Brooks (Base) Square (BS) 101
~ The Architecture of SpaceTime (TAOST)
&
The Conspicuous Absence of Primes (TCAOP) ~
VI. Appendix A
A Brief Introduction
I. TAOST  the network
II. TCAOP  everything minus the network
III. Interconnectedness <
TAOST: Rules 150  Rules 5180  Rules 8199  Rules 100107  Rules 108153  TCAOP: Rule 154  Rules 155157  Rules 158159  Rule 160  Interconnectedness: Rules 161175  Appendix A: Rules 176181  Appendix B: Rules 182200  
VI. Appendix A : Additions
The well of discoveries to be had in Brooks (Base) Square is enormous ... perhaps infinite. As new discoveries are made we will attempt to bring those highlights here under the banner of "Appendix" ... providing a focused look at a some aspect of the matrix that reveals yet another deep connection between the Inverse Square Law (ISL) and the numbers of the Universe.
Rule 176: Progressive Inner Grid Triangle Sets inform the Prime Diagonal (PD)
Here we are taking a look at how sequentially larger "parent" Triangle Sets ... and their "child" triangle sets contained within ... all work in concert to support the next ISL number on the PD. Yes it is true, once the pattern is discerned it only seems inevitable that the progression works as it does. But don't be fooled into thinking that now what is obvious must be mundane, for it is the very nature of the matrix that complex patterns are simply simple patterns in a choreographed dance. Learn the simple steps so that you may be able to realize the full dance. See Rules 176A176J.
Rule 177: Prime Triangle Sets within the Periodic Table of Primes (PTOP)
In the spirit of discovery ... especially in that one discovery often leads to another ... a quick look at the triangles inside the PTOP embedded in Brooks (Base) Square has revealed yet another prime pattern. See Rules 177A177E.
Rules 178180: SubMatrix: The distillation of the Brooks (Base) Square
The simplest and most basic pattern of the simple whole number sequence ... 1,2,3,... forming a truly fundamental base layer ... a submatrix ... lying below the original grid. Every number on that original grid is predicated on this simple pattern of 1,2,3,....
Rules 181: Conservation (Laws) in the Brooks (Base) Square
Fixed numbers of nodes, i.e. number units, forming a fixed length string, have conserved number differences when they lie over the two shorter sides of a right triangle ...suggesting a natural link to how the conservation laws of Nature might numerically unfold.
But first, a quick review:
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Rule 176: Progressive Inner Grid Triangle Sets inform the Prime Diagonal (PD)
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176  BS Rule 176 Overview: Now with these review rules in mind, let's look at number patterns and weights (sums) of progressive Inner Grid Triangle Sets all under the banner of Rule 176. Common to all Inner Grid Triangle Sets in Brooks (Base) Square:


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176A  BS Rule 176Set A: Triangle Set A: 1 unit, with apex numbers: 3. 
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176B  BS Rule 176Set B: Triangle Set B: 3 units, with apex numbers: 358. 
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176C  BS Rule 176Set C: Triangle Set C: 6 units, with apex numbers: 3715. 
Note:


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176D 
BS Rule 176Set D: Triangle Set D: 9+1=10 units, with apex numbers: 3924. 
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176E  BS Rule 176Set E: Triangle Set E: 15 units, with apex numbers: 31135. 
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176F  BS Rule 176Set F: Triangle Set F: 21 units, with apex numbers: 31348. 
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176G  BS Rule 176Set G: Triangle Set G: 27+1=28 units, with apex numbers: 31563. 
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176H  BS Rule 176Set H: Triangle Set H: 36 units, with apex numbers: 31780. 
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176I  BS Rule 176Set I: Triangle Set I: 45 units, with apex numbers: 31999. 
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176J  BS Rule 176Set J: Triangle Set J: 54+1=55 units, with apex numbers:321120 . 
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From this analysis of the Brooks (Base) Square, we can see that progressively lager Triangle Sets within the Inner Grid increase their total sums ... and the patterns they contain ... in a simple and orderly manner, not the least of which is that the number progression of the Inverse Square Law Pd (1,4,9,16,25,36,...) is recoverable in those total sums.
Rule 177: Prime Triangle Sets within the Periodic Table of Primes (PTOP)
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177A  BS Rule 177Set A: The number sums, ∑, from a Triangle Set formed from two odd numbers from the upper horizontal axis and, the rightangle apex number of the isosceles triangle formed within the Inner Grid, are prime at least once in every row and every column in that grid. 
Note: Rows 13. Bold numbers are prime. Example shown is for Triangle Sets up to number 37 on the axis and PD.

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177B  BS Rule 177Set B: The number sums, ∑, from a Triangle Set formed from two odd numbers from the upper horizontal axis and, the rightangle apex number of the isosceles triangle formed within the Inner Grid, are prime at least once in every row and every column in that grid. 
Note: Rows 46. Bold numbers are prime. Example shown is for Triangle Sets up to number 37 on the axis and PD.

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177C  BS Rule 177Set C: The number sums, ∑, from a Triangle Set formed from two odd numbers from the upper horizontal axis and, the rightangle apex number of the isosceles triangle formed within the Inner Grid, are prime at least once in every row and every column in that grid. 
Note: Rows 79. Bold numbers are prime. Example shown is for Triangle Sets up to number 37 on the axis and PD.


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177D 
BS Rule 177Set D: The number sums, ∑, from a Triangle Set formed from two odd numbers from the upper horizontal axis and, the rightangle apex number of the isosceles triangle formed within the Inner Grid, are prime at least once in every row and every column in that grid. 
Note: Rows 1012. Bold numbers are prime. Example shown is for Triangle Sets up to number 37 on the axis and PD.

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177E  BS Rule 177Set E: The number sums, ∑, from a Triangle Set formed from two odd numbers from the upper horizontal axis and, the rightangle apex number of the isosceles triangle formed within the Inner Grid, are prime at least once in every row and every column in that grid. 
Note: Rows 1317. Bold numbers are prime. Example shown is for Triangle Sets up to number 37 on the axis and PD.

Rules 178180: SubMatrix: The distillation of the Brooks (Base) Square
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178  BS Rule 178 The SubMatrix: The sums, ∑, of all the contained numbers within progressively larger triangles in the Inner Grid ...starting with 3, 385, 15127, ..., are evenly divisible progressively by the whole number sequence ... starting with 3, 4, 5, ..., respectively. 

Note: Same occurs for Rule 176.

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179  BS Rule 179 The SubMatrix: Within the Inner Grid, every progressive number of every column or row is evenly divisible , progressively, by a whole number sequence. Plotting those divisors reveals a "submatrix" grid, underlying the original Inner Grid, of a simple whole number sequence pattern. 

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180  BS Rule 180 The SubMatrix: Within the Inner Grid, every progressive number of every column or row is evenly divisible , progressively, by a whole number sequence. Plotting those dividends reveals a truly fundamental "submatrix" grid, underlying the original Inner Grid, of repeating simple whole number sequences, i.e. 1,2,3, ..., horizontally and vertically ...forming the simplest, most basic grid pattern possible, that also includes the axis numbers. All numbers of the original Brooks (Base) Square are predicated on this simple whole number sequence pattern. 
Note: This amazing submatrix grid pattern explains the entire parent grid matrix and the Inverse Square Law, ISL, relationship that unfolds. See an animated example by CLICKING IMAGE above. Patience. Follow the bold numbers. 
Rule 181: Conservation (Laws) in the Brooks (Base) Square
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181  BS Rule 181 The Conservation Laws: Take any number of sequential nodes, i.e. number units, from a diagonal array (parallel to the Prime Diagonal) to form a fixed length string of nodes. Wrapping that fixed length string variously around the short sides of the right triangle so formed from this diagonal and the one 90 degrees perpendicular ... with the longside hypotenuse formed from the beginning and end of said string ... the differences between the numbers included in that fixed string length will remain a constant, conserved quantity. The fixed node length string acts like a conserved unit in that no matter how it is manipulated ... in this case wrapped around a corner vertice forming a triangle ... the difference between the end and the beginning of the strings is no different from the difference of the twolegged route around the shortsides of the triangle formed. Advance the string and the results are the same. It is the difference in the number unit values which implies the conservation laws. The loss in number value in going from LEFT to RIGHT equals the gain in number values going from TOP to BOTTOM when going through the same vertice node. Furthermore, for those fixed length node strings formed along the 1st diagonal from the Prime Diagonal, the values of those differences are actually delineated to the side and bottom of the shortside vertice number unit. See examples. 

Note: Restated: 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 Pythagoreanlike relationship whereby the sums of those vertice differences on the short side equal that of the long side: CA=(CB)+(BA). As "A", "B" and "C" are the number values of the vertices of a right triangle, we call this the Pythagorean Theorem of Vertices (PTOV). The difference, ∆, in the sums of those number values at A, B, & C remains a constant within any right triangle formed with a given string node length and passing through the same shortside vertice. See an animated example by CLICKING IMAGE above. Patience. Follow the bold numbers.
The Pythagorean Theorem of Vertices (PTOV) CA=(CB)+(BA). 
The Conservation Laws: Conservation of energy, momentum, angular momentum, electric charge, mass (as massenergy), 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.
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TAOST: Rules 150  Rules 5180  Rules 8199  Rules 100107  Rules 108153  TCAOP: Rule 154  Rules 155157  Rules 158159  Rule 160  Interconnectedness: Rules 161175  Appendix A: Rules 176181  Appendix B: Rules 182200  
NEXT: On to
Appendix B
Back to Conclusion
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 DarkDarkLight: 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 SpaceTime 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: IntroductionLayout & Rules
Page 7c The LUFE Matrix Supplement: References
Page 8a The LUFE Matrix: Infinite Dimensions
Page 9 The LUFE Matrix:E=mc^{2}
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 SpaceTime (TAOST) and The Conspicuous Absence of Primes (TCAOP)  a brief introduction to the series
Page 16 Brooks (Base) Square (BBS) interactive matrix: BASICS (Part I)~ 5 presentations
Page 17 The Architecture Of SpaceTime (TAOST) as defined by the Brooks (Base) Square matrix and the Inverse Square Law (ISL).
The LUFE Matrix  The LUFE Matrix Supplement  The LUFE Matrix: Infinite Dimensions  The LUFE Matrix: E=mc^{2}  Dark Matter=Dark Energy  The History of the Universe in Scalar Graphics  The History of the Universe_update: The Big Void  Quantum Gravity ...by the book  The Conservation of SpaceTime  LUFE: The Layman's Unified Field Expose`  
net.art index  netart01: RealSurReal...aClone  netart02: funk'n DNA/Creation GoDNA  netart03: 911_remembered  netart04: Naughty Physics (a.k.a. The LUFE Matrix)  netart05: Your sFace or Mine?  netart06: Butterfly Primes  netart07: Geometry of Music Color  net.games  Art Theory 101 / White Papers Index  