continue, anticipated are also harmonic functions that dynamically link each abutting notation and Notation-0 to Notation-202. There is a causal efficacy. Its inherent order predefines each emerging number. Its inherent relations create geometries, and its inherent dynamics define a relational nexus between those numbers and all possible geometries.
Clear plastic models of those Platonic geometries provided visualizations. Among our building blocks the tetrahedron, the octahedron, and the icosahedron  were primary. The emerging numbers of the Planck base units quickly filled out the first chart: https://81018.com/big-board/.
It took a few years to believe that our first chart was an original . Then more numbers from other Planck base units quickly filled out additional charts. Our first “desktop version” was obviously an original .
Our current working chart emerged in 2016. It was our first horizontal chart, 34 pages wide. It helped us to follow the numbers more readily. The most-original version of that chart, an upgrade on our tenth anniversary, December 19, 2021, includes a start with the infinitely-hot Planck Temperature. If an infinitesimally-short blast of light, we hypothesize that cooling follows the inverse square law so that the quark-gluon plasma appropriately manifests. 
2. Research Methods.
2a. Multiplication and division by 2. This model of the universe is a map with 202 layers that start with the Planck base units. The four basic continuity equations, starting at the Planck scale, go out to this current time. In so doing, mathematics encapsulates the universe from the first moment of time to the current moment of time, the Now.
2b. Embedded geometries. This project had begun by dividing the edges of a tetrahedron in half. To begin to grasp the origins of the Platonic solids starting with the tetrahedron and octahedron, we began our studies of cubic-close packing of equal spheres  (there’s so much more to learn).
To understand sphere dynamics, we need to more deeply understand pi, harmonic functions, infinitesimals, and the finite-infinite relation.
2c. Sphere dynamics. Key evocative questions opened even more searches well beyond our studies of geometry. The chart progressed from the study of emergence to structure formation (including Langlands programs and string theory), then to particles and their hypothetical particles, and onto chemistry, biology, systems theory, and cosmology. The foremost criterion within this research is the simple logic of continuity-symmetry-harmony 
Base-2 exponential notation opens a diversity of bifurcation and fractal studies. Sphere stacking and cubic-close packing of equal spheres open pathways to deeper tetrahedral and octahedral studies, including an analysis of the tetrahedral gap created by five tetrahedrons sharing a common edge. It is now hypothesized that the gap is the first instantiation of quantum physics and it is now hypothesized that quantum dynamics could manifest before Notation-64, however, today’s laboratory-based fluctuations are first measured at lengths and durations orders of magnitude larger than those within Notation-64.
The focus on Planck Time opened questions about a cosmological constant defined as “one Planck sphere per unit of PlanckTime and Planck Length.” Using Planck’s numbers, it computes to 539 tredecillion spheres per second.  Using George Johnstone Stoney’s 1874 base units, it would be as high as 4605 tredecillion spheres per second. 
In 2011 when we emerged with the 202 steps, we slowly learned that it was a base-2 progression. We had followed embedded geometries to get down into the Planck scale. It seemed a bit of simple logic to turn around and use the Planck Length (and in 2013 add Planck Time). There was a profound relation between the two. We knew there was a profound relation with Planck Mass and Planck Charge; Einstein had done that work. Now it seemed the right time to figure out how the four all related. The resulting numbers as the four base units expand together the same 13.81 billion years later stretched our understanding of things and strained our logic, yet there is a compelling coherence and our on-going analysis. 
We discovered the 1957 work of Kees Boeke, a base-10 map of the universe. It became apparent that his work, Cosmic View: The Universe in 40 Jumps, challenged people for the first time in history to see the universe holistically-and-mathematically from the first moment to the current time. Books were written about it. An IMAX movie was made. Yet, at no time was casual efficacy ascribed to these 40 jumps, as Boeke called them. Nature has not revealed any causal efficacies whereby a transition is naturally made by multiplying or dividing by 10.
In 2011 when we emerged with the 202 steps, it seemed to be the best-possible, high school STEM tool. It defined the outside boundaries. The parameters were dynamic. It logically included everything, everywhere for all time. Although puzzled by the first 64 notations, it seemed like the penultimate Science-Technology-Engineering-Math (STEM) tool. 
After four years searching for other base-2 studies within articles, books, and the worldwide web, we decided the thrust of this work was original and it was time to consolidate our many pages on many sites around the web into one website. We obtained the address —- http://81018.com — whereby “8” was for our understanding of infinity. “1” was for the finite, unity and singularity and “0” was for transformation. It became our on-going work area. Although entirely idiosyncratic and not readily embraced by the scholarly or scientific community, this study had an inherent logic and it was a simple map of the universe that even our 6th grade students understood.
The sphere or tetrahedron. If something manifests per each unit of PlanckTime and Planck Length, it would have to be simple. At that time I was in discussions with Brown University applied mathematics professor, Phil Davis. He had been a mathematician for the National Institute for Standards and Technology (and for its precursor organizations). Phil proposed the sphere. I had proposed the tetrahedron. Once introduced to cubic-close packing of equal spheres and by seeing just how tetrahedrons and octahedrons were generated, in May 2012, I finally agreed with Prof. Dr. Davis. For me, it was an essential moment in time. 
In 2015 Steve Strogatz of Cornell reintroduced me to the Fourier transform through his article in The New Yorker. It was for Pi Day 2015. Pi’s importance had never been so heightened. 
The place and importance of the sphere was established, so the question was asked, “How do spheres come to be?” Looking at the never-ending, never-repeating, always changing but-always-the-same, numbers resulting from an enigmatic-but-simple formula for a circle and sphere, it becomes obvious that this continuity equation is not of the physical world. It is not finite. Though it is all numbers, it is not just quantitative, but qualitative, too. So, if it is not fully finite, is it infinite? If it is not infinite, is it a facet of infinity or a bridge between the finite and infinite?
Looking further at the sphere, its symmetries are ever apparent. Although a facet of every sphere, symmetry per se is not just a geometric facet; it is qualitative facet. It’s not just quantitative. Is it finite or infinite? Although it may be both, it seems fundamentally more infinite than finite. That third facet of the sphere, its harmonic functions, are measurable, however, the very nature of harmony is not finite. It is always dynamic and appears more infinite than finite.
We will discuss and argue about these three facets of infinity. Each given within the simple sphere, it becomes obvious that we all need to re-engage the finite-infinite relation.
In 2011 the first chart, a 60″ by 11″ board that started with the Planck Length and went to the Observable Universe, was readily shared with other secondary school math teachers. It seemed like the perfect little STEM tool because it touched the primary academic disciplines. It had a simple logic. Yes, even our sixth-grade science classes understood it! But, Scientific American ignored this project and our questions. Wikipedia rejected our summary as “original research.” The AAAS and their Science magazine, as well as Nature magazine, all rejected us without comment. A diversity of scholars were reluctant to comment. So, assuming an egregious mistake within our logic and math, the question continues to be asked, “What are we doing wrong?” Our simple plea for help: “A small group of high school students and a few of their teachers has been trying to figure out what to do with an all-encompassing-but-simple mathematical and geometrical model. Findings to date are presented with the hope that the academic-scientific community can tell us how best to proceed with our very simple charts.”
We had dubbed our little project, Big Board-little universe.
In the earliest days of this exploration, it was not clear where to stop. Each notation had three of four numbers, so experts, NASA’s Joe Kolecki, and the director of the Paris Observatory, Jean-Pierre Luminet, helped us with our calculations.
The Planck Time numbers tracked well with the Planck Length. Also, the Age of the Universe, 13.81 to 14.1 billion years, was no longer a mysterious number and it gave us a place to stop “multiplying by 2.”
We then discovered that the ratio of of the Planck Length to Planck Time within each of the 202 notations was always within .1% of the speed of light. We quickly asked, “What does that mean?” and knew it would be an open question for a while.
Finally in 2015, we then added the other two Planck base units to the chart, Planck Mass and Planck Charge.
There were so many things to discover, our heads began spinning. It all continues to be a challenge. Each notation is filled with data to analyze. Although each chart is a highly efficient way to organize vast amounts of information, these charts raise rather fascinating questions.
At first we wondered why we couldn’t find some vestiges of these charts within our textbooks or someplace on the World Wide Web. Then, when Wikipedia rejected our article as “original research,” we asked, “Isn’t all this information somewhere within the academic world?” Stepping back from our charts, we asked, ”Isn’t each column of the chart a very basic continuity equation from a Planck base unit to its largest possible measurement? Isn’t continuity the bedrock of order? Shouldn’t this be the first principle within our work?”
The small numbers were impossibly small and the large numbers were impossibly large, yet the 202 notations were relatively manageable. There was always one nagging question: “Is there a problem with our logic and math?” It was exponential notation that helped us get comfortable with both extremes and it helped make these numbers more manageable. When we learned a little about Leonhard Euler’s equation, we decided that we lived in an exponential universe! It has taken time. It has been a steep learning curve; we now have feelings or intuitions about the very nature of a number!
We also realized that our universe was perpetually starting. Every notation was always active. The 202nd notation was the only one that was not perfectly symmetrical. It was in process; there’s an arrow of time within it. So, what does sleep and the mind have to do with it? Is it like a computer program in need of recompiling?
What more could these numbers tell us about the universe and ourselves? The geometries started simple, but became exceedingly complex. We asked, “What is geometry? How is space necessarily defined? Does it require all the Planck base units? Does it require the extended Planck units?”
The human family seems to dominate the middle of this chart, yet the time epoch for humanity’s existence is entirely within a very small slice within Notation-202. What is the correlation, the working relation, between the current time and the other notations? If all notations are concurrent, active and forever, what does it say about the nature of space and time? When the chart is divided into thirds, the small-scale universe is extremely small. It goes from the Planck Length to about the size of the quark.
This particular view of the small-scale universe is virtually unknown yet it has a substantial amount of data waiting to be properly analyzed. We reached out to many of the finest scholars for their inputs. Everybody seemed puzzled.
The human scale and large scale did not seem to challenge our simple logic until the “time line” was observed, particularly the figures at one second. What does it mean that the Planck Length multiple is the distance light travels in a second? Well over two-thirds of all the notations are within that first second and within an area defined by the earth to the moon. What does that tell us?
Nobody seemed to know what to do with these charts. So, to get some scrutiny, online articles, blogs and emails, were written. Feedback has been limited. How can that change?
We did get some very helpful suggestions. In 2013 Prof. Dr. Freeman Dyson recommended that we use dimensional analysis and scaling laws to determine the number of possible vertices starting at the Planck base units. We did. The numbers became extremely large rather quickly; nevertheless, because these first 64+ notations were not on anybody’s charts of known things within space and time, we concluded that these vertices must be shared by the entire universe and have something to do with homogeneity, isotropy, the very nature of symmetry and the symmetry of nature, and the cosmological constant.
It was easy to ask ourselves, “Are we crazy or what?”
These numbers, geometries, equations, and charts appear to be a reasonable STEM tool. Elementary students have followed these numbers and appreciated seeing the universe all inter-related on a single page. A holistic view of the universe replaced smaller worldviews. Yet, these numbers, geometries and equations hold much greater potential to address historic problems between the silos of information that divide the sciences. At the same time, it will be helpful to have a larger community to critically engage the intuitions, conjectures, and rather-wild speculations that this map-and-model seems to stimulate. We need to learn if-when-and-how our simple logic and simple math have failed us. We are now trying to understand how continuity-order and symmetry-relations, then harmony-dynamics, are truly bedrock principles of Science, Technology, Engineering, and Mathematics.
By December 2012 a few of our best students, recently-graduated seniors, now students at Tulane and Loyola just down the street, dropped back by their favorite high school. They told us that their professors had never seen a base-2 progression like ours. The students were confused and so were we. It did not take too long to realize that this work was indeed out of the mainstream. Our little mathematical and geometric model even seemed a bit seditious. It raised too many questions. Within a year, we became cautious. Within another year, even more cautious, and within three years, we stopped teaching our students about this model. At that time this project became a cause, “We’ve got to figure out what’s going on here.” So, we continue working at it. The future?
We see a day when every student is challenged to grasp the foundations of STEM so every one of them can become a STEM teacher. Everyone is empowered to expand the horizons of family, friends, and colleagues to be inclusive, creative, and open to the greatest possible diversities in life.
Let’s build a discussion group. It’s time to find a professional publication about STEM and submit an article. We found a few publications:
- Journal of Physics: Conference Series, IOP Physics, Trends and Research Issues of STEM Education: A Review of Academic Publications from 2007 to 2017, Paranee Chomphuphra, Pawat Chaipidech and Chokchai Yuenyong, IOP Publishing Ltd, 2018
- Journal of STEM Education: https://www.jstem.org/.
- Journal of STEM Teacher Education: https://ir.library.illinoisstate.edu/jste/
- International Journal of STEM Education, an article-processing charge of £1190.00/$1690.0
- International Journal of Science and Mathematics Education (Springer)
- and more to come…
Thank you very much. –BEC
Click on the bracketed number to return to the body of the footnote or endnote.
 Laurie E. Bass, Randall Charles, Dan Kennedy, Geometry, Pearson Prentice Hall, 2004, page 517
 Conway, John H., and Guy, Richard K., The Book of Numbers, Springer, 1996
 R. Buckminster Fuller, Synergetics I & II (PDF), Macmillan Co., 1979
 An original base-2 chart of the universe. Using Planck Length, this chart was first used in high school geometry classes, Monday, December 19, 2011.
 Universe Table: An original, the Universe Table was introduced on the web in December 2013.
 Planck Temperature: The original 2016, horizontally-scrolled, base-2 chart of the universe (using Planck Time, Planck Length, Planck Mass and Planck Charge) was updated so Planck Temperature, as calculated by Max Planck, began at Notation-0. It was first on the web on Sunday, December 19, 2021.
 Conway, John Horton; Sloane, Neil J. A., Sphere packings, lattices, and groups (PDF), Springer, Section 6.3. ISBN 9780387985855, 1999, ResearchGate PDF
Endnote: Cubic-close packing of equal spheres. Wikipedia: accessed March 4, 2022. Also, see the many references to cubic-close packing of equal spheres within this website and to the structure of elements in the periodic table.
 Continuity-symmetry-harmony: An original formulation for the primary facets of infinity.
 539 tredecillion spheres per second. An original interpretation based on Max Planck’s calculation of Planck Time at 5.391 16(13) × 10−44 seconds whereby one infinitesimal sphere manifests per Plancksecond. That sphere is sometimes referred to as a Plancksphere. That assumption renders 539 tredecillion spheres per second.
 4605 tredecillion spheres per second. An original interpretation based on the work of George Johnston Stoney in 1874, the earliest known calculation of the smallest-possible units of time.
 The basic logic of the expansion. This original model is subject to constant analysis and revision. There is an abundance of original insights and information that has come out of it and will continue to come out of it. This study of the logic of the expansion is an early analysis that was first presented here in August 2017: https://81018.com/planck_universe/
 Human logic imposed on nature. Edward Zalta, Mechanizing Principia Logico-Metaphysica in Functional Type Theory, Nov. 2017
 Science-Technology-Engineering-Mathematics (STEM). Retrieved from Wikipedia, March 7, 2022
 Essential Moment in Time. An original concept, as applied within the 202 base-2 notations whereby there is concurrence and transformation across notations for an individual, family, community, nation, or all people residing on earth.
 Steven Strogatz, Why Pi Matters, The New Yorker, March 13, 2015
• Bell, E. T. , Finite or Infinite?, Philosophy of Science, The University of Chicago, PressVol. 1, No. 1, 1934, pp. 30-49 (20 pages)
• O’Sullivan, Simon, “A Diagram of the Finite-Infinite Relation: Towards a Bergsonian Production of Subjectivity.” Bergson and the Art of Immanence: Painting, Photography, Film, Performance, edited by John Mullarkey and Charlotte de Mille, Edinburgh University Press, 2013, pp. 165–186, https://www.simonosullivan.net/articles/bergsonian-production-of-subjectivity.pdf
• Wang, Xijia, Cosmic Continuum Theory: A New Idea on Hilbert’s Sixth Problem, Journal of Modern Physics, Vol.9 No.6, 2018, DOI: 10.4236/jmp.2018.96074
Also see: Some Calculations and Thoughts Regarding Measurement by Joe Kolecki, NASA scientist, retired: https://81018.com/kolecki/ and Jean-Pierre Luminet.
• The original 2012 Wikipedia article: https://81018.com/2012/05/05/wikipedia/
• Euler, Poincare, Fourier, Gauss, Karl Schwarzschild and Arthur Holly Compton
• The Planck Temperature – Absolute Hot: What is the hottest temperature possible (YouTube)
Disclosures: There are no competing interests regarding this data or the use of this data. All data and charts originated with the authors and there have been no rights granted regarding any of this data or the resulting charts.
Data access: This project began with equations, numbers, charts and geometries. All the data is available online within the website: https://81018.com/
The working chart is here: https://81018.com/chart/
The first chart is here: https://81018.com/big-board/
Ethics: No human subjects or animals or insects or other living things have been used in these studies. The studies are based solely on logic, mathematics, equations, and geometries. Of interest, however, is that these studies have emerged with its own unique definition of ethics: https://81018.com/ethics/
Funding: This work has received no funding from any sources. It has been an intellectual exercise whereby the authors were glad to do their work without any payments whatsoever.
Many pages within this site start out as an email. The following email was sent to several key scholars around the world and it is now emerging as a page here.
General: We all need to see this world in a very different way:
1. There is a necessary and dynamic finite-infinite relation. Just look at pi. This oldest, most-used equation in the world, defines spheres and circles yet it is not entirely finite. There are three key dynamic facets of infinity within every sphere and circle: continuity-symmetry-harmony and that’s a paradigm shift.
2. The Planck scale is assumed to be the first manifestation of space-time. It begs the question, “What does it look like?” Necessarily the answer is an infinitesimal sphere.
3. One infinitesimal sphere per Planck Length and Planck Time. The universe starts and grows. Cubic-close packing of equal spheres provides a mechanism for generating the Platonic solids, Riemannian geometries, Langlands programs, strings, SUSY, and quantum fluctuations.
4. Apply base-2 to the expansion. We emerge with 202 notations that encapsulate the universe from the very beginning of time to this point in time.
8:14 PM · Mar 7, 2022 @elonmusk We all need to be working together: https://81018.com It is a simple model of the universe. It could make us a little wiser. Even Putin. -Bruce
10:17 AM · Mar 9, 2022. @randizuckerberg You may do well to see your tokens in light of an integrated, mathematical view of the universe. Using base-2, there are only 202 notations (doublings) from the Planck base units to this day. https://81018.com is a start.
10:38 AM · Mar 9, 2022 @dinakaplan We are held back because we do not know our boundaries from the first moment in space-time to this moment. To grasp those boundaries, apply base-2 notation to the Planck base units and in 202 doublings you’ve got the universe. Here’s a start: https://81018.com
- A key in a four-page cluster: (1) /begin-here/ (2) /ultimatum/ (3) /checklist/
- The URL: https://81018.com/stem/
- This page was initiated on February 25, 2022 (built on https://81018.com/stem-letter/ )
- Next homepage: https://81018.com/begin-here/
- Prior homepage: https://81018.com/pi day march 14/
- Another prior homepage: https://81018.com/primordial/
- First Tagline: We started our first chart in 2011 and quickly turned to scholars for help.
- Current tagline: A simple explanation about the start of the universe
- The most recent update of this page: Thursday, September 29, 2022
(for our visitors from Nottingham University today)