As much as viewing the **MPS** on a universal grid is absolutely amazing in its combination of simple geometry to visually reveal the basic and fundamental relationships in the form of AREAS -- areas that add and subtract to inform all the AREAS that define all three generations within-- it is truly when we examine the **BIMMPS** that the numerical, quantity based relationships of these AREAS are unfolded in an in-depth and repeating **NPS** of their own that the profound connections are realized!

As before, the **NPS** of the **BIMMPS** at first looks formidable as a group. Taking a few examples of the smaller **MPS** and drawing a clear **NPS** picture will establish a fractal-template that will be repeated with each and every **MPS** thereafter -- only the numbers will change.

Taken from the BIM-MPS_Details-I.html page back in Book I. The central x+y column will be the focus of this section. While x may seem, and indeed is, quite simple and straightforward, y reveals all the Butterfly Fractal 1 pattern parts previous to x. Just follow the examples and the whole secret to the Perfect Numbers, Odd Complements and the resulting Mersenne Prime Squares will be revealed. And such a simple, repetitive pattern it is!

As one can see, the color bands that make up the **PN** can form all three "generations" withing the **MPS** space. They do so in a completely fractal manner, based on the exponential power of 2. They do so in a repeating **NPS** template that is exactly the same for each **MPS**, with only the relative numbers being different.

BIM-MPS-PN-1--1.mp4

BIM-MPS-PN-1--1.gif

As an overview, let's look at a half a dozen examples before diving in to a more detailed look. To note: only the y values listed are illustrated within the much larger x+y=z central column shown.

BIM+BF-8.gif

BFsums-x=2_p=2

BFsums-x=4_p=3

BFsums-x=8_p=4

BFsums-x=16_p=5

BFsums-x=32_p=6

BFsums-x=64_p=7

Looking at BFsums-x=16_p=5MPS-0-22 in detail:

BFsums-x=16_p=5MPS-0

BFsums-x=16_p=5MPS-1

BFsums-x=16_p=5MPS-2

BFsums-x=16_p=5MPS-3

BFsums-x=16_p=5MPS-4

BFsums-x=16_p=5MPS-5

BFsums-x=16_p=5MPS-6

BFsums-x=16_p=5MPS-7

BFsums-x=16_p=5MPS-8

BFsums-x=16_p=5MPS-9

BFsums-x=16_p=5MPS-10

BFsums-x=16_p=5MPS-11

BFsums-x=16_p=5MPS-12

BFsums-x=16_p=5MPS-13

BFsums-x=16_p=5MPS-14

BFsums-x=16_p=5MPS-15

BFsums-x=16_p=5MPS-16

BFsums-x=16_p=5MPS-17

BFsums-x=16_p=5MPS-18

BFsums-x=16_p=5MPS-19

BFsums-x=16_p=5MPS-20

BFsums-x=16_p=5MPS-21

BFsums-x=16_p=5MPS-22

And finally, looking atBFsums-x=64_p=7MPS-0-22 in detail:

MPS-0-22/BFsums-x=64_p=7_0

MPS-0-22/BFsums-x=64_p=7_1

MPS-0-22/BFsums-x=64_p=7_2

MPS-0-22/BFsums-x=64_p=7_3

MPS-0-22/BFsums-x=64_p=7_4

MPS-0-22/BFsums-x=64_p=7_5

MPS-0-22/BFsums-x=64_p=7_6

MPS-0-22/BFsums-x=64_p=7_7

MPS-0-22/BFsums-x=64_p=7_8

MPS-0-22/BFsums-x=64_p=7_9

MPS-0-22/BFsums-x=64_p=7_10

MPS-0-22/BFsums-x=64_p=7_11

MPS-0-22/BFsums-x=64_p=7_12

MPS-0-22/BFsums-x=64_p=7_13

MPS-0-22/BFsums-x=64_p=7_14

MPS-0-22/BFsums-x=64_p=7_15

MPS-0-22/BFsums-x=64_p=7_16

MPS-0-22/BFsums-x=64_p=7_17

MPS-0-22/BFsums-x=64_p=7_18

MPS-0-22/BFsums-x=64_p=7_19

MPS-0-22/BFsums-x=64_p=7_20

MPS-0-22/BFsums-x=64_p=7_21

MPS-0-22/BFsums-x=64_p=7_22

Yet another advanced section. And it will be addressed in the Appendix I in Book VI. Here is just a little bit.

The MPS on the BIM (BIMMPS)Let's repeat a key concept covered in Part II and III:

The Perfect Numbers may be expressed as the sequential running sums (**∑**) of the cubes of the sequential ODD numbers as 1³ + 3³ + 5³ ... and we have the genesis of the * Butterfly Fractal~3~*.

In the * Butterfly Fractal~1~* and

This gave the **x** and **z**=**Mp** directly and the **y**, **xz**=**PN** and **xy**=**CR** indirectly by calculation.

Now, in the * Butterfly Fractal~3~*, we find the

of Numbers (Part I)

NEXT: BOOK III: An Artist, Mathematician and Choreographer...Oceans of Numbers (Part II)

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~

LINKS:

Mersenne Prime Squares (Part I) the Introductory white paper.

Mersenne Prime Squares (Part II) 3 Simple Intros.

Mersenne Prime Squares (Part III) the Advanced white paper.

The MPS Project The Mersenne Prime Squares Project.

LINKS:

MathspeedST: TPISC Media Center

Artist Link in iTunes Apple Books Store: Reginald Brooks

Back to Part I of the BIM-Goldbach_Conjecture.

Back to Part III of the BIM-Goldbach_Conjecture.

BACK: ---> PRIMES Index on a separate White Paper BACK: ---> Periodic Table Of PRIMES (PTOP) and the Goldbach Conjecture on a separate White Paper (REFERENCES found here.) BACK: ---> Periodic Table Of PRIMES (PTOP) - Goldbach Conjecture ebook on a separate White Paper BACK: ---> Simple Path BIM to PRIMES on a separate White Paper BACK: ---> PRIMES vs NO-PRIMES on a separate White Paper BACK: ---> TPISC_IV: Details_BIM+PTs+PRIMES on a separate White Paper BACK: ---> PRIME GAPS on a separate White PaperReginald Brooks

Brooks Design

Portland, OR

brooksdesign-ps.net

Art Theory 101 / White Papers Index