# A Study Of Notation #5  (still rough notes)

## From pentastars to quasicrystals to quantum fluctuations

###### by Bruce e Camber

Observations:  The number 5 is a prime number and in the systems of geometries we ask, “What is the next most simple geometrical or unique mathematical system?”

What about the Fibonacci numbers?  When might this sequence begin to apply to these notations?  Reflecting on this question and the nature of an exponential universe, addition appears to be derivative of multiplication by 2.  If the universe is fundamentally a multiplicative system, one might begin to think that addition occurs “within” notations, and in order to get “carried” across notations, it requires a mathematical function that provides the transport through the other notations.  Let us be imaging the spiral nebulae. So, as of this writing, we are projecting the Fibonacci sequences might begin to engage possibly as late as the 144th notation where processing is just over one second.
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From pentastars, tetrahedral rings, tetrahedral systems, to the icosahedral phase.

On April 8, 1982 Dan Shechtman actually saw for the first time what is now named the icosahedral phase; and as a result, he singlehandedly opened the new field of quasiperiodic crystals. Prior to that day, an icosahedral phase barely existed in the minds of a very few mathematicians and geometers. Many of Shechtman’s colleagues thought he was a bit crazy. Nevertheless, he persevered — he knew what he could see was real — and as a result, he opened a new field of study with immediate applications that’d never existed prior to his work.

I believe the very foundations of the icosahedral phase begins within this 5th notation. Five perfect tetrahedrons, as pictured above, share eight vertices. With no less than 4096 to 32,768 vertices and possibly many more, this could be the notation within which complexity earnest begins.

The icosahedral structures involve 20 tetrahedrons within what we call quantum geometry, a squishy geometry where there are many of degrees of freedom.  Within this model, there are two groups of five tetrahedrons with a band of ten tetrahedrons separating the two. All twenty all share the same centerpoint.  Or, there are also three groups of five tetrahedrons with a cluster of four and a single tetrahedron such that each cluster shares only one edge with another cluster and, of course,  they all share the same centerpoint.

It is a wonderful model.  I’d be glad to send you such a model as pictured above.  It is a transformative experience to feel the first instances of squishy quantum geometries. In my more adventuresome moments, I propose that by the 60th notation, these simple structures give us quantum fluctuations. For this posting, I will make a short video demonstrating how the five tetrahedrons actually have movement within confined spaces and how the 20 tetrahedrons seem to do their thing.

We are only on the fifth notation, five steps beyond the nexus of transformations between the finite and infinite.  That  amalgamation of equations and ratios are necessarily an intimate part of this notational definition. These most simple and most perfect equations are all using pi to create the geometries and symmetries that become our first forms with real numbers (Planck base units) and real formulas. 