Prepared by Bruce Camber for five classes of high school geometry students and a sixth-grade class of scientific savants. There are no less than 15 concepts reviewed here. All have been explored within a high school yet have been virtually ignored by the larger academic community. It begs the questions, “Are any of these concepts important? Which should we keep studying and which should be deleted?” And, of course, if we delete any, we need to know why.

Students have been known to ask a rather key question, i.e., “Can’t you make it easier to understand?”

So, in light of the universal pursuit for simplicity, beauty and wholeness, our geometry classes just may have stumbled onto a path where we begin to see all the forces of nature come together in a somewhat simple, beautiful, yet entirely idiosyncratic model. It feels a bit like Alice-in-Wonderland — the entire known universe in 201+ notations or doublings — all tied together with an inherent geometry, an ever-so-simple complexity. The students ask, “Can this somehow be embedded within every thing everywhere?”

#1 Key Question: Is there a deep-seated order within the universe?

Geometry 101: From the Planck Length to the Observable Universe
December 19, 2011: Defining our Parameters and Boundaries

Over 120 high school students and about twenty 6th graders have divided each of the edges  of a tetrahedron in half.  They connected the new vertices to discover four half-sized tetrahedrons in each of the corners and an octahedron in the middle. They did the same with that octahedron and observed the six half-sized octahedrons in each of the corners and eight tetrahedrons, one in each face. We continued this process mathematically about 116 times until we were in the range of the Planck length. We eventually learned that this process is known as base-2 exponential notation. When we discovered-then-compared our work to that of Kees Boeke (Cosmic View, Holland, 1957), we thought base-2 was much more informative, granular, and natural (as in biological reproduction and chemical bonding) than Boeke’s base-10. Plus, our work began with an inherent geometry, not just a process of adding and subtracting zeros.  More… (opens in new tab/window).

#2 What are the smallest and largest possible measurements of a length?

Doublings and Measurement
December 2011: Getting More Results

We had taken those same tetrahedrons with their embedded octahedrons and multiplied them by 2. Within about 90 steps (doublings), we thought we were in the range of the recently-reported findings from Hubble Space Telescope and the Sloan Digital Sky Survey (SDSS III), Baryon Oscillation Spectroscopic Survey (BOSS) measurements (opens in new tab/window) to bring us out to the edges of the observable or known universe. It appeared to us that this perfect conceptual progression of embedded tetrahedrons and octahedrons could readily go from the smallest possible measurement to the largest in less than 209 notations. We decided at the very least it was an excellent way to organize the data in the entire universe.

More questions:  What are the most-simple parameters with which to engage the universe?  Do the geometries (relation/symmetry), base-2 (operations of multiplication or division), and sequence (order/continuity) provide an operational formula for expansion of the operand?

#3  Do these charts in any way reflect the realities within our universe?

 Big Board – little universe and our first Universe Table
2011 -2012

We had also develop a big board (1′ by 5 ‘) upon which to display this progression so we could begin inserting and updating examples from the real world within each notation (domain, doubling, or step). To simplify the look and feel of those listings, we also made a much smaller table (8.5″ x 11″) in September 2012.   The very first, very rough board (December 2011): http://smallbusinessschool.org/page2790.html and within a blog (May 2012): http://doublings.wordpress.com/  Then, we developed the Universe Table based on the board: http://utable.wordpress.com/2013/11/01/1/

Another question: What are the necessary relations between adjacent notations?

#4 How do we prioritize data (calculations), information and insight?  What is wisdom?

202.34 to 205.11: From Joe Kolecki to Jean-Pierre Luminet
May 2012: Getting Some Professional Insight and Confirmation

We consulted with Joe Kolecki, a retired NASA scientist involved with the education of school children. He did a calculation for us and found about 202.34 notations from the smallest to the largest (based on the age of the universe).

We had also consulted with Jean-Pierre Luminet, a French astrophysicist and research director for the CNRS (Centre National de la Recherche Scientifique) of the observatory of Paris-Meudon. He calculated 205.11 notations: http://doublings.wordpress.com/2013/07/09/1/#Footnotes See footnote 5 on this page within doublings.wordpress.com.

The nagging question: What are the necessary relations between adjacent notations (or doublings, layers or steps)?

#5 How does each notation build off the prior notation? Is it geometrical?
An Encounter with Wikipedia
April-May 2012: Grasping the New Realities

We wrote it up for Wikipedia to have a place to collaborate and build out the document with other schools and even universities. But, in May 2012, their review group told us that it was original research. Though there was a clear analogue to base-10 notation from Kees Boeke from 1957, an MIT professor, Steven G. Johnson (he reviews entries for Wikipedia) said that it was “original” research. We begrudgingly accepted his critique:
http://bigboardlittleuniverse.wordpress.com/2013/02/26/1/
The simple math: http://doublings.wordpress.com/2013/04/17/60/

#6 What is perfect and what is imperfect?

Pentastar, Icosahedron, Pentakis Dodecahedron

December 2011 to December 2012: One Year of Insights

We then observed some curious things. First, geometries can get messy very quickly. We were using the five Platonic solids. Starting with the tetrahedron, we quickly discovered that these objects rarely fit perfectly together. The pentastar, five tetrahedrons clustered tightly together, do not perfectly tile space, but leave a gap. This gap has been thoroughly documented yet to the best of our knowledge it was first written up by two mineralogists, Frank & Kaspers, in 1958. In its simplicity, we concluded that this was the beginning of imperfections and it extended out to the 20 tetrahedron cluster also known as the icosahedron, and then out to the 60 tetrahedron cluster (just the outer shell), which is called a Pentakis Dodecahedron. We dubbed these figures, “squishy geometry” because you could actual squish the tetrahedrons together. In a more temperate moment, we dubbed this category of figures a bit more appropriately, “quantum geometry.”

#7   What is the Planck Length? Is it a legitimate concept?

Frank Wilczek, Encouragement from an Authority, December 2012

We consulted Prof. Dr. Frank Wilczek (MIT) regarding his many articles in “Physics Today” about the Planck Length. He assured us that it was a good concept and that the Planck Length could be multiplied by 2. We titled our next entry, “Everything Starts Most Simply. Therefore, Might It Follow That The Planck Length Becomes The Next Big Thing? The current state of affairs in the physics of CERN Labs is anything but simple. We figure if we built things up simply, we might gain a few new insights on the nature of things.

#8  Is life a ratio?  Does it begin with Pi and the circumference of a circle?

Steve Waterman’s polyhedra and mathematics
March 2014: Discovering Others Searching the Boundaries

In December 2013, I sent a note around to an online group of mathematicians, mainly geometers; and of those who responded, Steve Waterman had done some truly original, rather-daunting, work that had certain similarities to Max Planck’s work a century earlier. It was not until a lengthy discussion in April 2014 that I began to understand the simplicity and uniqueness of his extensive work. He had emerged with many, if not most, of the 300+ NIST constants, the gold standard of the sciences. He had used constants in a similar way that Max Planck used the speed of light and the gravitational constant to begin his quest for the Planck Length. Waterman provokes the ratios of known constants to come ever so close to the NIST measurements. His math implies an inherent universal wholeness and he does it with a series of “what if” questions. It took me awhile to grasp his fascinating, far-reaching results:
http://watermanpolyhedron.com/abequalsc2.html
http://www.watermanpolyhedron.com/smallFOUR.html

#9 Is there anybody doing mathematics in any way related to these notations?

 Edward Frenkel and his book, “Love & Math: The Heart of Hidden Reality”

In October 2013, Edward Frenkel’s book, “Love & Math: The Heart of Hidden Reality” became part of our picture. Perhaps this remarkable mathematician can shed light on those areas where we all are weakest. We let him know we had his book and would be reading it to answer simple questions, “Why doesn’t anybody care about this construction? What are we missing? Why are people so sure that the fermion and its extended family represent the smallest-possible measurement of a length, especially in the face of the Planck Length? Why shouldn’t we attempt to think of the Mind and mathematics as representations of those steps between the Planck Length and those within the particle families?”

Through Frenkel’s work we have begun to discover the Langlands Program and its progenitors (i.e. Frobenius) and the current work in areas like sheaves, the categorifications of numbers, and the correlation functions. We have begun to learn about the work of other remarkable mathematicians like Grothendieck, Drinfield, Witten, Kapustin, and so many more.

The most important first-impression was that we could begin to discern the transformations from one notation to the next and possibly even discern the very nature of a vertex.

The expansion of vertices within each doubling has been a challenge for our imaginations and conceptual limitations. Yet, it could be an even greater challenge and far more complex if we were to follow Freeman Dyson’s suggestion. Using base-4 notation for the expansion of the tetrahedrons and base-6 notation for the expansion of the octahedrons, at the 60th notation, there would be a subtotal of 1.329228×10³⁶ for the tetrahedrons and 4.8873678×10⁴⁶ vertices from the octahedrons. Using simple addition that would be:

488,736,780,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000

+                    1,329,228,000,000,000,000,000,000,000,000,000,000,000,000

488,736,780,001,329,228,000,000,000,000,000,000,000,000,000,000,000,000

The base-2 exponentiation is the “simple math” starting point. It is a simple focus on the process called doubling and only accounts for number of times the original Planck Length has been doubled for each notation. If the focus is on objects, after the fourth doubling, there are four expansions to track, base-1 for the sole octahedron within the tetrahedron, base-4 for the tetrahedrons within the tetrahedrons, and base-6 for the octahedrons within the octahedron and base-8 for the tetrahedrons within the octahedron.  Addressing that schema and the results are:

Base-8 tetrahedrons:   1.5324955×10⁵⁴ units

Base-6 octahedrons in the octahedron:  4.8873678×10⁴⁶

#11 What are scaling laws and dimensional analysis?

Freeman Dyson

Mon, Oct 22, 2012

Freeman Dyson, in an email to me (for which he gave me permission to share), suggested the following: “Since space has three dimensions, the number of points goes up by a factor eight (scaling laws and dimensional analysis), not two, when you double the scale.” Of course, we felt we had more than enough vertices with which to contend, so we just multiplied by 2, using the simple analogue from biology or chemistry. Yet, we readily acknowledge that his advice could readily open even more doors for new explorations, so this question is raised and another dimension of our work has been set out before us by a sage of our time!

#12 Key Question: Is the inherent structure of the first 60 notations shared by everything in the universe?

January 2013 to today

#12a   January 2013:  Speculations about the first 60 notations

With our simple logic, it seems that with the diversity of particles and the uniqueness of identity, that the structure could continue to expand right up to the 201+ notations.

However, below that emergence of measurable particles, and their aggregate structures, a simple logic would tell us that there is a cutoff point as you go toward the Planck Length where a deep-seated Form (perhaps notations 3-to-10) and Structure (perhaps notations 11-20) might somehow be shared by every thing in the known universe. With vertices rapidly increasing with every doubling, options begin to manifest for types of Substances (possibly notations 21-to-30), then types of Qualities (perhaps notations 31-to-40), then types of Relations (possibly 41 to 50), and finally types of Systems (possibly 51-to-60). What does that mean? How are we to interpret it? It is on our list to continue to ponder.

#12b   February 2013   Literature survey

We’ve thought about this very, very small reality from the first notation to the 60th. Perhaps it is what Frank Wilczek (MIT) calls the Grid and Roger Penrose (Oxford) calls Conformal Cyclic Cosmology. We just call it the Small-Scale Universe. Actually, in deference to one of my early mentors, we call it the “really-real” Small-Scale Universe. And, because we started with simple geometries, our imaginative notions of this part of our universe appear to be historically explored yet relatively unexplored as a current scientific framework. First, we turned to our six sections: Forms (Eidos), Structures (Ousia), Substances, Qualities, Relations, and Systems (The Mind).

Also, picking up on a suggestion by Philip Davis (NIST, Brown), that the sphere is more fundamental than the tetrahedron, we start with a one-dimensional length, the Planck Length. When it doubles, it becomes a two-dimensional sphere. When it doubles again (4), it becomes a three-dimensional sphere with a tetrahedron within it. When it doubles again (8), we see the octahedron within the tetrahedron. When it doubles again (16), we begin to see the four hexagonal plates within the octahedron. We are projecting all these forms-structures, substances-qualities, relations-and-systems are complexifications of the first two vertices within the first doubling. We further project that there is a transitional area between each of the three scales, Small-Scale Universe, Human-Scale Universe, and Large-Scale Universe and each would include somewhere between 67-to-69 notations.

#12c.     Discovering Quanta Magazine

       May 2014

Amplituhedrons, Euler, and geometries mixing within necessary relations with geometries

We discovered the writings of Natalie Wolchover within Quanta Magazine, quantum geometries, and on the work of Andrew Hodges (Oxford), Jacob Bourjaily (Harvard) and Jeremy England (MIT). We believe these young academics are opening important doors so our simple work that began in and around December 2011 has a larger, current scientific context, not just simple mathematics. Within the excitement and continuing evolution of the Langlands programs, we perceive it all in light of defining a science of transformations between notations. We are now pursuing all the primary references for people working within quantum geometries.

The simplest, smallest, largest experiment, albeit a  thought experiment based on logic, the simplest mathematics (base-2 notation and platonic geometry), and the base Planck Units, quickly opened doors to look at this data in a radically new way. It will slowly become the basis for many new science fair projects.  The question is asked, “Could This Be The Smallest-Biggest-Simplest Scientific Experiment?” http://walktheplanck.wordpress.com/2014/03/03/domain/

#13a  October 2013 to February 2014: A National Science Fair Project

Some students wanted to take the project further. Here was an initial entry of one of our brighter students:

http://walktheplanck.wordpress.com/2013/12/03/welcome/

#13b January 2012: Is there a concrescence in the middle?

Is the ratio, 1:2, somehow special? Approximately between 101 and 103, clustered in the middle by the width of a hair, are paper upon which we document our history and the human egg. Perfectly human representations in the middle of this scale became a source for some reflections.

http://walktheplanck.wordpress.com/2013/12/03/c/

#13c   October 2013: Considering the Thirds, 1:3

Between Notation 66-to-67 and from 132-to-134:

The significance of the first third, particularly the transformation from the small scale to the human scale, was obvious — particles and atoms. The last third, the human scale to the large scale, we played with ideas, then made an hypothesis. In a most speculative gesture, turning to the Einstein-Rosen bridges and tunnels, we posited that range as a place to begin looking for wormholes.

http://walktheplanck.wordpress.com/2013/12/03/j/

We are now studying the fourths, fifths, sixths and sevenths… wondering in what ways are there parallels to music. How do things combine, mix, and move together to create a specific thing or a new thing? We began studying the notational ranges defined by simple mathematics and music to see what we could see.

Notational range for The Fourths: 50.6 – 51.3, 101.17 – 102.6, and 151.7 – 153.8 and finally 202.34 – 205.11 Notational range for The Fifths: 40.47 – 41.2, 80.94 – 82.4, 121.41-123.6, 161.86 – 164.8… Notational range for The Sixths: 33.72 – 34.35, 67.44 – 68.70, 101.17 -102.6, 134.89 – 136.95, 168.61 – 171.30… Notational range for The Sevenths: 28.62 – 29.30, 57.24 – 58.60, 86.46 – 87.90, 114.48 – 117.20…

To date, our very cursory, initial observations have not opened up more wild-and-crazy speculations! However, the obvious parallel to music has us thinking about the nature of chord, half notes and ratios (July 2014).

#14 Who are we and where did we come from?
1971-1973: Synectics, Polymorphs, Colloquiums, and more
Continuity-Order, Symmetry-Relations, Harmony-Dynamics

We are products of our experience. In 1971, when I (Bruce Camber) was just 24 years old, though active in the radical-liberal political community, my longstanding intellectual curiosity was the nature of creativity, the processes for problem-solving, the nature of a paradigm, and the stuff of scientific revolutions. At a think tank in Cambridge, I focused on interiority, analogies, empathy, and processes to open pathways to a deeper sense of knowing and insight. Within a Harvard study group, the Philomorphs, I studied basic geometric structures with Arthur Loeb. At Boston University, I was deeply involved with the weekly sessions of the Boston Studies in the Philosophy of Science with Robert S. Cohen, chairman of the Physics Department. It was within this mix, that the form-and-function of a momentary perfected state in space and time was engaged (continuity-order, symmetry-relations, harmony-dynamics). For many years, that formulation drove my studies to the point of ignoring all else. Now, years later, that work continues.

#15 Where are we going?  What is the meaning and value of life?

This Day and beyond   The Derivative Nature of Space and Time

Some of us have come to believe that space is derivative of geometry and time derivative of number… and all things as things are unique ratios between the two. Of course, we continue to ask ourselves, “So? What does that mean and what do we do with it?” And, as you might suspect, we have far more questions than we have insights. We are way out on the edges looking for new meaning in this universe. The inquiring minds of our most inquisitive students, want to go further,”Maybe we can find a path to a multiverse! “