Center for Perfection Studies • The Big Board–Little Universe Project • New Orleans • USA
May 2007. 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 answered three very simple, basic questions:
- What are the simplest three-dimensional structures?
- What is most simply and perfectly enclosed within those structures?
- What is most simply and perfectly enclosed within each of those parts?
When David Bohm died in 1992, I took down his little book, Fragmentation & Wholeness, he had given me in class and I started reading just one more time. Then, it hit me. “What is perfectly enclosed within the tetrahedron?” I did not know. “Four half-sized tetrahedrons and an octahedron.” Discovering what was inside the octahedron was a major breakthrough for me. 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, John Conway, 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 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 that 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.
Within each corner there is an octahedron. There are six corners. With each face is a tetrahedron. There are eight faces. The tape inside define four hexagonal plates that share a common center point. Notice the tape comes in four different colors.
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 (Dan 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 deep 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 picture on the right is a tetrahedral-octahedral-tetrahedral truss or 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. Yet, it seems that this tetrahedral-octahedral chain has not been 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 these three-dimensional interactions and then to the multi-dimensional complexity when correlated within the necessary plates of a yet deeper, internal tetrahedron, octahedron, or tetrahedral-octahedral structure.
Here we open the very nature of chemical bonding to new possibilities. The bonding (the function) is interior to a pure structure (the form).
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 (and not yet using dimensional analysis), 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.
It was reduced to first principles.
A footnote and timeline: Yes, 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.