BBS-ISL Matrix Fundamentals:

10 Basic, fundamental rules of the symmetrical BBS-ISL Matrix

• Basic BBS-ISL Rule 1: All numbers (#s) related by the 1—4—9—...PD sequence




• Basic BBS-ISL Rule 2: Every # in the PD sequence is the square of an Axial #.




• Basic BBS-ISL Rule 3: The Odd-Number Summation sequence forms the PD sequence.




• Basic BBS-ISL Rule 4: Every EVEN Inner Grid (IG) # is divisible by 4 & all are present.




• Basic BBS-ISL Rule 5: Every IG# is:

  • A: The difference (∆) between its two PD-sequence #s. (Note: A=B=C=D=E, and, F.)

    • Ex: PD 25 - PD9 = 16

     

  • B: The sum (∑) of the ∆s of each of its PD#s between its two PD-sequence #s (as above).

    • Ex: (PD 25 - PD16) + (PD16 - PD9) = 16

     

  • C: The ∆ between the squares of the two Axial #s forming that IG# (as above).

    • Ex: 5² - 3² = 16

     

  • D: The product of the Addition & Subtraction of the two Axial #s forming that IG# (as above).

    • Ex: (5 + 3) x (5 - 3) = 16

     

  • E: The product of the Diagonal Axis # — STEPS from the PD — times the ∑ of Row + Column Axis #s.

    • Ex: 2 x (5 + 3) = 16

     

  • F: Also, the product of its 2 Axial #s intersected by that IG#'s 90° diagonals.

    • Ex: 2 x 8 = 16

     

• Basic BBS-ISL Rule 6: Every *ODD IG# is NOT PRIME & all are present.

  • Corollary: NO PRIME #s are present on the *IG.

  • Corollary: NO EVEN, NOT divisible by 4 #s are present on the IG.
    

  *Excepting the 3—5—7—… ODD #s of the 1st Parallel Diagonal.





• Basic BBS-ISL Rule 7: The ODD-Number sequence 3—5—7—9—..., and the 1—4—9-...PD sequence, form the sequential ∆ between ALL IG#s.




• Basic BBS-ISL Rule 8: The ∆ between #s within the Parallel Diagonals is a constant 2 x its Axial #.




• Basic BBS-ISL Rule 9: The ∆ between #s in the Perpendicular Diagonals follow:

  • A: From EVEN PD#s, √PD x 4 starts the sequence & follows x1—x2—x3—x4—....

  • B: From ODD PD#s, √PD x 4 starts the sequence & follows x1—x2—x3—x4—....
    

  • C: From ODD Perpendicular Diagonals between the EVEN-ODD diagonals (above), the sequence starts with the same value as the Axis number ending the diagonal, the sequence following x1—x3—x5—x7—....




• Basic BBS-ISL Rule 10: Every #, especially the #s in the ONEs Column, informs both smaller and larger Sub-set symmetries (much larger grids required to demonstrate).

BBS-ISL Matrix Inner Grid Golden Rules (IGGR)

5 Basic, fundamental rules of the symmetrical BBS-ISL Matrix Inner Grid

• IGGR Rule 1: The IG is formed of two equal & symmetrical 90°-right, isosceles triangles that are bilaterally symmetrical about the PD — and, infinitely expandable.




• IGGR Rule 2: The 90°-right-triangle — inherent to ALL squares and rectangles by definition — both forms the alternating EVEN-ODD square grid cells within the Matrix, and, is responsible for all major patterns and sequences, thereupon.




• IGGR Rule 3: Subtraction-Addition: Every IG# is simply the ∆ between its two PD#s (subtraction), and, the sum (∑) of any IG# + its PD# above = the PD# on the end of that Row (or, Column).




• IGGR Rule 4: Multiplication-Division: Every IG# is simply the product of the two AXIAL #s intersected by the two diagonals — of that said IG# — pointing back to the Axis at a 90° angle (multiplication), and, the dividend of the Axial divisor and quotient (division).




• IGGR Rule 5: The actual # of grid-cell steps — i.e., the actual # of STEPS from a given IG# to another by a strictly horizontal, vertical, or 45° diagonal path — forms a simple, yet often fundamental descriptor to the pattern-sequence templates that inform the more advanced patterns, e.i., Exponentials and especially the Pythagorean Triples (PTs). STEPS are particularly important in the geometric visualizations within the BBS-ISL Matrix (as alluded to in IGGR 2, above).

Pythagorean Triples and BBS-ISL Fundamentals (TPISC: The Pythagorean-Inverse Square Connection)

3 Basic, fundamental rules of the symmetrical BBS-ISL Matrix Inner Grid that encompass the PTs.

• TPISC-BBS-ISL Rule 1: Every IG EVEN Squared # is part of a Paired-Factor Set (PFS) that:

  • A: Has reciprical PFS members on the PD vertically above.

  • B: Both PFS members reside on the SAME Row.

  • C: They represent the a² and b² values of a PT, whose c² value is on the PD intersection.

  
  

• TPISC-BBS-ISL Rule 2: Every PT is found on the BBS-ISL Matrix and can be located by this intersection of EVERY PD (9>) and a Row with PFSs.




• TPISC-BBS-ISL Rule 3: Every PT — including its sides, perimeter, area and proof — can also be found and fully profiled (and, predicted) as r-set, s-,t-set members of the Dickson Method (DM), Expanded Dickson Method (EDM), and the Fully Expanded Dickson Method (FEDM), shown herein.