Concepts & Parameters



Sometimes the most simple concepts elude us.

Would you agree that two of the most simple parameters of science and mathematics are (1) smallest-to-largest and (2) multiplying-and-dividing by-2? Then, why, as a well-educated general public, do we not know the smallest and largest lengths of measurement?

We will assume that Max Planck was right in 1900 when he calculated the Planck Length. Certainly it is well-worth the time to study its derivation; however, for this discussion, take it as a given that the Planck length is the smallest unit of measurement of length.

The number is 1.61619926 × 10-35 meters.

The largest measurement of length is an on-going research effort; and although there are many different conclusions, there is a general direction and concurrence within the scientific community. Among the more recent calculations, we turned to the Sloan Digital Sky Surveys (SDSS-III), particularly her Baryon Oscillation Spectroscopic Survey (BOSS) measurements from March 2012 to establish a working range from the smallest-to-the-largest lengths. We are simply taking it as a given that the SDSS BOSS measurement is close enough. They have confirmed earlier statements, i.e. “The universe is 13.75 billion years old.” Others are as high as 13.798.

You would think it is a rather straightforward conversion, yet, even those calculations (convert years to a length) are diverse.

With their Scale of the Universe Cary & Michael Huang suggest 9.3×1026 meters (93 trillion light years). Paul Halpern, physics professor and author of the book, Edge of the Universe: Voyage to the Cosmic Horizon and Beyond, told me (email) that a better figure is 4.3×1026 m.

Yet, that end figure is not as important as the starting figure, it just helps to know when we are getting close to it.

To create a simple relation between everything in the universe, take the Planck Length and multiply it by two, then continue multiplying each result by 2 until we are out to that largest measurement. It is a natural progression much like the unfolding of life within cellular division.

So, based on their calculations, how many times will we multiply by 2 to go from the smallest to the largest measurement lengths in the universe?

That process is called base-2 exponential notation and the answers are quite surprising:

Notwithstanding, that range, 202.34 to 206, is a very small number of doublings (notations, layers or steps) from the smallest to the largest measurement of a unit of length.

It is a cause for wonder. At the 202nd doubling of the Planck Length, the measurement is 1.03885326×1026 meters. At the 203rd doubling it is 2.07770658×1026 meters. And, at the 204th it is 4.15541315×1026 meters and the 205th doubled to 8.31082608×1026 meters. Within respectable universities they have used a number as high as 2×1028 meters!

202.34 to 206 doublings from the smallest measurement of a length to the largest. It is quite fascinating to find the thickness of a human hair (around 40 microns) at notation 101, the thickness of paper at 102, and diameter of an egg cell at 103.

Step back and look at the board. Here we have the universe, all of it, within a natural ordering sequence and we can begin to see relations between everything. A high school geometry class explored this simple model in December 2011 and we have been trying to understand why we have not found it anywhere in the academic community. It seems to be an oversight. We were studying nested geometries, starting with the tetrahedron, discovering the octahedron and four tetrahedrons inside it. Then the students discovered the half-sized objects inside the octahedron — six octahedrons and eight tetrahedrons. It was this progression that opened the smallest-largest question.

The purpose of the many discussions within these pages of the Big Board – little universe is to encourage students to explore each notation to see what is unique within it, to grasp the parameters and boundary conditions that could define each notation, to consider possible transformations between notations, to see how the constants and universals work within each notation, to grasp as many new concepts, ideas and insights as possible, and then to attempt to relate those insights to its smaller notation and then to its larger notation.

That is a lot of work.

So first, we will use it as a simple way to order information. Then, we will see if each notation opens new areas for speculations and analysis — the first sixty steps have never been critically explored. If just the simplest geometry of tetrahedrons and octahedrons is used, we have a simple structure for coherence throughout the universe in less than 206 notations. If we systematically add and analyze layers of complexity, more areas are opened to explore.

That is why these pages are here. Let’s explore the universe in the simplest ways possible, then ask, “What difference does it make?”

Editor’s Note: This page was last updated on May 12, 2013.

1. Go to an Introduction & Overview

2. A Colorful Image of the Big Board – little universe

3. A Wiki-like Overview

4. Current analysis

5. Big Board – little universe, the chart


An Architecture for Integrative Systems

MITA speculative conceptual frame of reference

by Bruce Camber (this page was first posted on the web in 2008)

Who would disagree with the observation that our world has deep and seemingly unsolvable problems? It is obvious there is something missing. So, what is it? Is it ethics, morality, common sense, patience, virtues like charity, hope and love?  We have hundreds of thousands of books, organizations and thoughtful people who extol all of these and more.  The lists are robust.  The work is compelling, but obviously it is not quite compelling enough.

Everybody seems to have their own unique spin to solve the world’s problems. Yet, we have discovered that one person’s spin does not easily integrate with another. Listen to those with their finger on nuclear triggers and those who are trying to be among them.  Thoughtful people in every part of the globe are deeply concerned.

In 1977 I was smitten with some of the insights of a theoretical physicist, David Bohm. He gathered a group of graduate students together to be like a child to examine everything we knew about points, lines, triangles and tetrahedrons. We were trying to discern what makes for fragmentation and what makes for wholeness.

In 1979 I proposed and developed a display project at MIT to focus on first principles within the major academic disciplines. For that project, I wrote, “The human future is becoming increasingly complex and problematical.  Proposals for redirecting human energies toward basic, realizable, and global values appear simplistic. Nevertheless, the need for such a vision is obvious.”

The focus was on cross-disciplinary scholarship of leading thinkers around the world who were attempting to define a more integrative and comprehensive understanding of physical nature and of human nature.  There were 77 scholars selected.

Now, well over 30 years later,  progress has been slow. Many of that original group have died without having actualized their dream of defining a bigger, more inclusive vision. There is an obvious bottleneck somewhere. And,  that is what this article seeks to address.

I believe a simple conceptual bottleneck that has been starring at us for many, many centuries exists in pure geometry.  I may be totally mistaken, but  I do not believe our best scholars throughout time and around the world have not fully analyzed three very simple, basic questions:

1.  What are the simplest three-dimensional structures?

2.  What is most simply and perfectly enclosed within those structures?

3.  What is most simply and perfectly enclosed within each of those parts?

Since 1994 I have asked literally hundreds of people those three questions. Chemists, biologists, architects, mathematicians, physicists, crystallographers, geologists, and geometers — few had quick answers.  Only one had an answer to the third question.

The tetrahedron.  The answer to the first question is the basic building block of biology, chemistry, geometry and physics.  The answer is the tetrahedron. Many, many people answered that question.  The tetrahedron has four sides and is made of four equilateral triangles.  It is not a pyramid  (that has a square base and it is half of an octahedron).

What is perfectly enclosed within the tetrahedron? The answer to this second question eluded most people.  To figure out the simple-perfect answer,  divide each of the six edges of the tetrahedron in half and connect the points.  You will quickly see a tetrahedron in each of the four corners, but, there is a middle object and it often requires a model to see it.  You will discover the octahedron, four of its faces are the “middle”  face of the tetrahedron, and four are interior.

The octahedron. The answer to this third question requires a quick analysis of the octahedron .  Only one person knew the answer to the question, “What is perfectly enclosed within an octahedron?”  Yet, he hesitated and said, “Let’s figure it out.”  That was Princeton professor, John Conway, who invented surreal numbers and is one of the most renown geometers living in the world today.

Here are the two most basic structures in the physical world and most people do not know what objects are most simply enclosed by each.  Yet, these are simple exercises. School children should have quick answers to all three questions.

When questioned about my focus on this gateway to interior space, my standard answer is, “…because we do not know.”  And, as I look through the history of knowledge, I do not know why it hasn’t been part of our education. It is too simple.

This simplicity became the basis for my first principles.

Why pursue this domain of information?

First, it is there to be examined. It is what is. This is not speculative. It just is. Second, it is truly rich with more information. Third, and here I’ll be speculative, it just may open a door to some of the most basic, unanswered academic questions that, if answered, might build bridges and open new ways to an integrative understanding of life (this link goes to such a door that opened on December 19, 2011).

I will predict that once more of the complexity-yet-simplicity of these basic interior relations are discerned, the mathematics will follow and these forms will beget new functions as we discovered within nanotechnologies, i.e. nanoparticles   (buckyballs  or fullerenes) and quasiparticles (Shechtman’s work).  I believe the results will impact every major discipline, including religion, ethics, ontology, epistemology and cosmology.

In physics we’ll have a new look at the weak and strong interactions,  gravity and polarity or electromagnetism, and even the deeper internal symmetry transformations.

In chemistry, the four hexagonal plates crisscrossing the center point should open a new understanding of bonding.  I even believe there will be a new science of “cross-dimensional bonding” in quantum chemistries.

Within biology, the sciences of RNA/DNA sequencing, genomics, applied biosystems, and even quantum biology will go deeper and become more cohesive.

In psychology, learning, memory, and even identity can be more richly addressed.

This apparent intellectual oversight does not seem to know any physical, cultural, religious or political boundaries.  I have not been able to find references to the interiority of simple structures in any culture to date.

Surely my friends who have worked with R. Buckminster Fuller and Arthur Loeb, would take exception to the comment.  Yet,  Bucky’s two volumes, Synergetics I and Synergetics II, are virtually impermeable to the average person and neither work has been widely used for common tasks or applied sciences.  Buckyballs or fullerenes are now being used widely within nanotechnologies, but that is all in its earliest stages of development as a reduction-to-practice.

The answer to the question about the octahedron renders a model with a profound complexity and simplicity.  Again, if you can picture an eight-sided object, essentially the two square bases of the pyramid pushed together, you’ll have an image of an  octahedron.

Divide each of the edges in half and connect the points.  You will find an octahedron in each of the four corners of the base square and an octahedron on the top and bottom.  In each of the eight faces is a tetrahedron.

There are very few models of the parts and whole relation.  There are fewer still that describe the interior relations of these objects.

Let us take a look.

This third picture from the top in the right column is of a tetrahedron.  There is a tetrahedron in each of the four corners and an octahedron in the middle.

The fourth picture is the octahedron.  Again, there is an octahedron in each of the six corners and a tetrahedron in each face.

The TOT.  The fifth picture from the top in the right column  is a tetrahedral-octahedral-tetrahedral chain. I dubbed it a TOT line. The first time I thought I was observing it in action as a trusss system to support the undulating roof system of the Kansai Airport in Japan. In February 2007, I realized that truss was actually just half a TOT when I actually made the model pictured here. It is a simple parallelogram that can be found in many basic geometry textbooks. However, I have not yet found this tetrahedral-octahedral chain examined in depth.

Geologists have been studying natural tetrahedral-octahedral layers within nature that is known as a TOT layer.  We will look extensively at the natural occurrences of TOT formations much later in this work.

In the photograph, it is two tetrahedrons facing on an edge with an octahedron in the middle.  Each face of the TOT is an equilateral triangle on the surface which, of course, opens to the inner cavity of either an octahedron  or a tetrahedron.

These are simple models that have been largely unexamined by the academic communities.

Towards a Theory of Everything Similar

With the TOT line, I believe we are looking at the structure of perfection.  Pure geometry.  And, I believe that geometry once expressed in the physical world, manifested within space and time, becomes rather randomly quantized and infinitely variegated.

I believe our chemists should look into chemical bonding that goes beyond  the usual two-dimensional diagrams to these three-dimensional interactions and then to the multi-dimensional complexity when correlated within the necessary plates of an internal tetrahedron or octahedron.

Here we open the very nature of chemical bonding to new possibilities. The bonding (the function) is interior to a pure structure (the form).

It is simple complexity.  If you were to keep going deeper within each octahedron and tetrahedron, as you might guess, the number of cells or objects expands quickly. By the tenth step within, there are 131,323,456 tetrahedrons and 10,730,656 octahedrons for a total of 142 million objects.

At the eleventh step there are over a billion tetrahedrons and 63,859,648 octahedrons within.  The total, just taking 11 steps within, are 1,110,412,992 objects.

At the twelfth step there are over  8 billion  tetrahedrons and 381 million octahedrons. That level of complexity within such simplicity allows for a wide range of diversity.

As a reduction to first principles…

A footnote and timeline:  This particular document was written in May 2007.  The first iterations that lead up to this document were written in 1994.

The precursor to it all was that display project pictured in the top right.  That was simply called, “A Display Project of First Principles.” It began as a list of some of the most-speculative, integrative thinkers within the major academic disciplines.

I wanted to invite them to a conference in July 1979 at MIT for the World Council of Churches.  Over 4000 people would gather to discuss, Faith, Science, and the Future.  Being on the organizing committee, it seemed to me that the ideas of the finest scholars from the area, and then from the world, should be part of that discussion.

At that time, those leading scholars were not invited.  The committee thought they would dominate and possibly overwhelm the discussions; so as a consolation, they allowed me to organize this display project.

The display project was titled What is Life? after Erwin Schrödinger’s book of the same title.  This work is being renewed.  Early stages of it can be found on other pages within these websites.  BEC

In Search of An Architecture for Integrative Systems

This page evolves from a 1979 global dialogue between scholars in the natural sciences, the humanities and theology. It was originally formulated to provide discussion materials for a conference at MIT entitled, Faith, Science and the Human Future.

The work and writings of leading, living scholars and most-speculative thinkers were examined. Though hardly anticipated, this display project became a template for the Big Board – little universe. The purpose of the display project was to summarize those comprehensive worldviews and powerfully suggestive ideas of living scholars.  All vetted were within their community as leading thinkers; the hope was that there might be a dynamic exchange and synthesis of ideas and information that would open the way to new and deeper insights and wisdom. Based on their experiences, observations, historical analysis, hypotheses and testing, informed speculations, and even visionary insights, each person’s  work was placed within one of three perspectives: The Small-Scale Universe, The Human-Scale Universe, and The Large-Scale Universe.  And then, with each perspective, there were three groups of scholars: (1) Natural Scientists, (2) Philosophers/Theologians and (3) The Boldly Speculative.

The Small-Scale Universe
To Be  – Reality

Scholars seek to define fundamental units of reality, experience and/or being.

The Human Scale Universe
To Know Ways of Knowing

Scholars seek to understand basic interactions from cells to populations of people. What makes life human? What gives life meaning?

The Large-Scale Universe
To Envision the Cosmos

Scholars seek to understand cosmology, the parts, laws, and operations of the universe. They seek to know the origin and nature of the universe.

All Living Scholars in 1979
(all listings, all columns are alphabetical)
Scientists to Theologians
(all listings are followed by a school designation)
Living Scholars in 1979
(listings of published work are linked)
•  Ian Barbour, Carleton, Northfield (MN)
Issues in Science and Religion
•  Michael Arbib, Massachusetts, UCLA
Brains, Machines and Mathematics
•  Hannes Alfven, Uppsala, Stockholm
Cosmic Plasma
•  Ted Bastin, Cambridge
Quantum Theory & Beyond
•  Peter Berger, Boston College
The Sacred Canopy
•  Hermann Bondi, London
The cosmological scene
 •  Charles Birch, Sydney
Biology and the Riddle of Life
•  Percy Brand Blanshard, Yale
The Nature of Thought
•  Margaret & Geoffrey Burbidge, UCSD (CA)
The Abundances of the Elements
•  David Bohm, Birbeck, London
Fragmentation & Wholeness
•  Kenneth Boulding, Colorado
The World as a Total System
•  Buckminster Fuller, Pennsylvania
Synergetics I & II
•  Mario Bunge, McGill, Montreal
Treatise on Basic Philosophy
•  Erwin Chargaff,Columbia
Heraclitean Fire
•  Stephen Hawking, Cambridge
On the Shoulders of Giants
•  Fritjof Capra, Lawrence Berkeley
The Tao of Physics
•  Noam Chomsky, MIT
Language and Mind
•  Fred Hoyle, Cambridge, Cal Tech
Ten Faces of the Universe
•  John Cobb, Claremont (CA)
Process Studies
•  Freeman Dyson, Princeton
Disturbing the Universe
•  Stanley Jaki, Seton Hall (NJ)
Science and Creation
•  Richard Feynman, Cal Tech
Theory of Fundamental Processes
•  John Eccles, SUNY-Buffalo
Understanding of the Brain
•  Bernard Lovell, Manchester, JBO
Emerging Cosmology: Convergence
•  Lewis Ford, Old Dominion, Norfolk (VA)
Lure of God
•  Richard Falk, Princeton
A Study of Future Worlds
•  Roger Penrose
The Emperor’s New Mind
•  Sheldon Glashow, Harvard
The charm of physics
•  Paul K. Feyerabend, Berkeley
Science in a Free Society
•  Arno Penzias,  Bell Labs (NJ)
The Origin of the Elements
•  David Griffin, Claremont (CA)
Archetypal Process
•  John N. Findlay, Oxford, Boston
Plato: The Written and Unwritten
•  Carl Sagan, Cornell
Contact  and  Cosmos
•  Charles Hartshorne, Chicago
The Zero Fallacy
•  Hans-Georg Gadamer, Heidelberg
Truth and Method
•  Fred A. Wolf
The Dreaming Universe
•  Krishnamurti, California
The First and Last Freedom
•  Langdon Gilkey, Chicago
Maker of Heaven and Earth
•  Tarthang Tulku (Berkeley, CA)
Time, Space, and Knowledge
•  H. Pierre Noyes, Stanford
Bit-String Physics
•  Steven Grossberg, Boston
Studies of Mind and Brain
•  Steven Weinberg, Harvard, Texas
The First Three Minutes
•  Shubert Ogden, SMU, Dallas (TX)
On Theology
•  Jürgen Habermas, Max Planck, Starnberg
The Fear of Freedom
•  Yakov B. Zel’dovich
Creation of particles in cosmology
•  Harold Oliver, Boston
A Relational Metaphysic
• Gerald Holton, Harvard
Scientific Imagination
Living Scholars Today Who shall we add in each category? Who are today’s
leading living scholars?
•  Gian-Carlo Rota,  MIT
Foundations of Combinatorics
•  William Johnston, Sophia, Japan
Still Point
•  Julian Schwinger, UCLA
Einstein’s Legacy
•  Gustavo Lagos, Chile
(in process)
•  John Baez, UCR (CA)
Knots and quantum gravity
•  Henry P. Stapp, Lawrence Berkeley
Mindful Universe
•  Erwin Laszlo, UN
Systems View of the World
•  Lisa Randall, Harvard
Warped Passages
•  Victor Weisskopf, MIT
The Joy of Insight
•  Bernard Lonergan, Regis
Insight: A Study of Human Understanding
•  Richard Dawkins, Oxford
The Magic of Reality
•  Carl F. von Weizsäcker, Max-Planck (Starnberg)
The Structure of Physics
•  Lynn Margulis, Massachusetts (Amherst)
Early Life
•  Daniel Shechtman, Technion
Icosahedral Quasiperiodic Phase
•  John Wheeler, Princeton, Texas
Spacetime Physics
•  Ali A. Mazrui, Michigan, SUNY-Binghamton
A World Federation of Cultures
•  Jim Yong Kim, World Bank,
Dartmouth, Toward a Golden Age
•  Eugene Wigner, Princeton
Symmetries & Reflections
•  Marvin Minsky, MIT
The Society of Mind
•  Ben J. Green, Cambridge
On arithmetic structures…
•  Jürgen Moltmann, Tübingen
The Spirit of Life
•  Brian Green, Columbia (NYC)
The Elegant Universe
The selection committeeIncluded Marx Wartofsky,
J. Robert Nelson, Alan Olson,
and Bill Henneman, all of
Boston University.
•  Wolfhart Pannenburg, Munich
Theology and the Philosophy of Science
 Agnieszka Zalewska, Krakow, CERN
Large Hadron Collider
•  Karl Popper, London
All Life Is Problem Solving
More to come…
Every scholar selected was also
invited to nominate others.
•  Karl Pribam, Stanford
The End of Certainty
Every scholar was also invited
to critique the selections.
•  Ilya Prigogine, Brussels
The End of Certainty
Bruce Camber initiated and
coordinated this effort.
• Karl Rahner
Theological Investigations
A back story of its development
is linked here
• Theodore Roszak, San Francisco State
The Making of a Counter Culture.
More to come…
• Huston Smith, Syracuse
The World’s Religions
• William I. Thomson, Lindisfarne
Passages about Earth

Does anybody know?

Can the Big Board – little universe become conceptually-rich enough to be a working environment that more deeply informs the concepts of continuity, symmetry and harmony?  Could all the notations, anywhere from 202-to-206  starting with the Planck length, the smallest measurement, to the largest measurement — the size of the observable universe — inform our understanding of order, relations, and dynamics?

We believe the answer is “Yes” yet time will tell.  We  will be carefully exploring each notation to see if there are boundary conditions and natural transitions (or transformations) from one notation to the next.

Let’s start with a  simple observation.  At notations 111 and 112 we are in the ranges of sizes from 4.19589 centimeters (cm) to 8.39178 cm or about 1.65 inches to 3.3 inches.  These ranges would include the sizes of the  plastic models displayed at the top of the Big Board. The tetrahedron measures around 2.5 inches; and if  it were divided in half 112 times, we would be in the range of the Planck Length, 1.616199(97)×10-35 meters.

Is that useful information? Simply as an ordering tool for data, the high school students (and many of the scholars with whom we have chatted) believe that it is.  A few of the kids thought it was just too much information and were overwhelmed by it.

Observations and Questions:  At notation 69 we are at diameter of an electron. At notation 66 we are at the one of the best guesses regarding the diameter of a proton. Smaller yet are the neutrinos (a type of lepton) and quarks (part of the fermion family). The now-famous Higgs particle is part of the boson family; and with the fermion family, comprise the elementary particles.

And here we have a gross infrastructure for a standard model for physics and most of science. But, is it complete?  We all know painfully well that it is not.

Is it the right time to test some other hypotheses?  Is there an inherent structure for these hypotheses within those first 60 doublings, layers, notations, or steps?

What is the universe made of?  In 1887 Michelson–Morley took an age-old concept,  the luminous aether, and soundly put it to sleep.  But, perhaps we need to take a second look.  MIT Nobel Laureate, Frank Wilczek in his book, The Lightness of Being (pp 74ff), reviews six different approaches to answer the question, yet comments that the aether “…is the  old concept that comes closest, but it bears the stigma of dead ideas and lacks several of the new ones.”  Another Nobel Laureate in Physics, Robert B. Laughlin (Stanford), is quoted as saying, “The word ‘ether’ has extremely negative connotations in theoretical physics because of its past association with opposition to relativity.”  The concept of a relativistic aether has been carried forward in many forms, nobody but the most idiosyncratic would dare risk one’s reputation actually using the word. Of course, those of us who have no reputation to risk could open that discussion.

Ask most any scholar today about the parts-whole relations within a simple tetrahedron and the response is a blank stare. Fewer still know what is most simply and perfectly enclosed within an octahedron even after dividing each edge in half and connecting those new vertices for them. That octahedrons and tetrahedrons embed perfectly and without obvious limit is well-known among our better geometers, but still very little discussed.

Ideas, concepts, reifications, instantiations, and hypostatizations.  It does seem true, “There is very little new under the sun,” yet, some ideas need to be re-tried under different conditions. And given that the 202+ notations had not been discussed prior to December 19, 2011 in that high school geometry class, why not take a second look?

Possible starting area:  There is a concrescence of thinking, research and development that is in need of an expansive area within which to work and  to develop continuity equations and real relations with all other scholarship.   Perhaps the first 60 notations create such an area.  Let us consider a few disciplines that could benefit:   (not in an necessary order)

  • Point-free geometry envisioned by Alfred North Whitehead (1919) continues to be developed under the headings of connection theory,  mereology and general systems theory.
  • Discrete and combinatorial geometries are getting substantial attention and work as a result of nanoscale research.
  • The Mind, Consciousness, Psychology, and Epistemology are age-old discussions that seem to have no grounding within the sciences.  Perhaps as Roger Penrose and so many others, with their yearning-burning desire to emerge with a science of consciousness, can find a home within some part of those 60 steps.
  • Mathematics, including Number Theory, Gödel’s incompleteness theorem, Lie superalgebra (and Lie groups) and so much more need a larger conceptual playing field for exquisitely small numbers.
  • More to come

Perhaps the concepts of perfection-imperfection and multiverses (from William James to Lisa Randall to Max Tegmark) can help.  We will see.

  1. Introduction & Overview

  2. Big Board – little universe

  3. Just an image of the Big Board – little universe Version

  4. Wikipedia Article, April 2012.

  5. An Unfinished Work, An On-going Study.

  6. 202.34:  the calculations by Joe Kolecki, retired, NASA scientist

  7. Just the numbers

  8. A little story

An addendum:
Earlier work, the scholars and speculative thinkers
Early questions, 1979 Display project at MIT