###### Center for Perfection Studies • The Big Board–Little Universe Project • USA • May 2017 •

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**A simple, mathematical model of our Universe**

*• Measuring an Expanding Universe Using Planck’s Units *

• Thrust of the Universe: What is it? (* work in progress*)

###### by Bruce Camber (first draft)

**Background**: The purpose of this article is to create a visualization of the current data within our chart. It starts with the Planck units and goes out to the current time all within 202 notations by applying base-2 exponentiation, also known as the powers of 2, and doublings.

**Format**: This visualization will be created on a canvas first within your mind. It is a very different frame of reference where space and time are derivative, finite and quantized. Continuity, symmetry and harmony are infinite such that the transformation nexus is the first notation, and then synchronously with all other notations within a given moment.

Trying to create any visualization with mostly words alone requires your help to paint the picture. To help formulate that picture, our visualization will be within a block quote and any image we can share and then it will be followed by comments to attempt to put some muscle on those bones. Flesh it up. Your comments are welcomed and will be added.

**Notations #1 & 2**:** Picture a circle**. It has just two vertices and it is defined by an equation that contains an irrational number called *pi. *When it is used to generate anything, it instantly becomes transcendental. The infinite becomes finite and you have a circle. In the first notation, each of the Planck base units double. However, these base units are defined by an array of formulas. Though some call these Planck units “a singularity,” it is truly a diversity of absolutes, constants (both physical and mathematical), transfinites, and universals. If it is called a singularity in a gross sense of the word, it is unlike any other singularity heretofore defined. There is an infusion of numbers that have specificity and it is the first introduction of physicality, but this physicality is not like any physicality we know. It just may be the base root of everything physical. It is the most simple but pervasive. Of course, when it first doubles, there is not much to it. Yet, this most basic facet and thrust of life, doubling, has begun.

The third doubling creates the basis for going from curves to lines. It begins to approximate our very first image that comes from a dynamic gif within Wikipedia under close-packing of equal spheres.

With the third notation there are 64 scaling vertices. We begin with a single circle that becomes a sphere. With eight scaling vertices there are four base spheres. With 64 scaling vertices, there are as many additional spheres as is required to do sphere stacking such that lattice is generated, and the first triangles emerge. It would require all nine spheres as pictured and the additional nine green spheres as demonstrated. One might say that the lattice generation to create the lines, triangles, tetrahedrons, and octahedrons is part of *the opportunistic thrust of creation*.

**Comments**: You know the old formula, π r^{2}; however, it appears that not many have thought about it as a possible beginning point of creation. There are other possibilities and we will explore these as we progress. But π (pi) is simple-but-complex; analyzed now out well over a trillion places, it is never-ending and never-repeating. And, it certainly is deeply rooted within our common history. The recognition of the numbers defined by *pi *predates Pythagoras (ca. 570 to ca. 495 BCE). Within Wikipedia’s explanation of pi, there is a reference to the book, *Pi Unleashed*, by Jörg Arndt and Christoph Haenel (Springer-Verlag, 2006 ISBN 978-3-540-66572-4/ English translation by Catriona and David Lischka); they claim that the earliest written approximations of π come from Egypt (1850 BCE) and Babylon (1900–1600 BCE).

We will have to examine the nature of numbers and how numbers shape geometries. To do so, we’ll continue our review of the insights of the greatest mathematicians and philosophers throughout our history. Perhaps within our Quiet Expansion model we can begin seeing numbers in new ways.

**Notation #3**: One working hypothesis is that with every prime number another basic mathematical progression could begin. Or, with this third notation we might go from circles to lines and triangles. No decision has been made.

Projective geometries, divisibility rules, lattice multiplication, and grid and matrix multiplication are all being studied.

A resource for our inquiries, beside Wikipedia, is a book, *Projective Geometry: From Foundations to Applications*, by Albrecht Beutelspacher and Ute Rosenbaum.

**Notation #4**:** ** **What might happen with the 512 new scaling vertices? **The tetrahedrons and octahedrons could readily begin to double. One might also conclude that the spheres continue to double as well. Given that the Planck Length doubles, these new objects are twice as large as the objects within notation #3. The build out might begin looking for something like those tetrahedrons and octahedrons scaling up three times within this first illustration. Inside each tetrahedron is a half-sized tetrahedron in each of the four corners and an octahedron in the middle.

Consider the image on the right. The purple number, 1, is the smallest tetrahedron. The green number, 2, contains #1 and it is the next larger. The yellow, number 3, contains #1 and #2, and it is the entire figure displayed here. Containing just the bottom loop ( ⊃) of the #3 and going over to each edge and then down to the center of the base, is the octahedron. It is in the middle, surrounded by tetrahedrons and it has four external faces and four internal faces.

That octahedron is comprised of a half-sized octahedron in each of the six corners and a tetrahedron in each of the eight faces. Also, within the octahedron are four hexagonal plates and everything shares a common center point.

Here is a simple beginning of complexity.

The 1-2-3 image of the tetrahedron enclosing tetrahedrons and octahedrons only has 33 vertices so quite literally this perfectly-fitting form could be about 20 times more complex assuming 512 vertices. Each face could share 15 vertices so you might picture this object with a similar object attached to each of the four faces. Each of these would require just additional 18 vertices each for a total of just 72 vertices for the addition of these four new tetrahedrons. Just to keep some focus on it all, we have used just 105 of the 512 new vertices.

At this point within this analysis, the resulting object appears to be the beginning of the tilings and tessellations discovered with Zeno in December 2011 by those high school geometry classes.

Visualization: Merge the first two images. Make them dynamic and no less than 20 times more complex.

**Sister article**” *Measuring an Expanding Universe Using Planck Units***: **Another article focuses on just six sets of numbers from the 202+ notations that define the horizontally-scrolled chart. The focus is on prime numbers and the possibilities that each prime opens a new data stream with new math**. **Another focus is to discern cycles, frequencies, and periodicity across the entire chart. Also, key components of the chart are to identify the reasons for imperfections, quantum geometries, quantum fluctuations and quantum gravity.

**Note**: The number, 1, is not a prime number. It may be that 1 (and not 0) should be the listing of the Planck base units. Today it is assumed that these base units constitute 0, and 1 is for the first doubling. That assumption will be reviewed over and over again.

**Notation #5**: Among all the numbers that define this notation the construction vertices, now 4096, is the only number that has a familiar sound. The other numbers are still infinitesimal. Yet, the scaling vertices, line 9 within the horizontally-scrolled chart, are now beginning to dramatically expand. Although called *construction vertices*, guides to understanding these first 67 notations can be found within the work of Alfred North Whitehead (point-free vertices) and the study of mereology or gunk theory. Further insights about these vertices can also be found within the manifold studies of topology and more generally, within systems theory and ontology.

**This notation is a prime**. If the logic of the first four notations are in the logic of pi and Euclid, perhaps in this fifth notation, a new mathematics is introduced with the introduction of five tetrahedrons sharing a common center point. Here is the first potential for motion given the gap of 7.36° or 1.54 steradians (see Figure 2). Yes, with the addition of just two more tetrahedrons, a radically new concept emerges: five tetrahedral clusters. For our purposes we call this figure a pentastar. The old Chrysler logo is an artistic rendering of it. Though we suspect this gap only creates a potential, at some point, perhaps within the notations 50 and higher, the mathematics of quantum fluctuations may emerge.

One face of an icosahedron is pictured here where every face can be seen as five-tetrahedral clusters. If taken as a whole, there can only be three such tetrahedral clusters (all similarly flexible) grouped at the same time whereby one of the five-tetrahedral clusters is broken apart with one tetrahedron sharing a common vertex with nine other tetrahedrons, a common counterpoint with all other tetrahedrons, and a common face with each of the three five tetrahedral clusters. The other four remaining tetrahedrons are grouped together so each shares two faces with each of the three three five tetrahedral clusters.

Another possible configuration is to identify any two groups of five tetrahedral clusters that directly opposite to each other and a band of ten-tetrahedrons is created between them.

The high school students know this as *squishy geometry*. At this point in time we believe it is the beginnings of quantum geometry and the beginnings of quantum fluctuations.

– – – – – – – – – – – MORE TO COME – – – – – – – – – – WORKING DRAFT – – – – – – – – – –

**This is our working area, a repository of ideas and concepts…**

*• Measuring an Expanding Universe Using Planck Units (still a work in progress)*

• The Thrust of the Universe: What is it? (still a *work in progress*)*• This page is “Visualizing the Universe”* *(and, still a work in progress)*

The first doubling… What is the nature of 1? What is the nature of 2?

**Cycles, Frequencies, Periodicity**: 1/2, 1/3, 1/4, 1/5, 1/6, 1/7 1/8, 1/9, 1/10

**Dark Energy, Dark Matter references**: Within the horizontally-scrolled chart, just follow the line 5, Planck Mass, and line 6, Planck Charge, through to the 67th notation.

http://cosmology.berkeley.edu/Education/FAQ/faq.html

There are many books and articles about these constants, however, our primary reference is the 2006 article by Tegmark, Aquirre, Rees, Wilczek (TARW)*, “Dimensionless constants, cosmology and other dark matters” *where they identify 31 dimensionless physical constants (PDF). The Planck Length (space) and Planck Time are two of their 31.

Martin Rees, author, *Just Six Numbers: The Deep Forces That Shape The Universe*

Point processes: https://arxiv.org/pdf/1703.10000.pdf