Brooks (Base) Square (BS) 101
~ The Architecture of SpaceTime (TAOST)
&
The Conspicuous Absence of Primes (TCAOP) ~
I. TAOST / C. Geometrics – relationships / 1. Pythagorean
I. TAOST  the network
1. Pythagorean <
II. TCAOP  everything minus the network
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  
I C. Geometrics  Relationships_1. Pythagorean
As we head deeper into the matrix, examining numerical relationships that include both lines and shapes that often move beyond the simple considerations we have touched on so far ... looking for those larger patterns. Hang on, for some of these new patterns are simply mind blowing in the majesty and beauty of their presence. What started as a simple table of relationships between the axial numbers and their square number counterparts depicting the Inverse Square Law (ISL) soon became much, much more.
How is it so? What is the mystery here in this magical mystery tour?
How does the Pythagorean Theorem (a^{2}+b^{2}=c^{2}) appear?
Part 1Pythagorean ((a^{2}+b^{2}=c^{2})
It does so in the form of triples ... Pythagorean triples ... of whole integer numbers. From Wikipedia (http://en.wikipedia.org/wiki/Pythagorean_theorem) we see a list of “Primitive Pythagorean triples up to 100.”
3,4,5 16,63,65
5,12,13 20,21,29
7,24,25 28,45,53
8,15,17 33,56,65
9,40,41 36,77,85
11,60,61 39,80,89
12,35,37 48,55,73
13,84,85 65,72,97
(listed by http://creativecommons.org/licenses/bysa/3.0/)
Yet it is multiples of the simplest ratio ... 3, 4, 5 ... that appear redundantly on the grid as an ever expanding pattern. Perhaps it is the very simplicity of the matrix that lends itself to the simplest of patterns.
It was the pattern itself that led down the path of parallelograms, in general, ultimately to that watershed of numerical relationships ... the penta ... and the symmetry of numbers.
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83  BS Rule 83: The 1^{st} generation of Pythagorean parallelograms that extend perpendicular to the Prime Diagonal (PD) and consist of all, even and odd, multiples of 12, 16, 20 separated by one step. The 2^{nd} generation by two steps, 3^{rd} generation by four steps, the 4^{th} generation by eight steps, and so on. 

Note: The sequential multiples of 12, 16, 20 continue downward from the PD as 4x, 5x, 6x, 7x, 8x, 9x, 10x, ....

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84  BS Rule 84: The oddmultiple 1^{st}, 2^{nd}, 3^{rd}, ... generations of Pythagorean triples also extend and overlap along the 2^{nd} diagonal parallel to the PD. 

Note:

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85  BS Rule 85: Plotting out single instances of the Pythagorean triple multiples (based on 3, 4, 5) reveals parallelogramlike shapes which step out from the PD in sequential steps, as 2, 4, 6,...steps. 

Note: This pattern focuses on multiples of 5. Difference, ∆, steps.

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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  
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86  BS Rule 86: Plotting out single instances of the Pythagorean Triple multiples (based on 3, 4, 5) in evennumbered rows across the PD reveals a natural progression of steps in going from side "a" to "b" to "c" as 111,222,333,..., respectively. 

Note: This pattern focuses on multiples of 2,3,4 and 5 in that each new Pythagorean Triple ... horizontally placed on a row, based on 345, that has side "a" in the lower SIG triangle, side "b" on the PD, and the hypotenuse, "c" in the upper SIG triangle... is located on the even numbered rows of the side axis. The first triple number ("a") is 3 and all subsequent "a's" are multiples of 3. The first "b" triple number is 4 and all subsequent "b's" on the PD are multiples of 4. The "c's" are all multiples of 5.

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While I told myself I was not going to look for ... and describe ... any other simple number patterns before stopping and writing up what has been found thus far, I must digress ... briefly. This pattern below is so blatantly obvious, yet I didn’t see it before. It, along with a few other others ought to be in your, dear reader, toolbox before we really get into the best part of this tour.
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87  BS Rule 87: Add the horizontal axis number to the vertical axis number to give the sum, ∑, at their intersecting grid point at the 1^{st} Diagonal. Note: The number next to it on the row at the 2nd diagonal is the sum of its horizontal and vertical axis number values x2. The number next to it on the row at the 3^{rd} diagonal is the sum of its horizontal and vertical axis number values x3. The number next to it on the row at the 4^{th} diagonal is the sum of its horizontal and vertical axis number values x4, and so on. 

Note:

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88  BS Rule 88: The difference, ∆, in the sums, ∑, of the axial numbers in sequence ...1,2,3,...follows as the sequence spacing, 2=2,3=3, 4=4,... 

Note:

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89  BS Rule 89: The difference, ∆, in the sums, ∑, of the PD numbers in sequence ...1,4,9,...follows as the double of the sequence spacing, 2=4,3=6, 4=8,... 

Note:

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90  BS Rule 90: Within the Inner Grid, the difference, ∆, in the sums, ∑, of the sums of successive rows reduces ultimately to 4. With the addition of the number values in the PD, the difference, ∆, in the sums, ∑, of the sums of the successive rows reduces ultimately to 4. 

Note: And includes every other odd # before reduction to 4.

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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  
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91  BS Rule 91: Within the Inner Grid, the difference, ∆, in the sums, ∑, of the running sums of successive number values in a given column reduces to 2. 

Note: And includes every odd # before reduction to 2.

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92  BS Rule 92: Within the Inner Grid, the difference, ∆, in the sum, ∑, of the last row minus the sum of the number values in Column B = the difference, ∆, in the sum of Column C number values minus the sum from the row above. 
Note: ∆(∑_{last row } ∑_{Col B}) = ∆(∑_{Col C }∑_{row above}) 
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93  BS Rule 93: Within the Inner Grid, the difference, ∆, in the sum, ∑, of the last row minus the sum of the number values in Column B = the difference, ∆, in the sum of Columns C+D number values minus the sum from their respective rows above. 
Note: ∆(∑_{last row } ∑_{Col B}) = ∆(∑_{Col C+D }∑_{respective rows above}) 
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94  BS Rule 94:Within the Inner Grid, the difference, ∆, in the sum, ∑, of the last row minus the sum of the number values in Column B = the difference, ∆, in the sum of Columns C+D+E number values minus the sum of their respective rows above. 
Note: ∆(∑_{last row } ∑_{Col B}) = ∆(∑_{Col C+D+E }∑_{respective rows above}) 
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95  BS Rule 95: Within the Inner Grid, the difference, ∆, in the sum, ∑, of the last row minus the sum of the number values in Column B = the difference, ∆, in the sum of Columns C+D+E+F number values minus the sum of their respective rows above.

Note: ∆(∑_{last row } ∑_{Col B}) = ∆(∑_{Col C+D+E+F}∑_{respective rows above}) 
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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  
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96  BS Rule 96: Within the Inner Grid, the difference, ∆, in the sum, ∑, of the last row minus the sum of the number values in Column B = the difference, ∆, in the sum of Columns C+D+E+F+G number values minus the sum of their respective rows above.

Note: ∆(∑_{last row } ∑_{Col B}) = ∆(∑_{Col C+D+E+F+G }∑_{respective rows above}) 
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97  BS Rule 97: Within the Inner, the difference, ∆, in the sum, ∑, of the last row minus the sum of the number values in Column B = the difference, ∆, in the sum of Column C+D+E+F+G+H number values minus the sum of their respective rows above. 
Note: ∆(∑_{last row } ∑_{Col B}) = ∆(∑_{Col C+D+E+F+G+H }∑_{respective rows above}) 
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98  BS Rule 98: Within the Inner Grid, the difference, ∆, in the sum, ∑, of the last row minus the sum of the number values in Column B = the difference, ∆, in the sum of Columns C+D+E+F+G+H+I number values minus the sum of their respective rows above.

Note: ∆(∑_{last row } ∑_{Col B}) = ∆(∑_{Col C+D+E+F+G+H+I }∑_{respective rowsabove}) 
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99  BS Rule 99: On the Strict Inner Grid (SIG), the number,#, of diagonal steps to reach the numerical value that is the sum, ∑, of the two evennumbered grid values further down that even diagonal continues in an orderly and predictable pattern. 

Note: Likewise, the # of steps down the diagonal to the first addition number grid value (starting with 2, it always being counted as two steps; starting with 4, it always being counted as 3 steps; starting with 6, it always being counted as 4 steps, and so on for each successive even diagonal) = the number of steps from the last addition number to the sum number of the two number values. Example: On the 2 diagonal: 28+32=60 and 60 is 7 steps down the diagonal from 32. It is also 7 steps from 2 ... the axial number topping the 8, 12, 16... diagonal, and remember it has a builtin count of 2, so 2 + 5=7 steps. If you go to the next even diagonal starting at axial number 4, its builtin step count is 3. Axial 6 has a builtin step count of 4, and so on. As the axial number grows by 2, the builtin step count grows by 1.

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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  
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I. TAOST>IC. Geometricsrelationships>IC2. Parallelograms  Brooks (Base) Square
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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 interactive (BBSi) matrix: Part I "BASICS" a step by step, multimedia interactive
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  