Last Update: Wednesday February 21, 2018

## Basic Observation Tells Us That Everything Starts Most Simply. |

Observing how the simplest geometric objects are readily embedded within each other, a high school geometry class
We went inside again. At each notation or step we simply selected an object and divided the edges in half and connected the dots. Perfectly enclosed within the octahedron are six half-sized octahedrons in each of the six corners and eight half-sized tetrahedrons in each of the eight faces. Selecting either a tetrahedron or octahedron, it would seem that one could divide-by-2 or multiply-by-2 each of the edges without limit. If we take the Planck Length as a given, it is not possible at the smallest scale. And, there are also apparent limits within the large-scale universe -- the Observable Universe. Also, observe how the total number of tetrahedrons and octahedrons increases at each doubling. At the next doubling there are a total of 10 octahedrons and 24 tetrahedrons. On the third doubling, there are 84 octahedrons and 176 tetrahedrons, and then on the fourth, 680 octahedrons and 1376 tetrahedrons. On the fifth step within, there are 10944 tetrahedrons and 5456 octahedrons. The numbers become astronomically large within 101 steps. It is more aggressive than the base-2 exponential notation used with the classic wheat and chessboard story The following day we followed the simple math going out to the edges of the Observable Universe. There were somewhere between 101 to 105 steps (doublings or notations) to get out in the range of that exceeding large measurement, 1.03885326x10 Not long into this exploration it was realized that to achieve a consistent framework for measurements, this simple model for our universe ought to begin with the Planck Length (ℓP). It was a very straightforward project to multiply by 2 from the ℓP to the edges of the Observable Universe (OU). That model first became a rather long chart that was dubbed the This simple construction raised questions about which we had no answers: Starting at the Planck Length, a possible tetrahedron can manifest at the second doubling and an octahedron could manifest at the third doubling. Thereafter, growth is exponential, base-4 and base-1 within the tetrahedron and base-8 and base-6 within the octahedron. To begin to understand what these numbers, the simple math, and the geometry could possibly mean, we turned to the history of scholarship particularly focusing on the Planck Length.
^{8} Though formulated in 1899 and 1900, the Planck Length received very little attention until C. Alden Mead in 1959 submitted a paper proposing that the Planck Length and Planck Time should "...play a more fundamental role in physics." Though published in Physical Review in 1964, very little positive feedback was forthcoming. Frank Wilczek in that 2001 Physics Today article comments that "...C. Alden Mead's discussion is the earliest that I am aware of." He posited the Planck constants as real realities within experimental constructs whereby these constants became more than mathematical curiosities.Frank Wilczek continued his analysis in several papers and books and he has personally encouraged the students and me to continue to focus on the Planck Length. We are.
A very simple logic suggests that things are always simple before they become complex. Growing up as a child, my father would ask, "Is there an even more simple solution?" Complex solutions make us feel smarter and wiser, yet the opposite is most often true. When teaching students from ages 12 to 18, one must always start with the simplest new concepts and build on them slowly. Then, a good teacher might challenge the students to see something new, "If you can, find a more simple solution." Our class was basic science and mathematics, focusing on geometry. My assignment was to introduce the students to the five platonic solids. Yet, by our third time together, we were engaging the Planck Length. Is it a single point? Is it a vertex making the simplest space? What else could it be? Can it be more than just a physical measurement? Are we looking at point-free geometry? We knew we would be coming back to those questions over and over again, so we went on. We had to assume that the measurement could be multiplied by 2. We attributed that doubling to the
- Within the first ten doublings, there are over 1000 vertices. Perhaps we might think about Plato's Eidos, the Forms.
- Within twenty doublings, there are over a million vertices. Could this be a domain for Aristotle's Ousia or Categories?
- Within thirty steps, there are over a billion vertices. One could hypostatize
*Substances*, a fundamental layer that anticipates the table of elements or periodic table. - Within forty layers, there are over a trillion vertices. We might intuit that Qualities begin to emerge.
- Within 50, there over a quadrillion vertices. Might this be a layer for Primary Relations, the precursors of subjects and objects?
- Between the 50th and 60th notation, still much smaller than the proton, there are over a quintillion vertices. Perhaps Systems and The Mind, and every possible manifestation of a mind, awaits its place within this ever-growing matrix or grid.
With so many vertices, one could build a diversity of constructions, then ask the question, "What does it mean?" Our exercise with the simplest math and simple concepts is the We knew our efforts were naive, surely a bit idiosyncratic (as prominent physicist had characterized them), but we were attempting to create a path that would take us from the simplest to the most complex. If we stayed with our simple math and simple geometries, we figured that we did not have to understand the dynamics of protons, fermions, scalar constraints and modes, gravitational fields, and so so much more. That could come later. Although not studied Oxford physicist-philosopher Roger Penrose We know with just two years of work on Big Board - little universe chart and much less time on our compact table, we will be exploring those 60-to-65 initial steps most closely for years to come. This project will be in an
This construction with its simple nested geometries and simple calculations (multiplying the Planck Length by 2 as few as 202.34 times to as many as 205.11 times) puts the entire universe in an mathematically ordered set and a geometrically homogeneous group. Although functionally interesting, quite simple and rather novel, is it useful? Some of the students thought it was. This author thought it was. And, a few scholars with whom we have spoken encouraged us. So the issue now is to continue to build on it until it has some real practical philosophical, mathematical, and scientific applicability. Taking our three simple parameters just as they have been given, (1) the Planck Length, (2) multiplication by 2 and (3) Plato’s simplest geometry, what more can we say about this simple construct? 1. 2. 3. Please note that our use of the double modal,
The third doubling renders eight vertices. With just seven of those vertices, a pentagonal cluster of five tetrahedrons can be inscribed (Illustration 3), however, there is a gap of about 7.36° or less than 1.5° between each face. There are many other configurations of a five-tetrahedral construction that can be created with those seven vertices. These will be addressed in a separate article. For our discussions here, it seems that each suggests a necessarily imperfect construction. The parts only fit together by stretching them out of their simple perfection. One might speculate that the spaces created within these imperfections could also provide room for movement or fluctuation.
With all eight vertices, a rather simple-but-complex figure can be readily constructed with six tetrahedrons, three on either side of a rather-stretched pyramid filling an empty space between each group. This figure has many different manifestations using just eight vertices. Between seven and eight vertices is a key step in this simple evolution. Both figures can morph and change in many different ways, breaking-and-making perfect constructions.
In one's most speculative intuitive moments, one "might-could" possibly begin to see the early start of Within the all the following notations simplicity begets complexity. Structures become diverse. And, grids of potential and a matrix of possibilities are unlocked.
************************************************************************************************************** Should we have further references for each of the three Illustrations? Should there be a section just for References? Speculative philosophy |