Brooks (Base) Square (BS) 101

~ The Architecture of Space-Time (TAOST)

&

The Conspicuous Absence of Primes (TCAOP) ~

Conclusion

A Brief Introduction

I. TAOST - the network

II. TCAOP - everything minus the network

III. Interconnectedness

Conclusion <---

References

Appendix A

Appendix B

 TAOST: Rules 1-50 | Rules 51-80 | Rules 81-99 | Rules 100-107 | Rules 108-153 | TCAOP: Rule 154 | Rules 155-157 | Rules 158-159 | Rule 160 | Interconnectedness: Rules 161-175 | Appendix A: Rules 176-181 | Appendix B: Rules 182-200 |

IV. Conclusion

Using numbers as symbols for quantities, i.e. number values, we purposely sought to discover new relationships between the sequential whole number integers ... 0, 1 ,2 ,3, ... along the axis of a square grid matrix and the corresponding diagonal of the squares of those same number values ... 0, 1, 4, 9, .... This diagonal is a core description of the Inverse Square Law (ISL) in that it reflects on that simple relationship of doubling the distance, r, from a source reduces its influence, f, by ¼, the square of that separating distance, as f1/r2. This diagonal is simply referred to as the Prime Diagonal (PD) so as to separate it out from the many other diagonals that run parallel or perpendicular to it within the grid.

The 1st Diagonal to run parallel to the PD is none other than 1, 3, 5, 7, ... the odd numbers ... that form the basis of the odd-number summation series ... 0+1=1, 1=3=4, 3+5=8, .... which, of course, directly informs the PD, and thus, the ISL.

Since there are two 1st Diagonals ... one on either side of the PD ... we elected to convert the upper one to all even numbers by simply adding 1 to the lower values. This variation came to be known as Brooks Square or Brooks (Base) Square ... the latter reference is the generic form to remind the viewer of the original Base Square form from which it is derived.

With this variation, we now have all the whole number integers on each axis, their squares on the PD, all the whole number odd integers on the lower and all the whole number even integers on the upper 1st Diagonals, respectively, and the rest of the inner, blank grid. That blank inner grid ... once filled out with the rest of the number values ... is referred to as the Strict Inner Grid (SIG), to distinguish it from the general Inner Grid. The Inner Grid includes the 1st Diagonals. The axes and PD, as a unit, are referred to as the Bare Bones (BB) ... and add the 1st Diagonals to give BB+1. The classification of the parts became important as a tool to see and separate out the patterns of the matrix table.

Two major findings resulted from this treatment of the square:

1. A Pandora’s Box of inter-relational number value patterns was discovered between the SIG, Inner Grid, PD, and BB. These included:

• horizontal (row), vertical (column) and diagonal step and linear sequence patterns and shapes;
• geometrical shapes;
• Pythagorean, Fibonacci and the Golden Mean, Φ, relationships;
• and, a hugh, matrix-anchoring penta-based networking pattern.

The “Penta” patterns ... forming pervasive squares and diamond-squares ... became the dominant pattern in that they spotlighted the integration of the PD number values into the Inner Grid in a highly complex, but symmetrical manner. They completely underscored the unbelievable interconnectedness of the entire matrix.

2. With all the numbers forming such intricate patterns across the square, the question of what were the exceptions came to mind. Prime numbers are the only numbers without such patterns ... and they are completely absent from the SIG. Any odd number not in the SIG ... and not on the PD ... is prime. Since Brooks Square is infinite in that it encompasses all whole numbers, one can simply look to see if any questionable odd number is on the SIG and if it is not there ... and not on the PD ... it is prime! (Any competent programmer should be able to hack out the code to spit out every prime number from here to eternity. Please donate your prize money to the health, care and education of the underprivileged children of the world! Thank you!) Findings included:

• horizontal (rows), vertical (columns) and diagonals can be modulated to reveal the primes;
• a Periodic Table of Primes (PTOP) emerges naturally from the grid;
• utilizing the PTOP, proof of Euler's version of the Goldback Conjecture is presented.

In closing, let us review the logical dictates that lie at the foundational core of this paper and its assertion of relevance to The Architecture of SpaceTime:

1. The Conservation Laws (known- energy, mass-energy; charge; momentum; angular momentum; and, the more esoteric- force; spacetime and the velocity of light) are inviolate;

2. The conservation of the spin, charge and mass identifying parameters of subatomic particles must also be reconciled with the the above Conservation Laws;

3. The Four Forces of Nature (known- strong; weak; electromagnetic; gravity; and, the more esoteric Fifth Force- velocity of light...mediates the properties of SpaceTime, the latter being intimately informed by the Higgs Boson) ... must be invoked to direct and/or be the consequence of the above Conservation Laws;

4. Quantum Gravity, the merging of the heretofore irreconcilable differences of the granular, particulate and discrete quanta of Quantum Mechanics with that of the smooth, curvaceous, analog-like presentation of General Relativity (Gravity) must be more than just allowed ... it must be placed within a context in which it is inevitable, as are the melding of all the forces;

5. Dark Energy and Dark Matter, together accounting for over 90+% of the known Universe, must also be shown to be a major, natural, and logical ... and thus inevitable ... constituent of the Universe, addressing the larger concerns above as well as being part and parcel of that foundational core of SpaceTime itself.

Each of these five inviolate concerns listed above (and addressed individually and together in a number of white papers by this author, see below) requires both a reduction to the fundamental core of physical existence ... that is, a reduction down to the actual SpaceTime that all matter and energy reside in and are made up of ... and to a means, a vector, of communication between those SpaceTime units allowing them to flexibly inform, build and respond to their neighboring SpaceTime units in the ultimate display of group behavior. The unit is SpaceTime.

The vector is numbers. Quantitative numbers. The "how much" of "who-what-when-where-why-how." Such numbers must incorporate the entire spectrum of quantitation from zero to infinity. It must be discrete in and of itself (quanta), but be so inter-related ... interconnected ... to not just its immediate neighbor ... but ultimately to all its neighbors ... that the action at a distance "butterfly effect" of large space and time effects may be naturally born. It must be a template of potential ... from the basic to the complex, and yet, such exquisite diversity and intricateness is never more than combinations of its simple building blocks. While it obviously informs the ISL, Brooks (Base) Square, it is suggested, ultimately informs The Architecture of SpaceTime. Geometry for the twenty-first century.

The implications of this interconnectedness to the very fabric ... the architecture ... of spacetime was discussed. That the blissful, beautiful, sublime harmony of the natural Universe is fundamentally built on the abstract abyss of simple number relationships becomes more than sustainable ... it become undeniable ... inevitable.

Finally, within the Appendix of new findings, perhaps none is more truly amazing than Rules 179-180, under the heading "Sub-Matrix: The distillation of the Brooks (Base) Square." Here we find the simplest and most basic pattern of the simple whole number sequence ... 1,2,3,... forming a truly fundamental base layer ... a sub-matrix ... lying below the original grid. And every number on that original grid is predicated on this simple pattern of 1,2,3,.... We have come full circle. And it begs the question: Who is the parent and who is the child. It seems the parent has become the child.

The physical laws of Nature are gounded on a few unwaivering principles. One is that of the Universal Constants ... the velocity of light in a vacuum, the rest mass and spin of the electron, proton, neutron and other subatomic particles, the Planck constant, the fine structure constant, etc. All measured quantities. Another set are the forces of Nature. Currently, four are recoginized: gravitation, electromagnetic, weak and strong. A fifth has been postulated.

Tying all this together are the Conservation Laws: Conservation of energy, momentum, angular momentum, electric charge, mass (as mass-energy), leptons, baryons, and, spacetime (as proposed by the author). In essence, the total energy of spacetime, as revealed by the matter and energy fields that are its expression, is a constant ... it can neither be created or destroyed.

That the quantities, as expressed by numbers, that hold, tie, interconnect, transform, transcend and ultimately account for these principles ... these laws ... of Nature, should be contained within the Inverse Square Law matrix ... i.e. "Brooks (Base) Square" is increasingly becoming an obvious conclusion.

Rule 181 in the Appendix points to just such an obvious leap. Yes a leap, but the numerical connection to the Conservation Laws has to be built into the same numerical connection to that law of Nature that fundamentally defines the relationship of light, electromagnetism and gravity (amongst others). And that law, of course, is the ISL.

In Rule 181, the Conservation Rule, we see that a fixed length string of numbers, when wrapped around a fixed vertice of a right triangle, maintains a constant difference between the numbers, despite the number's values changing. This occurs throughout the grid.

Examination of the simple rules of the triangles so formed reveals not only the source of this constancy ... it's built into the sub-matrix ... but a straightforward connection to that most famous of mathematical (geometrical) laws, the Pythagorean Theorem. c2=a2+b2, the hypotenuse of a right triangle equals the square root of the sums of its sides.

In Rule 181, the Conservation Rule, we see that the sum of the short sides of a right triangle equal that of the long side (hypotenuse), presented in the most basic form. The difference in the progressive right triangles, all with the same vertice at the short sides, remains a constant, conserved quantity. The Pythagorean Theorem in the context of the ISL matrix becomes the source description of the Conservation Laws.

The Pythagorean Theorem relates the two short sides to the long side of a right trianglel It is the Pythagorean Theorem of Sides (PTOS): c2=a2+b2 (see TOAST>Geometric-Relationships>Pythagorean).

Now, in rule 181, the Conservation Rule, a new form of the Pythagorean Theory has been found. Within Brooks (Base) Square, it is the difference between the vertices of the right triangle that reveals a Pythagorean-like relationship whereby the sums of those vertice differences on the short side equal that of the long side: C-A=(C-B)+(B-A). We call this the Pythagorean Theorem of Vertices (PTOV). See Appendix ~click to enlarge image BS 1.00 Brooks (Base) Square (BS 1.00):Identical to BS 1.01 except for 1st upper diagonal from the Prime Diagonal is ODD, mirroring that of the lower 1st diagonal.. ~click to enlarge image BS 1.01 Brooks (Base) Square (BS 1.01):Identical to BS 1.00 except for 1st upper diagonal from the Prime Diagonal is EVEN. This is the first variation version from the Base Square.

NEXT: On to References

Back to III. Interconnectedness

Page 2a- PIN: Pattern in Number...from primes to DNA.

Page 2b- PIN: Butterfly Primes...let the beauty seep in..

Page 2c- PIN: Butterfly Prime Directive...metamorphosis.

Page 2d- PIN: Butterfly Prime Determinant Number Array (DNA) ~conspicuous abstinence~.

Page 3- GoDNA: the Geometry of DNA (axial view) revealed.

Page 4- SCoDNA: the Structure and Chemistry of DNA (axial view).

Page 5a- Dark-Dark-Light: Dark Matter = Dark Energy

Page 5b- The History of the Universe in Scalar Graphics

Page 5c- The History of the Universe_update: The Big Void

Page 6a- Geometry- Layout

Page 6b- Geometry- Space Or Time Area (SOTA)

Page 6c- Geometry- Space-Time Interactional Dimensions(STID)

Page 6d- Distillation of SI units into ST dimensions

Page 6e- Distillation of SI quantities into ST dimensions

Page 7- The LUFE Matrix Supplement: Examples and Proofs: Introduction-Layout & Rules

Page 7c- The LUFE Matrix Supplement: References

Page 8a- The LUFE Matrix: Infinite Dimensions

Page 9- The LUFE Matrix:E=mc2

Page 10- Quantum Gravity ...by the book

Page 11- Conservation of SpaceTime

Page 12- LUFE: The Layman's Unified Field Expose`

Page 13- GoMAS: The Geometry of Music, Art and Structure ...linking science, art and esthetics. Part I

Page 14- GoMAS: The Geometry of Music, Art and Structure ...linking science, art and esthetics. Part II

Page 15- Brooks (Base) Square (BS): The Architecture of Space-Time (TAOST) and The Conspicuous Absence of Primes (TCAOP) - a brief introduction to the series

Page 16- Brooks (Base) Square interactive (BBSi) matrix: Part I "BASICS"- a step by step, multi-media interactive

Page 17- The Architecture Of SpaceTime (TAOST) as defined by the Brooks (Base) Square matrix and the Inverse Square Law (ISL).