#ppsets

Part II of III. Every EVEN is composed of two PRIMES. As shown here, every grid cell has three ID’ers.

The Middle is the Cell value and is always the difference between its two PD interects.

The Top # is the Axis Diagonal #—two Axis #s that meet at their 90° apex on the given cell, being the product of the respective Axis D #s.

The Bottom # of each Cell is the Axis Coordinate #-one from the horizontal and one from the vertical Axis. Their sums = EVEN # connected by the Diagonal line from said EVENS on each Axis.

There are ALWAYS one or more of the RED Cells intersected by the Diagonals of any EVEN. The full proof is found on the earlier Periodic Table Of Primes (PTOP) and PRIMES On the BIM (POB) works.

“PTOP (Periodic Table Of PRIMES) on the BIM: PART IV: #27”

Fractal — Symmetry — Inverse Square Law (ISL). These are not the usual descriptive terms associated with the Prime numbers (PRIMES). Yet that is exactly what best describes the PRIMES!

The PRIMES, when brought together as P1 & P2 members of PRIME Pair Sets (PPsets) demonstrate a robust symmetry and an intimate relationship with the ISL when shown on the BIM (BBS-ISL Matrix grid of the Inverse Square Law).PRIMES Index

This comes about as the well established PRIMES Sequence (PS) — 3-5-7-11-13-17-19-23-29-31-... acts in a fractal-like manner, i.e., it demonstrates redundant, repetitive and re-iterative behavior in presenting self-similar reflection of itself as it constructs PPsets and PPset “TRAILS.”

The TRAILS are formed from the concatenation of PS’s progressively onto each successive PRIME of a given PS — forming a series of PPsets along the way. The TRAILS can also be seen to be formed directly as individual PPsets on the BIM. Here, each PPset is easily seen to be intersection of 1 PPset member from the Horizontal AXIS and 1 PPset member from the Vertical AXIS, together forming the P1, P2 members. The TRAIL is simply all those PPsets thus formed across a Row — or down a Column — on the bilaterally symmetrical BIM. All this is easily visible on the BIM.

That leads us to our story about the PRIMES—Fractals—Symmetry—and the intimate relationship they have with the ISL as seen on the BIM.

~~~

That leads us to our story about the PRIMES—Fractals—Symmetry—and the intimate relationship they have with the ISL as seen on the BIM.

2020, Reginald Brooks

“PRIMES on the BIM (formerly “PTOP on the BIM): PART IV: #33”

cont’d...

Clarifications: The BIM is NOT a traditional multiplication table. NO WAY. While the central Prime Diagonal (PD) does represent the squares of the AXIS #s, ALL the Inner Grid (IG) values are determined by the difference (∆) in their horizontal and vertical PD intercept values, ei., 16 is PD 25 - PD 9 (25-9=16).

That said, in the PPsets as presented in the PTOP on the BIM, we will go from this standard approach, to overlaying key matrix values with the coordinates of their AXIS, i.e., 9 is overlaid with AXIS 3, AXIS 3 or simply 3,3 and 16 is overlaid with 3,5 and so on. This forms a “Sub-Matrix” on the BIM of overlaid values that will be quite useful. Once you see how simple this overlay is, a great simplification overall will reveal itself

One other note: the PPsets as 3,3–3,5-5,5–... form the EVENS by adding the set members, i.e. 3,3 = 3 + 3 = 6, and, a diagonal line from a given PPset back to the AXIS will reveal that EVEN.

Object AREAS:

AREAS — as Object Areas — are introduced here. Object Areas simply are the # of objects within an “area.” Here that AREA is a square on the BIM. Counting the # of objects within each AREA will give the number. More surprises await.

The PPset Areas are NOT traditional areas. Instead, they represent the number (#) of PPsets captured within the boundaries of the squared AXIS #s, i.e., the squared values of the PS-fractal 3-5-7-11-13-17-19-23-29-… are 9-25-49-121-169-289-361-529-841-… but they are represented, as a “Sub-Matrix” on the BIM, as 3,3–5,5–7,7–11,11–13,13–17,17–19,19–23,23–29,29—…PPsets.

It is the # of these PPsets within the squared PS-fractal AXIS value treated as area. One is really capturing the # of “objects” or Object AREAS.

~~~

2020, Reginald Brooks

PRIMES Index

PRIMES on the BIM ebook

“PRIMES on the BIM (formerly “PTOP on the BIM”): Part IV; #29”

cont’d...

A simplification of BIM: Part I formed the basis of the PTOP (Periodic Table Of PRIMES, as first described in MathspeedST, 2010).

The PTOP extracts the PPsets and EVENS from the BIM and relates them to each other in a straightforward and graphical manner.

A distinct PPset “TRAIL” is seen for each PPset starting with the P1=3 and P2 = each successive number in the natural PRIMES sequence—(2),3,5,7,11,13,17,19,23,29,31,37,…

In fact, this natural PRIMES Sequence (PS) acts in every way just like a fractal, whereby a fractal set consisting of the PS is “added” —i.e. concatenated—to each successive P1 value down the Column 3 on the PTOP.

This fractal PS property ensures that there are more overlapping “TRAILS” of PPsets than PRIME Gaps, ensuring that ALL EVENS = the sum of two PRIMES. Euler’s Strong Form of the Goldbach Conjecture is proven. More importantly, a definitive relationship between PRIMES and the Inverse Square Law is defined. This joins the ISL, Primitive Pythagorean Triples and PRIMES together under the umbrella of the Inverse Square Law as presented on the BIM.

~~~ISL~~~

The Inverse Square Law (ISL) defines how a force, an influence — energy — becomes diluted as the distance from the source increases: 1/r^2 where r = radius.

Double the distance and the energy (influence) is diluted by 1/4.

If you’re distance, r, from a person with the virus is 1’ you are one with the person. If you’re distance, r, from a person with the virus is 2’ you are 25% with the person. If you’re distance, r, from a person with the virus is 3’ you are ~11% with the person. If you’re distance, r, from a person with the virus is 4’ you are ~6% with the person. If you’re distance, r, from a person with the virus is 5’ you are 4% with the person. If you’re distance, r, from a person with the virus is 6’ you are ~3% with the person. If you’re distance, r, from a person with the virus is 7’ you are ~2% with the person. If you’re distance, r, from a person with the virus is 8’ you are ~1.6% with the person. If you’re distance, r, from a person with the virus is 9’, you are ~1.2% with the person.If you’re distance, r, from a person with the virus is 10’ you are 1% with the person.

1^2 = 1 2^2=4 3^2=9 4^2=16 5^2=25 6^2=36 7^2=49 8^2=64 9^2=81 10^2=100

100% 25% 11.1% 6.25% 4% 2.8% 2% 1.6% 1.2% 1%

This reference describes an evenly distributed cloud, not a direct focused source.

If you are safe and then become exposed to unsafe conditions you quickly move back up, depending on the degree and length of time of your exposure.

Masks help keep you to the right. It also keeps the total # of virions to the right.

Keeping the total # down is good.

~~~

2020, Reginald Brooks

“PRIMES on the BIM, #35”

2020, Reginald Brooks

PRIMES Index

PRIMES on the BIM ebook

“PTOP (Periodic Table Of PRIMES) on the BIM: PART IV: #5,6,7.”

Notice the pattern unfolding when the PRIMES Sequence (PS)-Fractals—3-5-7-11-13-17-19-23-29-...—from the horizontal and vertical AXIS combine on the BIM.

Clarifications: The BIM is NOT a traditional multiplication table. NO WAY. While the central Prime Diagonal (PD) does represent the squares of the AXIS #s, ALL the Inner Grid (IG) values are determined by the difference (∆) in their horizontal and vertical PD intercept values, ei., 16 is PD 25 - PD 9 (25-9=16).

That said, in the PPsets as presented in the PTOP on the BIM, we will go from this standard approach, to overlaying key matrix values with the coordinates of their AXIS, i.e., 9 is overlaid with AXIS 3, AXIS 3 or simply 3,3 and 16 is overlaid with 3,5 and so on. This forms a “Sub-Matrix” on the BIM of overlaid values that will be quite useful. Once you see how simple this overlay is, a great simplification overall will reveal itself. Coming up.

One other note: the PPsets as 3,3–3,5-5,5–... form the EVENS by adding the set members, i.e. 3,3 = 3 + 3 = 6, and, a diagonal line from a given PPset back to the AXIS will reveal that EVEN.

~~~~~

2020, Reginald Brooks

“PRIMES on the BIM (formerly “PTOP on the BIM): PART IV: #30”

cont’d...

In BIM: Part II, the focus was shifted to looking at individual PPsets on the BIM—those making one EVEN number at a time. It is quite interesting how once again those stealthy PRIMES only appear on the AXIS and 1st Parallel Diagonal, yet make a clear, distinctive Number Pattern Sequence (NPS) by their interactions.

In this case, for every EVEN/2 on the AXIS, an isosceles triangle is formed with the base along the AXIS, one side along the PD to the apex of the (EVEN/2)2 and the other side back to the base on the AXIS.

The PPset P1s are found on the Upper Diagonal PD as the √ of where the PS # on the AXIS intersects the PD—there the square root of that PD #=P1, or more easily, just take its Axial value.

The Lower Diagonal (LD) ALWAYS points back to the EVEN on the AXIS. Along the way, it intercepts with the PS values on the AXIS above, to give the P2 values. Dividing the various grid values along the LD by 4x(Even/2) will select out and confirm which values house the P2 PRIMES.

Incredibly, in turns of the number of STEPS from one apex to another around each of the triangles formed, will show them to ALWAYS be equal, thus equilateral. The larger base:PD:LD isosceles triangle is actually made of two equilateral triangles positioned back-to-back with the centerline being the line from the EVEN/2 back to it squared value on the PD.

Table 51 was formed by plotting all the LD values for each EVEN as seen in BIM: Part II. These values were then plotted back on to the BIM and BIM: Part III was formed.

~~~

2020, Reginald Brooks

PRIMES Index

“(Periodic Table Of PRIMES) on the BIM: PART IV: #25”

SYMMETRY, STEPS, EVENS, EVENS/2, PPsets, PRIME SEQUENCE FRACTALS, ISOSCELES & EQUILATERAL TRIANGLES, INVERSE SQUARE LAW, PYTHAGOREAN TRIPLES; ALL ON THE BIM.

~~~~~~

Notice the pattern unfolding when the PRIMES Sequence (PS)-Fractals—3-5-7-11-13-17-19-23-29-...—from the horizontal and vertical AXIS combine on the BIM.

Try and see if you can fill in the PPsets for each circled dot value. Simply combine the Horizontal AXIS value and the Vertical AXIS value, separated by a comma. The smaller value is listed first. Example, @circled 16 = 3,5 PPset.

The 3,5 PPset resides on a Lower Diagaonal that points down and to the left, ending up at Vertical Axis =8. The sum of the two PRIME members of the PPset = 8 = 3+5.

Yes, other values on that same Lower Diagonal will equal 8 — e.i. 2,6 — 1,7, but they are NOT PPsets as one or more members are NOT PRIME.

ALL members of a PPset, by definition, are PRIME. P1 = PRIME member 1. P2 = PRIME member 2.

When you add together the P1 and P2 members of a PPset you get an EVEN.

ALL EVENS = sum of one or more PPsets.

ALL EVENS (≥6) = P1 + P2, where P1 and P2 are ODD # PRIMES (≥3).

This is the Euler’s Strong Form of the Goldbach Conjecture.

Equally significant: all PRIMES — as PPsets — demonstrate Symmetry and a Fractal nature (as being composed of the intersection of two PRIMES Sequence chains) as shown on the BIM: Parts I-IV.

Counting the PPsets as “objects” within progressively larger sequenced squares — Object AREAS — reveals an intimate relationship with the 1-4-9-16-25-... ISL NPS of the Inverse Square Law on the BIM!

~~~

2020, Reginald Brooks