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.