Strange things can happen when one is invited to be a substitute teacher for a day, essentially just an assistant for students within five high school geometry classes and for their teacher who is part of our extended family.
My name is Bruce Camber and this is a little story about the unfolding of events that eventually resulted in these pages.
Have you ever seen the entire universe mathematically related and notated on one chart? We had not. In studying the platonic solids and base-2 notation, it seemed to be an interesting task to do the simple base-2 math to create a rather big board that related everything from the smallest measurement to the largest measurement, virtually every part of our universe. So, we created it. The first version of this board was printed down at the Office Max in Harahan, Louisiana on December 17, 2011. It measured 24″ by 120″ which we quickly discovered was a bit big and awkward. Two smaller charts, 12″ by 60″ were created the next day, December 18, for the classroom discussions on December 19.
The ten-foot board was cut about in half and the top section was put in the front of the class and the bottom section in the back. On the walls on the left and right were the two five foot charts. It all seemed a bit enchanting.
Those five high school geometry classes were challenged to see the universe using Plato’s five building blocks and base-2 exponential notation. Starting with just the tetrahedron and octahedron, we divided the edges in half, connected those new vertices and keep on going. In just over 101 steps we were in the range of the Planck Length, the smallest measurement in the universe. Then, multiplying by 2, in just over 100 steps we were out to the edges of the Observable Universe.
We found examples of base-10, yet clearly base-2 is more granular. One divides by 2 or multiplies by 2 instead of by 10. We also discovered and used a huge history of work done on the order of magnitude and the power of two. However, it seems that we had a different starting point than most. At first, we used an imaginary tetrahedron that was 1 meter on its side. Our actual models were 2.5 inches. We expected thousands of steps in either direction.
It was a bit unsettling to discover so few.
We reduced it to a chart with a color wheel as the background and dubbed it, a Big Board for our little universe. Not too much later, we decided to start at the Planck length and just multiply by two. It worked out pretty well and kind of, sort of confirmed our earlier work. It became Version 18.104.22.168 and you will find images of it throughout these pages.
This work all started with Plato’s five basic solids and thoughts about basic structure. Though most people do not give it much thought, these basic structures have been studied throughout time, probably starting in ancient Greece with Pythagoras and picked up later by Plato.
For many of the students, this encounter was our second time to explore these five basic solids. The very first time together in March 2011, the students explored models using clear plastic tetrahedrons and octahedrons. Both are pictured within the top rows of our working model of the Big Board. To go inside these models, essentially dividing them in half, requires a little finesse. Simply divide each edge in half and that point becomes a new vertex. With the tetrahedron there are six edges and within the octahedron there are eight edges. Connect all the new vertices and you have the simplest internal structure. Within the tetrahedron are four half-sized tetrahedra in each corner and an octahedron in the middle. Within the octahedron there are six half-sized octahedrons in each corner and a tetrahedron in each of the eight faces.
The students also made icosahedra out of 12 tetrahedrons. It was quite a lot of fun.
The second time with these kids would be more of a challenge. It would be the day just prior to their Christmas break.
The universe in 202.34 -to- 206 steps. When we began finding simple math errors, the number of notations increased from 206 to 215 (it became our fudge factor). Then a leading astrophysicist said, “There are 206 notations.” Then on May 2, 2012, a NASA physicist made the calculation based on the results of the SDSS-III Baryon Oscillation Spectroscopic Survey (BOSS). He reported 202.34 notations. At that time, looking throughout the web, we could only find bits and pieces of this work.
An earlier history began with the study of perfected states in space time.
Sometime around 2002, at Princeton with geometer, John Conway, the discussion focused on the work of David Bohm, once a physicist from Birbeck College, University of London. “What is a point? What is a line? What is a plane vis-a-vis the triangle? What is a tetrahedron?” Bohm’s book, Fragmentation & Wholeness, raised key questions about the nature of structure and thought. It occurred to me that I did not know what was perfectly and most simply enclosed by the tetrahedron. What were its most simple number of internal parts? Of course, John Conway, was amused by my simplicity. We talked about the four tetrahedrons and the octahedron in the center.
I said, “We all should know these things as easily as we know 2 times 2. The kids should be playing with tetrahedrons and octahedrons, not just blocks.”
“What is most simply and perfectly enclosed within the octahedron?” There are six octahedrons in each corner and the eight tetrahedrons within each face. Known by many, it was not in our geometry textbook. Professor Conway asked, “Now, why are you so hung up on the octahedron?” Of course, I was at the beginning of this discovery process, talking to a person who had studied and developed conceptual richness throughout his lifetime. I was taking baby steps, and was still surprised and delighted to find so much within both objects. Also, at that time I had asked thousands of professionals — teachers, including geometry teachers, architects, biologists, and chemists — and no one knew the answer that John Conway so easily articulated. It was not long thereafter that we began discovering communities of people in virtually every academic discipline who easily knew that answer and were shaping new discussions about facets of geometry we never imagined existed.
Of course, I blamed myself for getting hung up on the two most simple structures… “You’re just too simple and easily get hung up on simple things.”
My family knows about this curious hang up of mine. They have seen these models on my desk. We made a pseudo-Rubik’s cube type of game out of the octahedron. One of younger ones in the family is the geometry teacher in the family’s small private high school. “Come in and introduce the kids to Plato’s five basic solids.” That’s about my level. In so many ways, those kids were actually more advanced than me.
During one of my days with them, we made icosahedra with twenty tetrahedrons in each. It was not perfect geometry; the tetrahedrons had to be taped together and you could feel them moving. I called it squishy geometry, but told them that I have yet to find a good discussion about it under quantum geometry or imperfect geometries, “…but when I find it, I’ll report in.”
At first, our dodecahedron was a simple paper thing. We were trying to think of its simplest number of parts… “Could it be twelve odd objects coming into a center point, each with a pentagonal face and three triangular sides?” It didn’t seem like it would readily be extensible. On my desk was a “Chrysler logo” made up of five tetrahedrons. There was always a gap — squishy geometry — but it trigger a thought, “What would a pseudo-dodecahedron look like if it were made of twelve of those pentagonals (each made up of five tetrahedrons)?” Very quickly we had a model. A few hours later we were filling it with Play Doh to see what was within it. And just within, we found an icosahedron waiting.
Now that was fascinating to us, but is it? Is it common knowledge among all the best-of-the-best within mathematics, chemistry, and physics? We did not know and it was put on our list of things to do.
In thinking about a sequel class to that earlier time together, we began focusing on exponential notation. Having learned a little about base-2 notation — my first time over these grounds — we published pages deep within our business website to begin to share it with a wider audience: http://smallbusinessschool.org/page883.html
If you find it of some interest, there are links to more background pages from both.
Can Plato’s five most basic objects in some way hold each progression together in a mathematical relation? Is it meaningful in any way? We would all enjoy hearing from you.
Please drop us a note! – BEC