Brooks (Base) Square (BS) 101

 ~ The Architecture of Space-Time (TAOST)

 &

The Conspicuous Absence of Primes (TCAOP) ~


I.  TAOST / C. Geometrics – relationships / 3. Penta


Table of Contents




I. TAOST - the network

 

        A. Geometrics – lines

        B. Geometrics - shapes

        C. Geometrics - relationships

                1. Pythagorean

                2. Parallelogram

                3. Penta <---


II. TCAOP - everything minus the network




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 |





I C. Geometrics - Relationships - 3. Penta


Part 3- Penta 


Finally, we have arrived and entered the magic kingdom of the “Penta” ... number relationships centered around multiples of 5. This is quite a large section. The beauty ... and the devil ... are necessarily in the details ... and many of those details require looking at just the 1st (“ones”) and/or 2nd (“tens” together with the “ones”) number values throughout the pattern. The reader will be rewarded with a breathtaking panorama ... an exquisite choreography ... of intricate number relationships and the wonderful patterns that unfold. Ultimately, the larger vista will become apparent and one can delve into the structural details as one so desires.


Part of the sublime beauty of 5-based patterns is knowing that the pentagon contains the Golden Mean and together it implicitly informs the design structure of much of what we know as organic life ... including the very shape of our own DNA double helix spiral (axial view). The DNA Master Chart (from “GoDNA: The Geometry of DNA”) explicitly draws out the pentagon reference of the double helix ... most especially for the manner in which the concentric double-pentagons resonate and predictably inform the various parts of the structure to the whole of its axial patter. Such perfect reproducibility is, of course, the hallmark of this molecular species. 


Now finding that a similar resonating pattern based on 5’s is fundamentally built into the Brooks (Base) Square of the Inverse Square Law is so absolutely profound that one must entertain the idea that perhaps ... just perhaps ... it is the actual blueprint for the Architecture of SpaceTime itself!


Onward. 


Note: Reference to the 1st number = “ones” value and/or to the 2nd number = “ten” and “ones” number values follow as per this example: numbers 3, 42, 58. If we are referring to the 1st number, we have 3, 2, and 8, respectively. If referring to the 2nd number values, we have 03, 42, and 58, respectively.




Brooks (Base) Square
~click to enlarge image
 108  

BS Rule 108: Building on the “nodes” made from grid values that are multiples of 5, we find a natural unit on the grid to be a square grid of nodes that will be referred to as 5x5’s. Because the geometry of adjacent squares with crossing diagonals leads to the formation of diamond-squares ... and we will need to reference both shapes ... we will designate the former squares (with borders parallel to the axis borders) as simply 5x5’s, and, the latter diamond-squares (with borders 45to the axial borders) as 5x5’ds. Note the inclusion of the “d” for diamond.

 
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1st Number Penta Relationships


Brooks (Base) Square
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 109  

BS Rule 109: Take any 5x5’s from the Strict Inner Grid (SIG) and:

  • 6 numbers (#s) across ∑=25; two ends are nodes, four remaining ∑=20
  • 6 #s up and down the ∑=25; two ends are nodes, four remaining ∑=20
  • 6 #s on one diagonal ∑=0; two ends are nodes, all even 5-based ∑=0
  • 6 #s on other diagonal ∑=30; two ends are nodes, all odd 5-based, four remaining ∑=20
  • any row starting with 5 ends with 0, and vice versa
  • any column starting with 5 ends with 0, and vice versa
  • interior row ∑ between nodes always go as: 14-16-16-14
  • interior column ∑ between nodes always go as: 16-14-14-16
  • perimeter ∑ always = 90 (100 if nodes overlap)
  • nodes alternate odd/even in every direction
  • if node is 5, next column # on either side is 6
  • if node is 5, net row # on either side is 4, and, 4+6(above)=10
  • if node is 0, next column # on either side is 1
  • if node is 0, next row # on either side is 9, and 9+1(above)=10
  • if node is 5, next diagonal # on either side is 5, and 5+5=10
  • if node is 0, next diagonal # on either side is 0, and, 0+0=0
  • sequential diagonal #s symmetrically have reverse order and their ∑ always follow as: 5-10-15-20-25-0-25-20-15-10-5 one way and as 0-10-10-20-20-30-20-20-10-10-0 the other
  • column and row #s between nodes have reverse order and their perimeter cross-∑, column to row, always = 10
  • the ∑ of all # values = 150
  • 5x5’s next to the axis or PD are slightly different, as we shall see.


 

Brooks (Base)Square
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 110  

BS Rule 110: 5x5’s built next to the axis, incorporating Column A, are slightly different. The vertical axis of Column A values is, of course, a straight linear sequence 1, 2, 3, .... The ∑ of the first set of 1st #s ... 1-2-3-4-5=15, and that of the next set 5-6-7-8-9-0=35. Together ∑set 1 and ∑set 2 =50, with an average of 25. 25 is the same ∑, node to node, of all interior 5x5’s. The ∑vertical Column A 5x5’s alternate as 35-15-35-15-.... The ∑horizontal Column A 5x5’s, except the *first, go as 25-25-25-25-.... (If the # in Column A ends in 5, the ∑hor=25. If the # in Column A ends in 0, then the ∑hor=25 as the nodes alternate 5-0-5-0-... *except for the first 5x5’s which intercepts the PD. See below.)


 
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Brooks (Base) Square
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 111  

BS Rule 111: 5x5’s built with one node on the PD are also slightly different from those in the SIG in that those 5x5’s with a node on the PD ending in 5, the ∑hor=30 and those ending in 0, the ∑hor=25. The ∑hor  along the PD alternate as 30-25-30-25-... as the 5-based PD values 25, 100, 225, 400, ... alternate odd-even, respectively.


 

Brooks (Base)Square
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 112  

BS Rule 112: Combining 5x5’s from the SIG with those built into the Column A axis, and those built into the PD, we see that the sum of all, ∑all of every 1st number value within a given 5x5’s set equals:

  • 150 in the SIG
  • alternates 160-140-160-140-... within Column A
  • alternates 150-155-150-155-... within the PD, and
  • has one exception: the first 5x5’s that intercepts both Column A and the PD, equals 165.
  • the triangle that includes the PD always equals 100


Note: 



Column A

Col A+PD

PD

SIG

PD

triangle

 

165



100

140


150

 

100

160


155

150

100

140


150

150

100




150


160


155

150

100




150





150


140


150

150

100




150





150





150




 
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Brooks (Base) Square
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 113  

BS Rule 113: There are 36 grid number values contained in every 5x5’s.


Note: 



#/side

x

#/side

total

6

 

6

36



 
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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 |


Brooks (Base) Square
~click to enlarge image
 114  

BS Rule 114:  All but 8 of 36 grid number values are derived directly from the 1st number values of the PD ... exclusively 0-1-4-9-6-5.


Note: Restated: Six #s ...0-1-4-9-6-5 ... account for all the 1st #s of the PD, and, for all 28 of the possible 36 1st # values of any 5x5’s within the SIG. The 1st # values 2-3-7-8 (twice over) account for the remaining values. 


 

Brooks (Base)Square
~click to enlarge image
 115  

BS Rule 115: The perimeter of any SIG 5x5’s contains only 0-1-4-9-6-5 1st number values with opposite sides running in reverse order. Values in adjacent sides always add up to ∑=10 in one of two alternating patterns ... both with 180rotational symmetry.


 
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Brooks (Base) Square
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 116  

BS Rule 116: The interior of any SIG 5x5’s contains only 0-2-3-7-8-5 1st number values, and since the inner diagonals are either all 0 or all 5, the remaining 8 grid spots ... 2 in each quadrant ... are filled exclusively with non-PD 1st number values 2-3-7-8 in a rotational-symmetry array that always places 7 -3 and 8-2 kiddy-corner, respectively, so that their ∑=10.


 

Brooks (Base)Square
~click to enlarge image
 117  

BS Rule 117: Combining the two sets of alternating patterns (Rule 115-116), gives us the matrix grid of Brooks (Base) Square.


Note: Individual 5x5’s (except for Column A and PD variations) and cluster sets of 4, 16 and more 5x5’s have both mirror and 180rotational symmetry.


 
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Brooks (Base) Square
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 118  

BS Rule 118: Looking at the nodes, we see two patterns of 1st number values radiating outward and alternating within the SIG.


Note: 


center=0

center=5

5

6

9

4

1

5-4-1-6-9-0-9-6-1-4-5

1

4

9

6

5

0

1

4

9

6

0-9-6-1-4-5-4-1-6-9-0

6

9

4

1

0

∑ at 90〫 degrees=10

∑ at 90〫 degrees=10









 
TOP ]


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 |


Brooks (Base) Square
~click to enlarge image
 119  

BS Rule 119: Combining the two alternating node patterns (above) into this 5x5’s 4-cluster set, we can easily see the 180rotational symmetry in both the cluster and individual cells, as well as the mirror symmetry between adjacent cells.


Note: Also note that a given cell is repeated diagonally in either direction.

 

 

Brooks (Base)Square
~click to enlarge image
 120  

BS Rule 120: If the 1st node of a 5x5’s is 5, then its next two interior numbers ... taken from the non-PD interior number roster 2-3-7-8... will be 2-8. If the node is 0, its next two interior numbers will be 7-3.


Note: 



node=5

node=0

2

8

7

3



 
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Brooks (Base) Square
~click to enlarge image
 121  

BS Rule 121: The 1st number sums, ∑, of the interior 2-7-3-8 series, the cross-diagonal 5-0 series, and the net perimeter series (nodes + #s between) on any SIG 5x5’s always = 150.


Note: 



Perimeter

Interior

80+10=90

20+40=60

150




 

Brooks (Base)Square
~click to enlarge image
 122  

BS Rule 122: The 5-base triangles formed as a 5x5’s butts up to the PD at successive nodes have identical 1st  number values, as the PD is itself 1st number symmetrical about each 5-based node.


Note: Perimeter = 5+20+5+20+0+20 = 70 and interior = 30. The 1st #s are identical in value on all three sides. This pattern naturally fits in with the staggered, alternating 5x5’s pattern whose base and height 1st # values are identical to and derived from those of the PD!



Node

Interior

Base

Interior

Height

Interior

Hypotenuse

Interior

10

20

20

20

30

100




 
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Brooks (Base) Square
~click to enlarge image
 123  

BS Rule 123: The triangles formed from 5x5’s, node to node, may also be seen as quadrants of a larger diamond-square (5x5’ds). When doubled in size, they may be seen joining every other 5-based node as halves of larger 5x5’s squares or quadrants of 5x5’ds diamond-squares.


Note: Large 5x5’ds diamond-squares formed as vertices of even 5-based nodes are all identical, as to pattern and values, of their 1st number values.



 
TOP ]


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 |


Brooks (Base) Square
~click to enlarge image
 124  

BS Rule 124: Plotting only 1st # 5-based nodes and the 0-0 and 5-5 crisscrossing diagonals on Brooks (Base) Square shows us the 1st number pattern of all the 0’s and 5’s.


Note: This naturally reveals concentric diamond-squares alternating 5’s and 0’s.




 

Brooks (Base)Square
~click to enlarge image
 125  

BS Rule 125: Filling one such large diamond-square (225-625-1000-600) with its 1st number interior numbers (0-3-7-8-2-5), again demonstrates the mirror symmetry about the 25-625 horizontal line and the 180rotational symmetry for the 1st #s.


Note: Notice how any intermediate diamond-square (e.g. 300-400-800-600) has identical 1st numbers. Within these, any smaller diamond-square is mirrored across the center point. 





 
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Brooks (Base) Square
~click to enlarge image
 126  

BS Rule 126: Every diagonal consists of all odd or all even 1st number values and these alternate across the matrix in both diagonal directions. They repeat every 10th diagonal. There are always four values between 0 and 5, just like on the PD. The even diagonals have 1st # values of 0-2-4-6-8, the odds 1-3-5-7-9. While consistently repeating in a given diagonal, each odd and each even diagonal has a unique sequence (9 versions) until it repeats at every 10 diagonals.


 

Brooks (Base)Square
~click to enlarge image
 127  

BS Rule 127: The crisscross pattern of 1st # 5-based nodes described in Rule 124 can be shown to surround sets of odd/even diagonal 1st #s that are centered on 0 or 5, respectively. In between the crisscross of the 5-based nodes and the odd/even diagonal diamond sets is a perimeter diamond set... like a matt around an image in a picture frame ... containing the opposite odd/even diagonal numbers from that of the interior.


Note: Restated: If a diamond-square has a center of 5 (an odd #), it will be nested in a larger node diamond of 0’s (even #s), and in between, concentric diamond layers will radiate outward, alternating even-odd-even-odd and vice versa. A diamond-square with 0 at its center will radiate as 0-odd-even-odd-even-5.





 
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Brooks (Base) Square
~click to enlarge image
 128  

BS Rule 128: Combining rules 125 and 126 for 1st # diagonals, we can see the full pattern of 5-based nodes and their internal 1st # values as: 

  • alternating diamond-squares
  • the even diagonal diamond sets have their 1st # values (4-8-2-6) and sequence taken from the 2nd, 12th, 22nd, ... diagonal from the PD (every 10th diagonal)... and, are centered on 5 and surrounded by their opposite and alternating 1-3-5-7-9 1st # values
  • the odd diagonal diamond sets have their 1st # values (9-3-7-1) and sequence taken from the 7th, 17th, 27th, ... diagonal from the PD (every 10th diagonal) ... and are centered on and surrounded by their opposite and alternating 6-8-0-2-4 1st # values
  • both odd and even diagonal diamond sets repeat every 4 diagonal steps in both diagonal directions
  • both odd and even diagonal diamond sets are 180rotationally symmetric (mirror symmetry on center).


Note: The vertices #s for the even nodes are all PD #s (4-6), internal #s are not (2-8). Also, the vertices #s for the odd nodes are all PD #s (1-9), internal #s are not (3-7).



even

one quadrant

all four 

quadrants

∑ odd

one quadrant

all four

quadrants

6+4=10

8+2=10


1+9=10

5=5


6+4=10

5=5

50

125

1+9=10

3+7=10


6+4=10

0=0


9+1=10

0=0

40

100



 
TOP ]


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 |



2nd Number Penta Relationships


Brooks (Base) Square
~click to enlarge image
 129  

BS Rule 129: Adding the 2nd # ... that is the “ten” and “ones” number values ... to the diagonal diamond sets spotlights a larger mirror symmetry radiating out from the 625 horizontal and vertical symmetry line. The pattern above the 25-625 horizontal line is mirrored below. Additionally, the pattern to the left of the vertical 25-625 line is mirrored to its right.


Note: New, larger symmetries revel themselves as increasing squares of the 5-based unit. The crisscross of 0 and 5 diagonals alternates from 5x5’s, cell to cell, and are themselves interspersed within a larger grid pattern.


This is a brief introduction to the number relationships of the 2nd (“tens” and “ones”) # values. You may want to revisit this rule after reading through this next little section on the 1st and 2nd # values and the PD.



5-based PD nodes

minor symmetry lines

major symmetry lines

52=25


x

102=100

x


152=225

x


202=400

x


252=625


x

302=900

x


352=1225

x


402=1600

x


452=2025

x


502=2500


x



 

Brooks (Base)Square
~click to enlarge image
 130  

BS Rule 130: The Prime Diagonal (PD) is symmetric to its 1st # value about every 5-based node. 


Note: The 1st # valuses repeat every 10th #.



1st # Value

action

steps

∑ of steps

1s 

repeat 

8+2

10

4s

repeat

6+4

10

9s

repeat

4+6

10

6s

repeat

2+8

10


 
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Brooks (Base) Square
~click to enlarge image
 131  

BS Rule 131: The 1st # values of the PD ... 0-1-4-9-6-5 ...are present on every 5-based square or diamond-square, alternating their sequence as 0-1-4-9-6-5 or 5-6-9-4-1-0 vertically, and, 0-9-6-1-4-5 or 5-4-1-6-9-0 horizontally, between every 5-based node. 


Note: There are two, crisscross horizontal-vertical patterns. The one whose diamond center lines are intercepted by a vertical column 0-PD and a horizontal row 5-PD, and the other by the intercept of a vertical column 5-PD and horizontal row 0-PD. All patterns are equal (see next) when presented as diamond-squares in that they are naturally staggered. 1st # value patterns are identical across the grid. 2nd # value patterns are not.


 

Brooks (Base)Square
~click to enlarge image
 132  

BS Rule 132: The 1st # PD Pattern 1, based on vertical column 0-PD and horizontal row 5-PD, shows the 1st # values to be identical to those in Pattern 2 (Rule 133, below). 


Note: The 2nd # patterns, which overlap here, are different. If the diamond-square pattern butts up to include the actual PD values, exchange that 0 value for 5.



1st #s 

center=5

1

4

9

6

0

1

4

9

6

0-9-6-1-4-5-4-1-6-9-0

6

9

4

1

0

5

6

9

4

1

∑=45

∑ opposite axes pairs=10




 
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Brooks (Base) Square
~click to enlarge image
 133  

BS Rule 133: The 1st # PD Pattern 2, based on vertical column 5-PD and horizontal row 0-PD, shows the 1st # values to be identical to those in Pattern 1 (Rule 132, above). 


Note: The 2nd # patterns, which overlap here, are different. If the diamond-square pattern butts up to include the actual PD values, exchange that 0 value for 5.



1st #s 

center=5

1

4

9

6

0

1

4

9

6

0-9-6-1-4-5-4-1-6-9-0

6

9

4

1

0

5

6

9

4

1

∑=45

∑ opposite axes pairs=10








 
TOP ]


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 |


Brooks (Base) Square
~click to enlarge image
 134  

BS Rule 134: The Prime Diagonal (PD) is symmetric to its 2nd # values (‘ten” and “ones”) about the 5-based node 625. Specific PD-derived patterns inform the 1st and 2nd # values across all the 5-based rows and up and down all the 5-based columns within any square of diamond square. The patterns of the 1st # values are not the same as those of the 2nd.


Note: Because of this larger symmetry, the 2nd # values do not repeat until they span symmetrically across the 625 node. It follows this order as shown in this vertical column example from 100-PD and down across the 25-625 symmetry line. It is easiest to track these sequential progressions from either side of the symmetry lines, in this case from node 525 on the 25-625 line. Note that within this larger symmetry at the 625 node where the major pattern direction reverses and now mirrors in the opposite direction, smaller symmetries mirror at each 5-based node.  Again, tracking their sequences from either direction at each 5-based node is the simplest. The column patterns alternate their 2nd # patterns. Furthermore, the 2nd # values of the rows also come in two alternating patterns. That on the major 25-625 symmetry line, starting on the PD at 625 and considering it to be zero, count every 7th 2nd # value thereafter along the PD to fill in all the 2nd # values along that 25-625 row. And, of course, it completely mirrors on the other side of the 625 node. These two example are just an introduction to the major horizontal and vertical symmetry patterns coming from the 625 node. Alternating In between are the secondary horizontal and vertical patterns. More on this in the following rules and examples. 









2nd # Patterns -5-based columns with 0-PD


odd

even

odd/even

5-based node

even

odd



(0)1


 


(0)1

 

(0)4

 

(0)4

 

(0)9


 


(0)9

 

16

 

16

 



25




36


36


49




49


64


64


81




81



1 0-0



1 21




1 21


1 44


1 44


1 69




1 69


1 96


1 96




2 25




2 56


2 56


2 89




2 89


3 24


3 24


3 61




3 61



4 0-0



4 41




4 41


4 84


4 84


5 29




5 29


5 76


5 76




6 25






2nd # Row Values from  Symmetry                                                            Line at 625

2nd # Row

Values

84

41

96

49

625

49 

96

41

84

25





 



...

64

(0)1

36

69


... 

...

...

...




(0)0


...




(0)4

81

56

29


...

...

...

...

25








...

44

61

76

89


... 

...

...

...




(0)0


...




24

21

16

(0)9


...

...

...

...



 

Brooks (Base)Square
~click to enlarge image
 135  

BS Rule 135: The 2nd # values of the PD include all 25 PD #s up to 625, thereafter they repeat in a mirror symmetry fashion. 


Note: They, of course, are present on the same 5-based square and diamond-square patterns on the grid as where the 1st # values are. They, too, fall into two patterns, 1 and 2, over the grid depending on the intercept of their center axis (diamond). As will be shown over the next several rules, the big difference between 1st and 2nd # pattern distributions is that for the 2nd #s, the patterns 1 and 2 are different ... they alternate from 1 diamond-square to another, and, importantly, they are mirror symmetric to a much larger symmetry, the symmetry of 625.




 
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Brooks (Base) Square
~click to enlarge image
 136  

BS Rule 136: The 2nd # PD pattern 1, based on vertical column 0-PD and horizontal row 5-PD shows that the vertical 2nd # values are taken straight from the PD, and the horizontal 2nd # values are taken from every 7th PD # (see Rule 134).


Note: When counting the 14 steps from one 2nd # value to the next off the PD sequence 1-625, you must actually consider the sequence as going from 0-625 and count the 0 and 625 as steps as you “round the corners” so to speak, e.g. 4 to 25, count down 2 steps to 0, then back to 25 for 5 more steps, total = 7.




2nd #s 

Pattern 1: center=5, col 0-PD, row 5-PD

21

44

69

96

(0)0

21

44

69

96

(0)0-89-76-61-44-25-(0)4-81-56-29-25

56

89

24

61

(0)0

25

56

89

24

61




 

Brooks (Base)Square
~click to enlarge image
 137  

BS Rule 137: The 2nd # PD pattern 2, based on vertical column 5-PD and horizontal row 0-PD shows that the vertical 2nd # values are taken straight from other PD #, and the horizontal 2nd # values are taken from every 14th PD # (see Rule 134). In both cases, only every other value on the column or row has its 2nd # value directly from the PD. This is an important difference from pattern 1.



Note: When counting the 14 steps from one 2nd # value to the next off the PD sequence 1-625, you must actually consider the sequence as going from 0-625 and count the 0 and 625 as steps as you “round the corners” so to speak, e.g. 36 to 64, count down 6 steps to 0, then back to 64 for 8 more steps, total = 14.





2nd #s 

Pattern 2: center=5, col 5-PD, row 0-PD

1

24

9

56

25

1

24

9

56

(0)0-9-96-1-84-75-64-1-36-9-(0)0

96

9

44

1

(0)0

75

96

9

44

1








 
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Brooks (Base) Square
~click to enlarge image
 138  

BS Rule 138: Details of Rule 137: The 2nd # PD pattern 2, based on vertical column 5-PD and horizontal row 0-PD shows that the vertical 2nd # values are taken straight from other PD #, and the horizontal 2nd # values are taken from every 14th PD # (see Rule 134). In both cases, only every other value on the column or row has its 2nd # value directly from the PD. This is an important difference from pattern 1.



Note: When counting the 14 steps from one 2nd # value to the next off the PD sequence 1-625, you must actually consider the sequence as going from 0-625 and count the 0 and 625 as steps as you “round the corners” so to speak, e.g. 36 to 64, count down 6 steps to 0, then back to 64 for 8 more steps, total = 14.



Pattern 2: 2nd # Sums

Vertical

Column

Horizontal

Row

336

76

84

___

24

56

__

96

336

96

84

__

64

36

__

56

20

0

__

40

20

__

40

164

44

__

64

36

__

16

(0)4

164

(0)4

__

44

76

__

16

24

40

__

20

40

__

0

20


@625


@625


ver=500

hor=500


0

∑#s in

between=550


∑#s in

between=550


0

total=1050

total=1050


0

Note: 

This ∑total=1050 Pattern 2 + ∑total=950 Pattern 1=2000


∆ of 14 steps along PD for the horizontal row




 
TOP ]


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 |


Brooks (Base) Square
~click to enlarge image
 139  

BS Rule 139: Details of Rule 136: The 2nd # PD pattern 1, based on vertical column 0-PD and horizontal row 5-PD shows that the vertical 2nd # values are taken straight from the PD, and the horizontal 2nd # values are taken from every 7th PD # (see Rule 134).


Note: When counting the 14 steps from one 2nd # value to the next off the PD sequence 1-625, you must actually consider the sequence as going from 0-625 and count the 0 and 625 as steps as you “round the corners” so to speak, e.g. 4 to 25, count down 2 steps to 0, then back to 25 for 5 more steps, total = 7.



Pattern 1: 2nd # Sums

Vertical

Column

Horizontal

Row

30

(0)1

(0)4

(0)9

16

__


70

24

21

16

(0)9

__


40

230

36

49

64

81

__

270

89

76

61

44

__

40

230

21

44

69

96

__

170

(0)4

81

56

29

__


60

230

56

89

24

61

__

170

69

36

(0)1

64

__

60

230

41

84

29

76

__

270

84

41

96

49

__

40


@625


@625


ver=950

hor=950


0

∑#s in

between=0


∑#s in

between=0


0

total=950

total=950


0

Note: 

This ∑total=1050 Pattern 2 + ∑total=950 Pattern 1=2000


∆ of 7 steps along PD for the horizontal row



 

Brooks (Base)Square
~click to enlarge image
 140  

BS Rule 140: In a 5x5’s square, Pattern 1 of 2nd # values becomes easy to see as being symmetric to the 625 PD node symmetry lines. 


Note: The horizontal symmetry line runs from 25-625 (PD), and mirrors beyond. The vertical symmetry line mirrors  from the 25-625 (PD) line up to 100 (PD) and down to 2400 at the next major symmetry line of 50-2500.



Pattern 1: 2nd # Sums

Vertical

Column

Horizontal

Row

(0)1

(0)4

(0)9

16

24

21

16

(0)9

25

600

36

49

64

81

89

76

61

44

100

525

21

44

69

96

(0)4

81

56

29

125

400

56

89

24

61

69

36

(0)1

64

300

225

41

84

29

76

84

41

96

49

525⬆⬇

625

76

29

84

41

49

96

41

84

800

325/1125

61

24

89

56

64

(0)1

36

69

1125

800

96

69

44

21

29

56

81

(0)4

1500

425

81

64

49

36

44

61

76

89

1925

2025

16

(0)9

(0)4

(0)1

(0)9

16

21

24

Notice the minor symmetry pattern of the 1st #s at each 5-based node



 
TOP ]

Brooks (Base) Square
~click to enlarge image
 141  

BS Rule 141: In a 5x5’s square, Pattern 2 of 2nd # values becomes easy to see as being symmetric to the 625 PD node symmetry lines. 


Note: The horizontal symmetry line runs from 25-625 (PD), and mirrors beyond. The vertical symmetry line mirrors  from the 25-625 (PD) line up to 100 (PD) and down to 2400 at the next major symmetry line of 50-2500.


Pattern 2: 2nd # Sums

Vertical

Column

Horizontal

Row

(0)1

(0)4

(0)9

16

24

21

16

(0)9

25

600

36

49

64

81

89

76

61

44

100

525

21

44

69

96

(0)4

81

56

29

125

400

56

89

24

61

69

36

(0)1

64

300

225

41

84

29

76

84

41

96

49

525⬆⬇

625

76

29

84

41

49

96

41

84

800

325/1125

61

24

89

56

64

(0)1

36

69

1125

800

96

69

44

21

29

56

81

(0)4

1500

425

81

64

49

36

44

61

76

89

1925

2025

16

(0)9

(0)4

(0)1

(0)9

16

21

24

Notice the minor symmetry pattern of the 1st #s at each 5-based node










 

Brooks (Base)Square
~click to enlarge image
 142  

BS Rule 142: Combining the two 2nd # Pattern, 1 and 2, together as 5x5’s squares emphasizes the great symmetrical divide at the 625 symmetry line. 


 
TOP ]

Brooks (Base) Square
~click to enlarge image
 143  

BS Rule 143: Plotting the 2nd # 5-based nodes and their respective 2nd # values ... 0-0 and 5-5 ... crisscrossing diagonals ... similar to Rule 124 that was done with the 1st # only values ... gives yet another look at the large, major symmetry focused on the 625 symmetry line.


 
TOP ]


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 |


Brooks (Base) Square
~click to enlarge image
 144  

BS Rule 144: Taking the same plot from above (Rule 143), another pattern within the pattern emerges. In both diagonal directions, the difference, ∆, in the 2nd # values of a given diagonal are constant down the diagonal ... beginning at the 1st Diagonal from the PD ... and such ∆ grows in each subsequent diagonal by 10, as 20-30-40-... even ∆s are always centered on the diagonals that have 5-based nodes on the PD, odds in between. All this while maintaining strict mirror symmetry at the 625 symmetry line. 


 

Brooks (Base)Square
~click to enlarge image
 145  

BS Rule 145: The 2nd #s for any 5-based diagonal on any 5x5’ds (diamond-square) has the following properties:

  • there are 5 possible diagonal sequences internal to the nodes:
  • 20-40-60-80
  • 40-80-20-60
  • 60-20-80-40
  • 80-60-40-20
  • 00-00-00-00
  • each diagonal sequence repeats it sequence along the entire leg of that diagonal
  • the sum, ∑, of any sequence is, of course, equal to 0 or 200, with the former made of 00, 00, 00, 00 and the latter of 00, 20, 80, 60,40 combinations, respectively. 
  • any sequence will have 60-40 or 40-60 on the inside, and, 20-80 or 80-20 on the outside, and vice versa
  • the sum, ∑, of all sides of a given whole diamond-square, other than one that includes the PD, is always: 400, 600 or 800
  • where the diagonal sequences meet the left axis, the ending sequence 2nd # value follows a pattern:
  • 80/20 - 60/40 | 40/60 - 20/80 -00/00
  • this axis pattern is symmetrical to the 625 symmetry line
  • the entire diagonal pattern is symmetrical to the 625 symmetry line, both horizontally and vertically
  • another, larger symmetry line is revealed at the 50-2500 (502) symmetry line
  • (50)2 = 2500 symmetry (not fully shown in the graphics) is symmetry bordered by a 00-00-00 diamond-square


 
TOP ]

BS Rules 146-153  that follow are a series of 8 steps to building the square.


There are numerous ways to do this. This is just one example that reveals something about the interconnectedness of the elements of the square.


Brooks (Base) Square
~click to enlarge image
 146  

BS Rule 146: On a square grid, with 0 at the origin, place all the whole number integers ... 1, 2, 3, ... in sequence, and, their square number values along the Prime Diagonal (PD). Preferably, do this in BOLD.


Note: Go ahead and fill in the next 6 Inner Grid number values by taking the diagonal # and subtracting the PD number above the space, move over to the next blank column value and do the same procedure, and so on. Yes, you can always do this to fill in and/or double check your work. From this point on, we are only going to place 1st # values on the square ... later the 2nd #s ... and finally the remaining # values to complete the grid.



PD 

#

-

Column

#

value

4


1

3

9


1

8

16


1

15

9


4

5

16


4

12

16


9

7



 

Brooks (Base)Square
~click to enlarge image
 147  

BS Rule 147: Notice the first two diagonal patterns. On the 1st Diagonal from the PD, we have (1), 3, 5 ,7 and on the 2nd Diagonal we have (2), 8, 12. The former is all the odd numbers and each increases by a difference, ∆, of 2. The latter 8, 12 has a ∆ of 4. Could it be that this diagonal of even #s  increases sequentially by 4? Fill in the 1st # values only for these two diagonals.


Note: Notice we have two repeating sequences: 1-3-5-7-9 and 8-2-6-0-4. At this point, you may suggest that each subsequent diagonal away from the PD increases its number ∆s by 2 over its neighbor diagonal.


 
TOP ]

Brooks (Base) Square
~click to enlarge image
 148  

BS Rule 148: Locate all the 5-based nodes within the Strict Inner Grid (SIG) and fill them in BOLDLY. Circle, or highlight, those on the PD (25-100-225-400-625) and do the same on the axis (5-10-15-20-25-...). Forming a diamond-square, connect all the 0 and 5 nodes with the same value, respectively. Hint: You already have the first 5 on the 1st Diagonal. 


Note: See how the 0 and 5 nodes alternate across the grid.


 
TOP ]


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 |


Brooks (Base) Square
~click to enlarge image
 149 

BS Rule 149: Draw a light square around one or more of the 5-based 5x5’s squares with a 0 vertices at the start. You may also want to draw a faint diamond-square border around the vertices-nodes of a 5x5’s square. Remembering that the 5x5’s perimeter 1st # values are all derived from the PD ... that is 0-1-4-9-6-5 ... do the following: In the perimeter space of any square, place 1-4-9-6 ... the same 1st number values of the first four PD #s ... in the left perimeter column and again in the right perimeter column, only reverse the sequence order in the latter. The perimeter row is also filled with the same PD 1st # values ... but what is the order? We have learned that throughout the grid, any two 1st #s equidistant from, and at a right angle relationship to, a 0 or 5, always have a sum, ∑=10. So therefore, 9-6-1-4 satisfy the top row requirement and the reverse for the bottom row. We also remember that the 8 spaces internal to the perimeter 5x5’s are occupied by the remaining 1st numbers NOT on the PD ...2-3-7-8.


Satisfying the right angle relationship rule (above), we see that 3-7 and 2-8 are the only diagonal pair combinations possible. We have to decide which group ... 7-2 or 2-7 or 3-8 or 8-3 ... goes in the top remaining row. Once we pick even one space value all others will be automatically determined by the dictates of the pattern. While there are a number (no pun intended) of ways, the easiest rule to remember is the basic diagonal rule ... diagonals alternate odd and even and increase their ∆s by 2. Applying that rule makes 7-2 the only choice. Otherwise, you can remember this little rule: like James Bond 007, 0s like 7-3, 5s like 8-2. (Seems now, with that knowledge, you can also fill in the truncated diamond-square.)


 

Brooks (Base)Square
~click to enlarge image
 150  

BS Rule 150: Knowing that every diamond-square ... and the 5x5’s square it contains ... is identical, you can now fill in the rest of the grid 1st # values.


 
TOP ]

Brooks (Base) Square
~click to enlarge image
 151  

BS Rule 151: Applying Rule 143 for the crisscrossing diagonals of 0 and 5, and a little help from Rule 134, we can add quite a bit of the whole number and 2nd # values, respectively. The horizontal row of the 625 symmetry line, working from the PD outward, across the 25-625 row, uses the 2nd # values of the PD itself, but in a sequence that starts with 49 ... 7 steps from the origin on the PD ... and continues its sequence as every 7th PD 2nd # value thereafter. And we know from Rules 140-141, that this pattern of symmetry repeats every other row up and down the grid.


Note: The rows in between are also duplicated and based on every 14 steps starting from 96 ... 14 steps from the origin 0 on the PD. Notice that it moves opposite, from left to right, on the row, and fills in only every other grid space, starting with the second number in from the axis.









Pattern 1

PD 

#

even 

or 

odd

Vertical

Column

1st #

*order

**#

of 

steps

Vertical 

Column

2nd #

*order

**#

of 

steps

Horiz.

Row

1st #

*order

**#

of 

steps

Horiz.

Row

2nd #

*order

**#

of 

steps

0

even





20




(0)1


25

1

25

1

3

3

7

7

(0)4


24

1

24

1

4

3

14

7

(0)9


23

1

23

1

19/21

3

21

7

16


22

1

22

1

2

3

22

7

25

odd

21

1

21

1

5

3

15

7

36


20

1

20

1

18/22

3

8

7

49


19

1

19

1

1

3

1

7

64


18

1

18

1

6

3

6

7

81


17

1

17

1

17/23

3

13

7

100

even 

16

1

16

1


3

20

7

121


15

1

15

1

7

3

23

7

144


14

1

14

1

16/24

3

16

7

169


13

1

13

1


3

9

7

196


12

1

12

1

8

3

2

7

225

odd

11

1

11

1

15/25

3

5

7

256


10

1

10

1


3

12

7

289


9

1

9

1

9

3

19

7

324


8

1

8

1

14

3

24

7

361


7

1

7

1


3

17

7

400

even 

6

1

6

1

10

3

10

7

441


5

1

5

1

13

3

3

7

484


4

1

4

1


3

4

7

529


3

1

3

1

11

3

11

7

576


2

1

2

1

12

3

18

7

625

odd

1

1

1



3

25

7

*taken from PD, **steps from previous # on PD

Note: Because the 2nd # also contains the 1st #, the 1st # sequence and steps follows exactly that of the 2nd # with 7 steps between values. The 1st # values also follow a 3 step pattern between values as shown here. Select any 1st # and its next # in the sequence will be 3 steps away as indicated on the PD. The 1st # sequence in Pattern 1 is identical to that of Pattern 2, only the order is reversed.




Pattern 2

PD 

#

even 

or 

odd

Vertical

Column

1st #

*order

**#

of 

steps

Vertical 

Column

2nd #

*order

**#

of 

steps

Horiz.

Row

1st #

*order

**#

of 

steps

Horiz.

Row

2nd #

*order

**#

of 

steps

0

even

1

1

1

2

5

3



(0)1


2

1

-


23

3



(0)4


3

1

3

2

22

3

13

14

(0)9


4

1

-


4/6

3

-


16


5

1

5

2

24

3

21

14

25

odd

6

1

-


21

3

-


36


7

1

7

2

3/7

3

7

14

49


8

1

-


25

3

-


64


9

1

9

2

20

3

5

14

81


10

1

-


2/8

3

-


100

even 

11

1

11

2


3

19

14

121


12

1

-


19

3

-


144


13

1

13

2

1/9

3

15

14

169


14

1

-



3

-


196


15

1

15

2

18

3

1

14

225

odd

16

1

-


10

3

-


256


17

1

17

2


3

11

14

289


18

1

-


17

3

-


324


19

1

19

2

11

3

23

14

361


20

1

-



3

-


400

even 

21

1

21

2

16

3

9

14

441


22

1

-


12

3

-


484


23

1

23

2


3

3

14

529


24

1

-


15

3

-


576


25

1

24

2

13

3

17

14

625

odd









*taken from PD, **steps from previous # on PD

Note: Because the 2nd # also contains the 1st #, the 1st # sequence and steps follows exactly that of the 2nd # with 14 steps between values. The 1st # values also follow a 3 step pattern between values as shown here. Select any 1st # and its next # in the sequence will be 3 steps away as indicated on the PD. The 1st # sequence in Pattern 1 is identical to that of Pattern 2, only the order is reversed. 



 

Brooks (Base)Square
~click to enlarge image
 152  

BS Rule 152: The 2nd # values in the vertical columns are also symmetrical to the 625 symmetry line as they alternate the two patterns across the grid. 


Note: The vertical #s are easier because in the first pattern (Pattern 1), they simply mimic the 2nd # values of the PD, straight across. In the second pattern (Pattern 2), alternating between the first, they follow ... starting from the second space up from the 25-625 horizontal symmetry line ... every other PD # value starting at 04.



 
TOP ]

Brooks (Base) Square
~click to enlarge image
 153  

BS Rule 153: With this much of the matrix grid filled in, it is a simple matter to fill in the blanks and the 1st  and 2nd # values with the full number value for the space. Use your own method of choice. 


Note: Don’t forget: 

  • diagonals alternate, odd-even, and, ∆s increase by 2 both diagonal ways
  • numbers decrease across row from axis to PD as -1, -4, -9, ..., or,  of a ∆ of 1, 3, 5, ...
  • numbers increase down the columns as 5, 7, 9, ...


Now, on to “The Conspicuous Absence of Primes (TCAOP).



 
TOP ]


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 |


 


 

NEXT: On to II. TCAOP - Brooks (Base) Square

Back to I. TAOST>IC. Geometrics-relationships>IC2. Parallelograms - Brooks (Base) Square





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).



Copyright©2009-12 Reginald Brooks, BROOKS DESIGN. All Rights Reserved.
The LUFE Matrix | The LUFE Matrix Supplement | The LUFE Matrix: Infinite Dimensions | The LUFE Matrix: E=mc2 | 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: 9-11_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
 
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