Introducing MathspeedST, an interactive ebook. MathspeedST is the complement to LightspeedST (published in June, 2013).

While LightspeedST addressed the ultimate question: “Why is the speed of light constant?”MathspeedST addresses the fundamental, underlying question: “How can information be ubiquitous, effectively traveling FASTER than the speed of light?”

LightspeedST describes how the formation of spacetime (ST) is intimately tied to the speed of light—all within the Observable Universe.

In the Observable Universe, nothing can travel faster than the speed of light—and this includes any information contained within a tangible form.

In order to satisfy a higher rule, The Conservation of Energy—you know, energy (along with its progeny: momentum, charge, spin, etc.) can neither be net created nor destroyed—clearly a mechanism must be in place to account for the distribution of said energy. Not only that, but such mechanism must do the accounting before—not after—the fact. Enter MathspeedST.

MathspeedST: Leapfrogging Lightspeed FASTER than The Speed of Light” provides the fundamental mathematical basis—lying hidden below—for the accounting and distribution of all spacetime and its subsequent expression as energy, matter and fields in the Observable Universe above.

This freely available interactive ebook brings together under one publication the Brooks (Base) Square-Inverse Square Law, or BBS-ISL Matrix for short, under the title MathspeedST. It includes hundreds of images, a number of videos and lots of test-yourself, interactive widgets in presenting the derivation, layout, rules, patterns and ISL-derived mathematical basis of the matrix. Ages 8-10 and up.

A new paradigm in the visualization of basic mathematics is presented!.

Based on MathspeedST, an interactive ebook. MathspeedST is the complement to LightspeedST (published in June, 2013).

While LightspeedST addressed the ultimate question: “Why is the speed of light constant?”MathspeedST addresses the fundamental, underlying question: “How can information be ubiquitous, effectively traveling FASTER than the speed of light?”



MathspeedST: Leapfrogging Lightspeed FASTER than The Speed of Light” provides the fundamental mathematical basis—lying hidden below—for the accounting and distribution of all spacetime and its subsequent expression as energy, matter and fields in the Observable Universe above.



This freely available interactive ebook brings together under one publication the Brooks (Base) Square-Inverse Square Law, or BBS-ISL Matrix for short, under the title MathspeedST. It includes hundreds of images, a number of videos and lots of test-yourself, interactive widgets in presenting the derivation, layout, rules, patterns and ISL-derived mathematical basis of the matrix. Ages 8-10 and up.

A new paradigm in the visualization of basic mathematics is presented!

There is a simple whole number (integer) matrix grid table upon — and within — that every possible whole number Pythagorean Triangle — a.k.a. Pythagorean Triple — can be placed, and proved. The Brooks Base Square - Inverse Square Law (BBS-ISL Matrix) is an infinitely expandable grid that reveals ALL Pythagorean Triples — both Primitive Triples (PPT) and their non-Primitive multiples (nPTT). An extremely simple geometric AREA proof of the Pythagorean Theorem — c²=a²+b² — is built into the BBS-ISL Matrix.

Not only does the Inverse Square Law describe, define and quantify our most important energies (and their expressions as force) — gravity, light, sound, electromagnetism…, —the BBS-ISL Matrix grid is composed of Pythagorean Triples crisscrossing over much of the entire grid. An intimate inter-connection way beyond a simple, casual association has been revealed. The Dickson Method confirms, validates and provides insight into the generation of ALL Pythagorean Triples directly on the BBS-ISL Matrix.

Numereous animated gifs, slideshows and videos illustrate the compelling visual connection between the Inverse Square Law and the Pythagorean Theorem.

Based on MathspeedST, an interactive ebook. MathspeedST is the complement to LightspeedST (published in June, 2013).

While LightspeedST addressed the ultimate question: “Why is the speed of light constant?”MathspeedST addresses the fundamental, underlying question: “How can information be ubiquitous, effectively traveling FASTER than the speed of light?”



MathspeedST: Leapfrogging Lightspeed FASTER than The Speed of Light” provides the fundamental mathematical basis—lying hidden below—for the accounting and distribution of all spacetime and its subsequent expression as energy, matter and fields in the Observable Universe above.



This freely available interactive ebook brings together under one publication the Brooks (Base) Square-Inverse Square Law, or BBS-ISL Matrix for short, under the title MathspeedST. It includes hundreds of images, a number of videos and lots of test-yourself, interactive widgets in presenting the derivation, layout, rules, patterns and ISL-derived mathematical basis of the matrix. Ages 8-10 and up.

A new paradigm in the visualization of basic mathematics is presented!

This is an INTRO — OVERVIEW of the main white paper (page 31 above). It may also be found under the New MEDIA net.art section. It may be helpful to view this first or second, after the NEW, Simplified INTRODUCTION (see Page 35 below).

There is a simple whole number (integer) matrix grid table upon — and within — that every possible whole number Pythagorean Triangle — a.k.a. Pythagorean Triple — can be placed, and proved. The Brooks Base Square - Inverse Square Law (BBS-ISL) Matrix) is an infinitely expandable grid that reveals ALL Pythagorean Triples — both Primitive Triples (PPT) and their non-Primitive multiples (nPTT). An extremely simple geometric AREA proof of the Pythagorean Theorem — c²=a²+b² — is built into the BBS-ISL Matrix.

Not only does the Inverse Square Law describe, define and quantify our most important energies (and their expressions as force) — gravity, light, sound, electromagnetism…, —the BBS-ISL Matrix grid is composed of Pythagorean Triples crisscrossing over much of the entire grid. An intimate inter-connection way beyond a simple, casual association has been revealed. The Dickson Method confirms, validates and provides insight into the generation of ALL Pythagorean Triples directly on the BBS-ISL Matrix.

Numerous animated gifs, slideshows and videos illustrate the compelling visual connection between the Inverse Square Law and the Pythagorean Theorem.

Based on MathspeedST, an interactive ebook (published in November, 2013). MathspeedST is, itself, the complement to LightspeedST (published in June, 2013).

While LightspeedST addressed the ultimate question: “Why is the speed of light constant?”MathspeedST addresses the fundamental, underlying question: “How can information be ubiquitous, effectively traveling FASTER than the speed of light?”



MathspeedST: Leapfrogging Lightspeed FASTER than The Speed of Light” provides the fundamental mathematical basis—lying hidden below—for the accounting and distribution of all spacetime and its subsequent expression as energy, matter and fields in the Observable Universe above.



This freely available interactive ebook brings together under one publication the BBS-ISL Matrix and the Pythagorean Triangles (Triples) under the title TPISC: The Pythagorean — Inverse Square Connection. It includes hundreds of images, a number of videos and lots of test-yourself, interactive widgets in presenting the derivation, layout, proofs, and patterns connecting the two geometries. Ages 8-10 and up.

A new paradigm in the visualization of basic mathematics is presented!

TPISC II: Advanced picks up where TPISC I: Basics left off and takes you under the hood of just how dynamically the two geometries of the BBS-ISL Matrix and the Pythagorean Triples are related. In fact, the two are so inter-connected that either one could be employed to generate the other. This is not coincidence. The ISL and the Pythagorean Triples, together, inform the SpaceTime of the Cosmos!

There is a simple whole number (integer) matrix grid table upon — and within — that every possible whole number Pythagorean Triangle — a.k.a. Pythagorean Triple — can be placed, and proved. The Brooks Base Square - Inverse Square Law (BBS-ISL) Matrix is an infinitely expandable grid that reveals ALL Pythagorean Triples — both Primitive Triples (PPT) and their non-Primitive multiples (nPTT). An extremely simple geometric AREA proof of the Pythagorean Theorem — c²=a²+b² — is built into the BBS-ISL Matrix.

Not only does the Inverse Square Law describe, define and quantify our most important energies (and their expressions as force) — gravity, light, sound, electromagnetism…, —the BBS-ISL Matrix grid is composed of Pythagorean Triples crisscrossing over much of the entire grid. An intimate inter-connection way beyond a simple, casual association has been revealed. The Dickson Method confirms, validates and provides insight into the generation of ALL Pythagorean Triples directly on the BBS-ISL Matrix.

Numerous animated gifs, slideshows and videos illustrate the compelling visual connection between the Inverse Square Law and the Pythagorean Theorem.

Based on MathspeedST, an interactive ebook (published in November, 2013). MathspeedST is, itself, the complement to LightspeedST (published in June, 2013).

While LightspeedST addressed the ultimate question: “Why is the speed of light constant?”MathspeedST addresses the fundamental, underlying question: “How can information be ubiquitous, effectively traveling FASTER than the speed of light?”



MathspeedST: Leapfrogging Lightspeed FASTER than The Speed of Light” provides the fundamental mathematical basis—lying hidden below—for the accounting and distribution of all spacetime and its subsequent expression as energy, matter and fields in the Observable Universe above.



This interactive ebook brings together under one publication the BBS-ISL Matrix and the Pythagorean Triangles (Triples) under the title TPISC: The Pythagorean — Inverse Square Connection. It includes hundreds of images, a number of videos and lots of test-yourself, interactive widgets in presenting the derivation, layout, proofs, and extended fractal-like patterns connecting the two geometries. Ages 8-10 and up.

A new paradigm in the visualization of basic mathematics is presented!

This new and simplified INTRODUCTION to TPISC I: Basics & TPISC II: Advanced is exactly where you should start! In the BIG overview picture, it takes you under the hood of just how dynamically the two geometries of the BBS-ISL Matrix and the Pythagorean Triples are related. In fact, the two are so inter-connected that either one could be employed to generate the other. This is not coincidence. The ISL and the Pythagorean Triples, together, inform the SpaceTime of the Cosmos!

MSST-TPISC on the BBS-ISL Matrix movies on VIMEO. TPISC — MathspeedST-->BBS-ISL Matrix media center.

When we look at competing ideas on TIME IN COSMOLOGY, as wonderfully presented at the recent Perimeter Institute for Theoretical Physics conference (2016), one is struck by the inevitable conclusions drawn by the elegant Casual Set Theory of Rafael Sorkin (ref: https://www.perimeterinstitute.ca/people/rafael-sorkin):

"order + number = geometry."

And why is that? What does it mean to say that the progression of inter-connected number relations forms a geometry --- a geometry of spacetime (ST)?

Take away the overlay of advanced mathematics and look simply at whole integer numbers (WINs) and their relationships to each other. What do you have? Are not numbers simply quantities? So much of this or that? Is not geometry the representation — both in the abstract and physical manifestation state — of those number relationships and an expression of their inter-connectedness? Yes, of course!

But let us step back a moment and frame the concept of time within the cosmology of the universe. What tenants must be satisfied?

See how the order brought to the ISL by the BBS-ISL Matrix gives rise to the discrete geometry of the quantum Universe! Yes, of course, those Pythagorean Triples have a role here, too, as The Pythagorean – Inverse Connection (TPISC) continues!

Simplification

Take 5 or 10 minutes (more or less) and learn to make the 10x10 — or any other size —BBS-ISL Matrix!.

Once you do, you will open the door, get over any fears, and fly right on through to some amazing mathematics!

The BBS-ISL Matrix! is based on simple, whole integer numbers: 1- 2- 3- ,... and their squares: 1- 4- 9-,....

Simple Addition-Subtraction & Multiplication-Division will define the Geometry.

The BBS-ISL Matrix! combines the simplest of algebraic and geometric concepts into easy visualizations!

The BBS-ISL Matrix! is your doorway into the magnificent Pythagorean Theorem and the dynamic Pythagorean Triples!

TPISC the The Pythagorean – Inverse Square Connection, reveals the ubiquitous and intimately connected relationship between the simple, whole number Pythagorean Triple Triangles and the BBS-ISL Matrix!

Let's get started!

Simplification

Take the BBS-ISL Matrix (BIM)for short,

and reveal the Pythagorean Triples (PTs) embedded ubiquitously throughout.

The BBS-ISL Matrix not only reveals every possible PT — both Primitive PTs (PPTs) and their non-Primitive PTs (nPPTs) children — it also uniquely defines the Tree of Primitive Pythagorean Triples (ToPPT).

The ToPPT presents only PPTs and each is only represented once.

The ToPPT "grows" like a tree from its 3-4-5 PPT forming a Trunk. The Trunk forms a Tertiary Branch. This 1st Tertiary Branch/Tier has a Cluster of 3 PPTs. Each PPT member of a Cluster forms the next, 2nd Tertiary Branch/Tier, now with 3 Clusters of 3 PPT each, or 9 PPTs, and so on. But are they related? Are they predictable? Can you see them on the BIM?

The The Pythagorean – Inverse Square Connection, reveals the ubiquitous and intimately connected relationship between the simple, whole number Pythagorean Triple Triangles and the BBS-ISL Matrix! and the answers are YES, YES and YES!

The Quantum Entanglement Conjecture (QEC) is introduced at the end of this visual storyboard presentation. Let's get started!

TPISC III-V

12 Simple, but easy, AREA pieces.

This gives the visual geometry behind the equations.

Once you see how the AREAS — the whole BIM and BIMtree is about AREAS — reveal and give proof to many of the basic equations, you will be well on your way!

The Tree of Primitive Pythagorean Triples (ToPPT) grows outward to larger and larger PPTs and they are inter-connected by AREAS.

eBook

1 banana, 2 banana, 3 banana, 4 … and there’s so much more!

A simple curiosity on how even simpler Whole Integer Numbers (WINs), when squared, form a law that rules the Universe — indeed, SpaceTime (ST) itself — has been a source of awe and wonder for quite some time. This is the Inverse Square Law (ISL).

The “inverse” means it’s per area. When the WINs are squared, you get the area.

The forms and forces of Nature obey the ISL. The initial concentration of an impulse or influence of energy dissipates in a known manner as it expands out from its source. A wonderful and ubiquitous law in forming the ST that builds the Universe. And it doesn’t even need to travel to get there…it is known!

Now, on top of this beautiful law and all the magnificent forms and expressions it can offer, comes yet another sublimely beautiful law — or theorem — that also relates areas to significant form — the Pythagorean Theorem: the long, hypotenuse side squared = the sum of the two squared shorter sides of a 90°-right triangle.

When these triangles are made solely of WINs, we have Pythagorean Triples (PTs).

The PTs are a major subset of the ISL and as such are both infinite in number and ubiquitous in distribution along with the ISL. Their structural beauty lies in the fractal-nodes of slight asymmetry they give to the expanding expression of the ISL. They give new forms!

Not only can these PT fractals be located throughout the BBS-ISL Matrix, together they form a linked structure relating all the parts to the whole, in the form the Tree of Primitive Pythagorean Triples (ToPPT).

This clarity and simplification of form has become the focus of this work. The non-isosceles right triangles, rectangles and ovals of the ToPPT join with the isosceles right triangles, squares and circles of the ISL to generate our ST Universe — giving expression to the Universal mathematical constructs.

3 potato, 4 potato, 5 potato squared …

“BIM — BBS-ISL Matrix — is the geometry defining SpaceTime!”

  MathspeedST: Leapfrogging Lightspeed FASTER than The Speed of Light
The MathspeedST Supplements:

1. TPISC I: Basics
2. TPISC II: Advanced
3. TPISC III: Clarity & Simplification (ToPPT)
4. TPISC IV: Details
5. TPISC V: Exponentials (coming)

1. TPISC I: Basics
2. TPISC II: Advanced
3. TPISC III: Clarity & Simplification (ToPPT)
4. TPISC IV: Details
5. TPISC V: Exponentials (coming)

NEW! Connections between:

• 1. BIM (BBS-ISL Matrix) + PTs (Pythagorean Triples);
• 2. PT + Pentagon & Decagon (double pentagon of DNA);
• 3. BIM + PT + DNA;
• 4. BIM + PT + DNA + Zika Virus (cryo-em imagery).

It is suggested that the pentagonal geometry of the Zika virus allows it to insert itself within the decagonal geometry of our DNA, perhaps spiraling in along the DNA double-helical axis, looking for a simple match opening!

NEW! Connections between:

• 1. BIM (BBS-ISL Matrix) + PTs (Pythagorean Triples);
• 2. PT + Pentagon & Decagon (double pentagon of DNA);
• 3. BIM + PT + DNA;
• 4. BIM + PT + DNA + Zika Virus and Epstein-Barr Virus + many more (cryo-em imagery).

It is suggested that the pentagonal geometry of the Epstein-Barr virus allows it to insert itself within the decagonal geometry of our DNA, perhaps spiraling in along the DNA double-helical axis, looking for a simple match opening! The majority of human viruses display icosahedral pentameter geometry on their capsid surface.

NEW! Connections between:

• 1. BIM (BBS-ISL Matrix) + PTs (Pythagorean Triples);
• 2. PT + Pentagon & Decagon (double pentagon of DNA);
• 3. BIM + PT + DNA;
• 4. BIM + PT + DNA + Zika Virus (cryo-em imagery).

Here we continue with TPISC IV: The Pythagorean - Inverse Square Connection: Details. This time looking for a connection between the 3-4-5 Pythagorean Triple and the concentric, decagons (double pentagons) that form the geometric structure of the DNA double-helix molecule as viewed down its axis composited from 1 full, 360 degree spiral. While emphatically NOT AN EXACT MATCH, the 3-4-5 PT is ridiculously close to being an exact match both from the the angles of a given pentagon to that of the decagon and most especially connecting key angles and vertices between the concentric decagons. In 2001, I wrote up "GoDNA: the Geometry of DNA (axial view)," and, "SCoDNA: Structure & Chemistry of DNA" about the pure double-pentagon (decagon) geometry of the double-helix molecule. It includes a DNA Master Chart that specifically relates the concentric decagons.

NEW! Connections between:

• 1. BIM (BBS-ISL Matrix) + PTs (Pythagorean Triples);
• 2. PT + Pentagon & Decagon (double pentagon of DNA);
• 3. BIM + PT + DNA;
• 4. BIM + PT + DNA + Zika Virus (cryo-em imagery).

Here we continue with TPISC IV: The Pythagorean - Inverse Square Connection: Details. A special connection between the BBS-ISL Matrix + Pythagorean Triples + axial DNA decagon (double pentagonal) geometry was the topic of the previous white paper (Page 44).

Much to see here!

NEW! Connections between:

• 1. BIM (BBS-ISL Matrix) + PPTs (Primitive Pythagorean Triples);
• 2. BIM (BBS-ISL Matrix) + PRIMES;
• 3. PRIMES + PPTs (Primitive Pythagorean Triples);

Here’s the thing. Amongst a myriad of other connections, there exist an intimate connection between three number systems:

• 1. The ISL as laid out in the BIM;
• 2. PTs — and most especially PPTs — as laid out on the BIM;
• 3. The PRIME numbers — PRIMES — as laid out on the BIM.

The BIM is the FIXED GRID numerical array of the ISL.

Amongst its vast array of inter-connecting Number Pattern Sequences (NPS) — i.e., number systems — two such systems stick out and do so in such an overtly visual — as well as mathematical — way that their connection to each other is more than implied.

You see, both the PPTs and PRIMES strictly align themselves on the SAME paths within the BIM.

SEE Page 52 (below) for the UPDATE:PRIMES_vs_NO-PRIMES.

SEE Page 53 (below) for the SUMMARY: Simple Path: BIM-PRIMES.

Open letter addressing the "ER=EPR" work of Leonard Susskind, Juan Maldacena, and Mark Van Raamsdonk, …

The quantum universe begs the question: what is the quantum?

Perhaps we can say the basic pulse-propagation of ST from its singularity — a singularity connected to the larger pool of all singularities by the Conservation of Energy — out into full spacetime extension, and back, is the fundamental quantum. The ST so formed has articulation points, nodes of structural formation made by the embedded Pythagorean Triples. These nodes are also quantized as they both are dependent upon the unfolding Inverse Square Law-based ST, and they have distinct vector parameters inherently built into their asymmetric forms. They have both magnitude and direction — and as vectors their net expression can be the result of the total added or subtracted interfering ST units. The fact that every primitive PT comes in both its parent PPT form as well as its child nPPT form provides a built in fractal-like structure, that, when combined with the fact that each and every PT — PPT and nPPT — has 4 iterations of expression within the unfolding ST unit pulse-propagation, and we we now have a fractal-based, holographic quantum universe(s)!

Such profound harmony of form naturally begs the question: to what effect?

Several conjectures come to mind:

1. Would not the fractal-nodes of the PTs lead to structural articulation points along the expanding ST that would allow the geometry of the Squares to be enhanced by the embedded geometry of the non-Square Rectangles holding these PTs?
2. Would not these same fractal-nodes of the PTs be internal, integral and identifying structural parameters (Quantum State Numbers) of any wave-particle ST unit formed?
3. Would not the immensely inter-connecting linkage of the PTs, between the PTs and between the PTs and the BIM itself, provide the very basis of the wave-particle duality and entanglement phenomenon we see in the *Double-Slit Experiment and the Quantum Entanglement (DSEQEConjecture) that lies at the core of our understanding the Quantum Universe?

The ER = EPR conjecture is a very interesting insight into the joining of quantum mechanics with quantum gravity through the geometry of spacetime.

What is needed is the other, complementary side that seals the deal: any entanglement of spacetime (ST) and ST units/particles, i.e. photons, gravitons, Higgs Bosons,… must reckon with:

1. ST itself is formed in such a way that it obeys the Inverse Square Law (ISL), is always consistent with the Conservation of Energy, and its very formation must be one and the same with the formation of its progeny ST units and the built-in velocity of light, c;
2. ST itself, and thus its ST progeny, must be conserved;
3. ST itself, and thus it ST progeny, must be fully accountable for all intricacies of the Double-Slit Experiment.

SEE Page 49 (below) for Part II of II CaCoST - DSEQEC.

SEE Page 50 (below) for spreadsheets: BIM: How to Make.

Here is the working draft of the TPISC IV: Details: PPT Profiles; PPT and PRIMES Distribution-Active Rows_24; and, DSEQEC (Double-Slit Experiment – Quantum Entanglement Conjecture ebook. This includes the above Pages 42-48. A PDF version will be forthcoming.

ORGANIZATIONAL NOTE: While it may have made more sense to divide the entire work into three parts — PPT profiles followed by the ÷ 24 all by itself in the second part, and the Double-Slit Experiment—Quantum Entanglement Conjecture (DSEQEC) in the third — preference was given to keeping it all together under "Details" in that each part informs the other and keeping a constant "eye" on the bigger picture outweighs over-focusing on either part alone. In Nature, these "parts" are NOT separated! As before, the "discovery" phase of this work follows a loose chronological order. The storyline actually helps to see how the BIM reveals itself, little by little, by following its leads — bits of information acting like bread crumbs pointing to a path of even greater clarity — via greater detail!

The HTML Canvas feature has been extensively utilized to tell this story in a sequential series of interactive images and forms the outline basis of TPISC IV: Details. Along with additional images, animations, tables and charts, reference will be made to this interactive page: Page 42: TPISC III: Clarity & Simplification and TPISC IV: Details (~ A MathspeedST Supplement~).

Please keep this in mind: The quantum universe begs the question: what is the quantum?

Perhaps we can say the basic pulse-propagation of ST from its singularity — a singularity connected to the larger pool of all singularities by the Conservation of Energy — out into full spacetime extension, and back, is the fundamental quantum. The ST so formed has articulation points, nodes of structural formation made by the embedded Pythagorean Triples. These nodes are also quantized as they both are dependent upon the unfolding Inverse Square Law-based ST, and they have distinct vector parameters inherently built into their asymmetric forms. They have both magnitude and direction — and as vectors their net expression can be the result of the total added or subtracted interfering ST units. The fact that every primitive PT comes in both its parent PPT form as well as its child nPPT form provides a built in fractal-like structure, that, when combined with the fact that each and every PT — PPT and nPPT — has 4 iterations of expression within the unfolding ST unit pulse-propagation, and we we now have a fractal-based, holographic quantum universe(s)!

SEE Page 50 (below) for spreadsheets: BIM: How to Make.

SEE Page 51 (below) for Big OUTLINE: Big OUTLINE.

Part II of II on the DSEQEC (Summary) as a natural extension of the CaCoST

From Part I of DSEQEC: The quantum universe begs the question: what is the quantum?

Perhaps we can say the basic pulse-propagation of ST from its singularity — a singularity connected to the larger pool of all singularities by the Conservation of Energy — out into full spacetime extension, and back, is the fundamental quantum. The ST so formed has articulation points, nodes of structural formation made by the embedded Pythagorean Triples. These nodes are also quantized as they both are dependent upon the unfolding Inverse Square Law-based ST, and they have distinct vector parameters inherently built into their asymmetric forms. They have both magnitude and direction — and as vectors their net expression can be the result of the total added or subtracted interfering ST units. The fact that every primitive PT comes in both its parent PPT form as well as its child nPPT form provides a built in fractal-like structure, that, when combined with the fact that each and every PT — PPT and nPPT — has 4 iterations of expression within the unfolding ST unit pulse-propagation, and we we now have a fractal-based, holographic quantum universe(s)!

The ER = EPR conjecture is a very interesting insight into the joining of quantum mechanics with quantum gravity through the geometry of spacetime.

What is needed is the other, complementary side that seals the deal: any entanglement of spacetime (ST) and ST units/particles, i.e. photons, gravitons, Higgs Bosons,… must reckon with:

1. ST itself is formed in such a way that it obeys the Inverse Square Law (ISL), is always consistent with the Conservation of Energy, and its very formation must be one and the same with the formation of its progeny ST units and the built-in velocity of light, c;
2. ST itself, and thus its ST progeny, must be conserved;
3. ST itself, and thus it ST progeny, must be fully accountable for all intricacies of the Double-Slit Experiment.

SEE Page 47 (above) for Part I of II CaCoST - DSEQEC.

SEE Page 50 (below) for spreadsheets: BIM: How to Make.

Making the BIM

It is remarkably simple to make a BIM grid of any size right within your own spreadsheet app (Numbers, Excel,...). Only a few simple formulas to head up the columns and you can expand the BIM to whatever size you want, limited only by your processing power, file size limits and the practical means of filling the cells with very large numbers should you go way out there.

You can also selectively add ONLY THE ROWS and COLUMNS you desire. It works all across the grid. It is also searchable!

Although Python may be preferable for its depth and versatility in making and searching within the BIM, your spreadsheet-BIM will be just so easy to implement, display and image out that it may become your go to method for working on the BIM.

Requirements:

• spreadsheet app (I used Numbers)
• a few simple formulas (supplied)
• numbers list: 1,2,3,...
Date published: 12-11-2018. Updated: 12-28-2018, 05-23-2019
[ TOP ]

A simple, linked outline for the contents of
TPISC IV: Details (page 48 above) of PPT Profiles; PPTs and PRIMES Distribution-Active Rows_24; and, DSEQEC (Double-Slit Experiment – Quantum Entanglement Conjecture)

ABSTRACT

INTRODUCTION and PREMISE

CONCLUSION

BACKGROUND

SUMMARY

SUPPORTING GRAPHICS, TABLES, and REFERENCES

SEE Page 46 (above) for the History: BIM_PPT_PRIMES.

SEE Page 53 (below) for the SUMMARY: Simple Path: BIM-PRIMES.

They BOTH generate ALL the same NO-PRIMES (NP) that can be eliminated from the ODD#s to give ALL the PRIMES (≧3).

It helps to start with the visualization of the geometry on the BIM (BBS-ISL Matrix).

eBook

Gaps and Combinations (Conjectures):

1. Twin Primes
2. Primes separated by other EVEN numbers
3. Euler's 6n+1 and 6n-1
4. Fermat-Euler's 4n + 1= Sum of Two Squares Theorem (Pythagorean Primes)
5. Dickson's Conjectures: Sophie Germain Primes
6. Goldbach Conjecture (Euler's "strong" form)
7. Primes - BIM - Pythagorean Triples

Details of the PTOP and the patterns proving the Goldbach Conjecture (strong form):

1. Periodic Table of Primes (PTOP)
2. Goldbach Conjecture (Euler's "strong" form)

eBook

PRIMES, Periodic Table Of PRIMES (PTOP), PTOP on the BIM, Make the PTOP with Fractals

What if the secret of the primes were to be more than just atomic-base units of the natural whole numbers.

What if they really gained their constructive prowess when their natural sequence — the PRIMES Sequence (PS = 3,5,7,11,13,17,19,23,29,31,...) — were tied together as PSs.

Yes as FRACTALS! PS-Fractals.

Here we see just how easy it is to make the Periodic Table Of PRIMES (PTOP) from scratch without any direct reference to the PTOP on the BIM from which it was originally derived. Of course, the original derivation method is also described. The PS-Fractal method of making the PTOP showcases the repetitive, re-iterate, self-similar fractal construction right on the PTOP. On the other hand, the visualization of the symmetrical geometry the PPsets present is best seen directly with the PTOP on the BIM.

PRIMES on the BIM

ABSTRACT

The “PTOP (Periodic Table Of PRIMES) & the Goldbach Conjecture” (2019), also referred to as BIM: Part I, updated and clarified the PTOP, PPsets and their PPset TRAILS that were originally presented in MathspeedST (2010). BIM = BBS-ISL Matrix.

In this work, a major update and refinement has been made. Each of the three parts — BIM: II, III and IV presents new findings that visibly demonstrate the PRIMES on the BIM.

Each and every PRIME, when treated as part of a PPset (P₁, P₂), can be found and individually profiled DIRECTLY on the BIM as seen in BIM: Part II.

Plotting all the Lower Diagonal P₂ PRIMES on a table and then re-plotting those results back onto the BIM opened up a new vista. By substituting the AXIS values for each of those P₂ PRIMES from the table, one has now formed a new SubMatrix of the BIM with ALL the PPsets in place — BIM: Part III.

In BIM: Part I-III, we have mostly used just half of the bilaterally symmetric BIM to reveal and describe the geometry.

In BIM: Part IV, we now look at the whole BIM with the PPset SubMatrix in place.

Treating these PPsets as objects and counting them within progressively larger square areas — forming what is called “Object AREAS” — a pure ISL Number Pattern Sequence (NPS) is found. As the PS-Fractal series of each AXIS joins to form the PPsets, their actual numbers — as PPset TRAILS — progressively grows and sums up to quantities with Object AREAS that directly mirror the fundamental ISL NPS: 1—4—9—16—25–…

And while this intimate relationship of the PRIMES to the ISL can not predict the next largest PRIME, it can ABSOLUTELY account for each and every PRIME at, and below, any given PRIME, regardless of size.

While Euler’s Strong Form of the Goldbach Conjecture is proved along the way, the real significance is what we have seen unfolding in BIM: Parts I-IV. The role of Symmetry and Fractal underlie everything about the PRIMES.

The PRIMES on the BIM is all about how the fractal nature of the PS becomes expressed as symmetry on the BIM as isosceles and equilateral triangles, forming the PPsets that ultimately form ALL the EVEN numbers!

What is marvelous, incredible, mind-blowing in every way, is that this same symmetry—fractal—isosceles/equilateral triangle relationship is found — indeed, is part and parcel — throughout the BIM.

The ISL seems to reflect the most basic and fundamental relationship(s) between quantity and the numbers that account for it.

PRIMES on the BIM ebook

ABSTRACT

The “PTOP (Periodic Table Of PRIMES) & the Goldbach Conjecture” (2019), also referred to as BIM: Part I, updated and clarified the PTOP, PPsets and their PPset TRAILS that were originally presented in MathspeedST (2010). BIM = BBS-ISL Matrix.

In this work, a major update and refinement has been made. Each of the three parts — BIM: II, III and IV presents new findings that visibly demonstrate the PRIMES on the BIM.

Each and every PRIME, when treated as part of a PPset (P₁, P₂), can be found and individually profiled DIRECTLY on the BIM as seen in BIM: Part II.

Plotting all the Lower Diagonal P₂ PRIMES on a table and then re-plotting those results back onto the BIM opened up a new vista. By substituting the AXIS values for each of those P₂ PRIMES from the table, one has now formed a new SubMatrix of the BIM with ALL the PPsets in place — BIM: Part III.

In BIM: Part I-III, we have mostly used just half of the bilaterally symmetric BIM to reveal and describe the geometry.

In BIM: Part IV, we now look at the whole BIM with the PPset SubMatrix in place.

Treating these PPsets as objects and counting them within progressively larger square areas — forming what is called “Object AREAS” — a pure ISL Number Pattern Sequence (NPS) is found. As the PS-Fractal series of each AXIS joins to form the PPsets, their actual numbers — as PPset TRAILS — progressively grows and sums up to quantities with Object AREAS that directly mirror the fundamental ISL NPS: 1—4—9—16—25–…

And while this intimate relationship of the PRIMES to the ISL can not predict the next largest PRIME, it can ABSOLUTELY account for each and every PRIME at, and below, any given PRIME, regardless of size.

While Euler’s Strong Form of the Goldbach Conjecture is proved along the way, the real significance is what we have seen unfolding in BIM: Parts I-IV. The role of Symmetry and Fractal underlie everything about the PRIMES.

The PRIMES on the BIM is all about how the fractal nature of the PS becomes expressed as symmetry on the BIM as isosceles and equilateral triangles, forming the PPsets that ultimately form ALL the EVEN numbers!

What is marvelous, incredible, mind-blowing in every way, is that this same symmetry—fractal—isosceles/equilateral triangle relationship is found — indeed, is part and parcel — throughout the BIM.

The ISL seems to reflect the most basic and fundamental relationship(s) between quantity and the numbers that account for it.

DSEQEC (Double-Slit Experiment - Quantum Entanglement Conjecture) and CaCoST (Creation and Construction of SpaceTime) eBook ebook

ABSTRACT

DSEQEC & CaCoST

.

DSEQEC: The Double-Slit Experiment & Quantum Entanglement Conjecture.

The DSEQEC states that the two are really just two sides of the same coin. Both being explained by the BIM and BIMtree.

Aside from the more modern dilemma of Dark Matter and Dark Energy, there are two major — and largely unanswered — questions that have historically persisted to this day at the heart of the Standard Model and its description by Quantum Mechanics: the Double-Slit Experiment and Quantum Entanglement.

The geometry of ST formation generates descriptive parameters referred to as quantum numbers or their quantum number state — unique identifiers for each and every subatomic particle. These wave-particle ST units exhibit unusual and non-intuitive behaviors that reveal a connection to each other below the physical reality we observe. The Conjecture is that that underlying connection is the BIM and BIMtree geometry that informs the creation and disposition of ALL ST — including their quantum number state identifiers.

The implications for this dual Matter-Anti-Matter alternating "current" of p-p ST units — here it is the photon, but it holds for all ST units — are far and wide. The troubling and poorly understood Double-Slit Experiments were trying to tell us something all along. Our one-sided perception of "reality" was just that: one sided! Of course actual experimentation was showing us, or at least providing some clues, that there is more to the story than just one side. That a single photon can actually demonstrate interference is exactly what it should be knowing that the Universe is actually a dual, two-sided alternating "reality."

Where did all the anti-matter go? No where! Its "hiding" right where it has been since the inflationary, Big-Bang!

The other side of the DSE coin is Quantum Entanglement. As the model is demonstrating, every ST unit is both entangled as part of its generating p-p, and, that pulse itself alternately reveals itself in OV/AV. Entanglement is both part of the ST unit structure and part of the "expression", the latter p-p into and out of the OV-Matter and AV-Anti-Matter continuum as a necessary and most efficient means to balance the accounting ledgers of the Conservation Laws — not the least of which is that of the Creation and Conservation of SpaceTime (CaCoST).

As the Universe is a numbers game, we naturally must distill down to the fundamental numbers and the system that guides them, to get any real traction on the overall Cosmos. In LightspeedST, MathspeedST and all its supplemental works on Pythagorean Triples and PRIMES, that guiding system points back to the unifying Inverse Square Law — a simple law that relates number quantities such as amounts and distribution — densities — over time.

BIM (BBS-ISL Matrix) provides the "information" — ubiquitously known throughout the Cosmos — template that guides its formation and conservation.

What has been emphasized before, must again be restated: the BIM (BBS-ISL Matrix) that presents the Inverse Square Law (ISL), is so fundamentally at ground zero that the connections and interconnections it keeps on revealing between the Pythagorean Triples and the PRIMES more than suggests a key relationship to the Construction and Conservation of SpaceTime (CaCoST)— it is SpaceTime.

And as such, any deconstruction and elucidation of the fundamental nature of Quantum Mechanics — as in the Double-Slit Experiment and the Quantum Entanglement Conjecture (DSEQEC) — will necessarily be fundamentally about the ISL — and the ISL as seen on the BIM!

Resistance is futile! 

Of greater significance is the underlying theme here: Nature works with simple numbers, amounts and qualities to generate a rich, complex, inter-connected and harmonious “system” by means of symmetry, repetition, re-iteration and scale to inform what we think of as our “Universe!”

What is being suggested here is that our “Universe” is really only part of the larger Cosmos — or SpaceTime Continuum — that is fundamentally bi-polar.

Our “Universe” and all that “matter” is but half of the full Continuum, the “other” half being all that “anti-matter,” reverse charge and spin “-Universe” that co-exists in space and time.

Entanglement ensures that the “accounting” for the Conservation Laws — energy, momentum, angular momentum, charge — remains in balance, ultimately through the Conservation of SpaceTime!

TPISC IV: Details, DSEQEC and PRIMES ebook

Everything in one interactive ebook!

Math—Art—Physics (MAP)

Interactive Slideshow Links of White Papers and eBooks

An Extremely Simple Visualization, I

In Part I to the solution to Euler's Strong Form of the Goldbach Conjecture is shown as the simplest visualization: On the BIM, a diagonal line drawn between any two similar EVENS --- Axis to Axis --- intersects those cells whose coordinate values sum to said EVEN! Part II takes this simplification EVEN further!

An Extremely Simple Visualization, II

In Part II to the solution to Euler's Strong Form of the Goldbach Conjecture is shown as the simplest visualization: On the BIM, a diagonal line drawn between any two similar EVENS --- Axis to Axis --- intersects those cells whose coordinate values sum to said EVEN! And in Part II, that simplification is EVEN simpler!

A Not So Simple Visualization, III

In Part III to the solution to Euler's Strong Form of the Goldbach Conjecture the simplest visualizations of Part I and II now translate back to the original visualization of the Periodic Table Of Primes (PTOP): On the BIM, taking 1/2 of any given EVEN finds the PRIME Pair sets (PPsets) that inform it symmetrically placed on either side of said EVEN/2.

Mersenne Prime Squares: The Geometry of the Mersenne Prime-Perfect Number Relationship

In this first look into the newly discovered geometry of the inherent relationship between EVEN Perfect Numbers and ODD Mersenne Primes is revealed. This follows from, and in many ways evolved from Euler's Strong Form of the Goldbach Conjecture, the simplest visualizations of Part I, II and III on the BIM as translated back from the original visualization of the Periodic Table Of Primes (PTOP) found in the BIM-Goldbach_Conjecture series described above (Pages 66-68).

Every Mersenne Prime has a 1-to-1 relationship with a Perfect Number. The former are all ODD and the latter all EVEN numbers. When the Mersenne Prime is expanded into a Mersenne Prime Square area, it always contains its Perfect Number area, within. And it does so for each and every Mersenne Prime. But there is more...

Date published: 04-05-2021. Updated: 06-20-2021, 03-07-2023.(see BIMMPS) below

Redirected to BIMMPS-__Please see the BIMMPS Outline to follow the sequence in order.

Mersenne Prime Square: 3 simple intros

In this second look into the newly discovered geometry of the inherent relationship between EVEN Perfect Numbers and ODD Mersenne Primes is revealed with three simple intros:

1. Basics
2. Intermediate
3. Advanced

Every Mersenne Prime has a 1-to-1 relationship with a Perfect Number. The former are all ODD and the latter all EVEN numbers. When the Mersenne Prime is expanded into a Mersenne Prime Square area, it always contains its Perfect Number area, within. And it does so for each and every Mersenne Prime. But there is more...

Date published: 06-10-2021. Updated: 06-20-2021, 03-07-2023.(see BIMMPS) below

Redirected to BIMMPS-__Please see the BIMMPS Outline to follow the sequence in order.

Mersenne Prime Square: Advanced

In this third look into the newly discovered geometry of the inherent relationship between EVEN Perfect Numbers and ODD Mersenne Primes, an advanced, step-by-step analysis of that visualization will be presented. Basically, we will be taking the invisible Euclid-Euler Theorem and transforming it into the highly visible geometric components (lines, areas and shapes) that make up the Mersenne Prime Square.

1. MPS divides into:
2. PN + OC subdivide into:
3. PNS + CR + CR + OCS

Every Mersenne Prime has a 1-to-1 relationship with a Perfect Number. The former are all ODD and the latter all EVEN numbers. When the Mersenne Prime is expanded into a Mersenne Prime Square area, it always contains its Perfect Number area, within. And it does so for each and every Mersenne Prime. But there is more...

Date published: 06-15-2021. Updated: 06-20-2021, 03-07-2023.(see BIMMPS) below

Redirected to BIMMPS-__Please see the BIMMPS Outline to follow the sequence in order.

The MPS Project

In July 1-16, 2021 I was invited to participate in an artist residency at Bldg 5 of the NWMAW (NW Marine Art Works) in Portland, Oregon. I took this opportunity to create an installation of The Mersenne Prime Squares (MPS) Project.

One could say this is the fourth look into the newly discovered geometry of the inherent relationship between EVEN Perfect Numbers and ODD Mersenne Primes. Basically, we will be taking the invisible Euclid-Euler Theorem and transforming it into the highly visible geometric components (lines, areas and shapes) that make up the Mersenne Prime Square.The geometry is large. It is meant to be felt -- to be experienced. It occurs within a highly geometric, WWII era barrel-vault space. The three overlapping areas (below) will be stacked directly on top of each other.

1. MPS divides into:
2. PN + OC subdivide into:
3. PNS + CR + CR + OCS

Every Mersenne Prime has a 1-to-1 relationship with a Perfect Number. The former are all ODD and the latter all EVEN numbers. When the Mersenne Prime is expanded into a Mersenne Prime Square area, it always contains its Perfect Number area, within. And it does so for each and every Mersenne Prime. But there is more...

A Silent Auction of the 108 grid paintings and 28 butterflies will be held online here. The majority of the proceeds will be donated to North Pole Studio, a studio of artists with autism or I/DD.

Redirected to BIMMPS-__Please see the BIMMPS Outline to follow the sequence in order.

The Butterfly Fractal 1

Every Mersenne Prime has a 1-to-1 relationship with a Perfect Number. The former are all ODD and the latter all EVEN numbers. When the Mersenne Prime is expanded into a Mersenne Prime Square area, it always contains its Perfect Number (PN) area, within. And it does so for each and every Mersenne Prime. But there is more...

Remember that the Mersenne Primes Square is basically taking the invisible Euclid-Euler Theorem and transforming it into the highly visible geometric components (lines, areas and shapes) that make up the Mersenne Prime Square. Squaring the Mersenne Prime (Mp) gives the MPS, with the PN embedded within. Once you see the geometry, it becomes obvious -- inevitable -- that every Mp is associated with a PN. It is also associated with and ODD Complement area (OC), a Complement Rectangle area (CR) and the squares of the x- and y- short sides of the embedded rectangles, known as the Perfect Number Square (PNS) and the ODD Complement Square (OCS). All these acronyms become reduced to a simpler designation as x--x²--y--y²z--z²--xy--xz--xy.

1. MPS divides into:
2. PN + OC subdivide into:
3. PNS + CR + CR + OCS

As the Mersenne Prime (Mp = z) is based on the exponential power of 2 - 1, it follows that x--x²--y--y²z--z²--xy--xz--xy can be shown to be simple additive combinations -- better known as the running sums (∑) of 2ⁿ that adds onto the original 1. What does mean? Take the original number "1" and now double it -- ∑ = 1+2=3, and now add that to the next doubling of 2 -- ∑ = 1+2+4=7, add that to the next doubling of 4 -- ∑ = 1+2+4+8=15, next ∑ = 1+2+4+8+16=31,.... The running sums of ∑ give 3--7--15--31--.... The 3, 7 and 31 are the first 3 Mps while the 15 is a Mersenne number, but not a prime.

A visual depiction of the doubling of quantity "1" forms the basis of the Butterfly Fractal 1. It is a fractal in the sense that the 1--2--4--8--16--... repeats in a redundant, reiterative, self-similar manner in which any part of the resulting whole can be shown to be make up of the same identical sequence. This visualization expands to infinity and along the way the shape it forms resembles that of the butterfly! A number of such examples are shown herein.

Redirected to BIMMPS-__Please see the BIMMPS Outline to follow the sequence in order.

Ocean of Numbers

An overview of the relationship between the Mersenne Prime Squares and the Mersenne Primes -- Perfect Numbers -- Butterfly Fractal 1 -- and the Exponential Power of 2.

You may want to read through this first before engaging with the several introductions -- both simple and advanced -- and the Butterfly Fractal 1 pages. It is a bit of a catch 22. This overview will make a lot more sense after, yet it hopefully gives one the bigger sense of how all the parts fit together, and why. That is good to know, at least as an overview, before you dig deeper!

There are numerous images, animations, tables,... in those earlier pages. Here the description is simple text and simple math.

Date published: 07-28-2022. Updated: 07-30-2022, 03-07-2023.(see BIMMPS) below

Redirected to BIMMPS-__Please see the BIMMPS Outline to follow the sequence in order.

Ocean of Numbers II (BFM)

An overview of the relationship between the Mersenne Prime Squares and the Mersenne Primes -- Perfect Numbers -- Butterfly Fractal 1 -- and the Exponential Power of 2 via the Butterfly Fractal Matrix (BFM).

You may want to read through Oceans of Numbers Part I and the intro pages on the Butterfly Fractal and Mersenne Prime Square first before engaging with Part II. This overview, with the focus on the new Butterfly Fractal Matrix, will make a lot more sense after, yet it hopefully gives one and even bigger sense of how all the parts fit together, and why. That is good to know, at least as an overview, before you dig deeper!

There are numerous images, animations, tables,... in those earlier pages. Here the description is simple text and simple math.

Date published: 11-19-2022. Updated: 07-10-2022, 03-07-2023.(see BIMMPS) below

Redirected to BIMMPS-__Please see the BIMMPS Outline to follow the sequence in order.

BIMMPS

Work in progress - ongoing - see Updates below

BIM-Mersenne Prime Square (BIM-MPS_0)

An overview of the relationship between the Mersenne Prime Squares and the Mersenne Primes -- Perfect Numbers -- Exponential Power of 2 -- Butterfly Fractal 1 and Butterfly Fractal Matrix on the BIM.

You may want to familiarize yourself with this Introductory Outline, Ground "Rules" and Overview of the 10 Part "BIMMPS."

BIMMPS stands for BIM-Mersenne Prime Square connection that this series is all about. Everything you wanted to know about the Mersenne Prime Squares (MPS), Mersenne Primes (Mp), Perfect Numbers (PN), Complement Rectangles (CR), Butterfly Fractal 1 (BF1), Butterfly Fractal Matrix and the Exponential Power of 2 is found in plain sight right on the Inverse Square Law-driven BBS-ISL Matrix (BIM).

There are numerous images, animations, tables, simple text and simple math.

Each of the Parts I-VIII will have the same Outline and Links for navigation.

NOTE: Earlier published papers (pages 69-75) are incorporated within these works.

Date published: 03-04-2023. Updated: 03-17-2023.

Work in progress - ongoing - see Updates below

BIMMPS: Part I: BOOK I: Two Dots: The MPS.(BIM-MPS_1)

An introductory overview -- from from basics to slightly advanced -- of the relationship between the Mersenne Prime Squares and the Mersenne Primes -- Perfect Numbers on the BIM.

You may want to familiarize yourself with this Introductory Outline, Ground "Rules" and Overview of the 10 Part "BIMMPS."

BIMMPS stands for BIM-Mersenne Prime Square connection that this series is all about. Everything you wanted to know about the Mersenne Prime Squares (MPS), Mersenne Primes (Mp), Perfect Numbers (PN), Complement Rectangles (CR), Butterfly Fractal 1 (BF1), Butterfly Fractal Matrix and the Exponential Power of 2 is found in plain sight right on the Inverse Square Law-driven BBS-ISL Matrix (BIM).

Three very short stories weave together both the generative ideas that informed both the research and discovery, and, subsequent paths for presenting the scope of the findings. We start here with the first short story: Two dots...

There are numerous images, animations, tables, simple text and simple math.

Work in progress - ongoing - see Updates below

BOOK II: An ODD Tale: Ocean of Numbers (Part I).(BIM-MPS_2)

The overview of the relationship between the Mersenne Prime Squares and the Mersenne Primes -- Perfect Numbers -- Exponential Power of 2 -- Butterfly Fractal 1 and Butterfly Fractal Matrix on the BIM starts digging a bit deeper.

You may want to familiarize yourself with this Introductory Outline, Ground "Rules" and Overview of the 10 Part "BIMMPS."

BIMMPS stands for BIM-Mersenne Prime Square connection that this series is all about. Everything you wanted to know about the Mersenne Prime Squares (MPS), Mersenne Primes (Mp), Perfect Numbers (PN), Complement Rectangles (CR), Butterfly Fractal 1 (BF1), Butterfly Fractal Matrix and the Exponential Power of 2 is found in plain sight right on the Inverse Square Law-driven BBS-ISL Matrix (BIM).

Three very short stories weave together both the generative ideas that informed both the research and discovery, and, subsequent paths for presenting the scope of the findings. Here we move on with the second short story: Oceans of Numbers (Part I)...

There are numerous images, animations, tables, simple text and simple math.

Work in progress - ongoing - see Updates below

BOOK III: An Artist, Mathematician and Choreographer ... : Oceans of Numbers (Part II).(BIM-MPS_3)

An overview of the relationship between the Mersenne Prime Squares and the Mersenne Primes -- Perfect Numbers -- Exponential Power of 2 -- Butterfly Fractal 1 and Butterfly Fractal Matrix on the BIM now suggest that to really grasp the scope of this work we will greatly benefit from viewing it both from different perspectives, and even with different sensibilities.

You may want to familiarize yourself with this Introductory Outline, Ground "Rules" and Overview of the 10 Part "BIMMPS."

BIMMPS stands for BIM-Mersenne Prime Square connection that this series is all about. Everything you wanted to know about the Mersenne Prime Squares (MPS), Mersenne Primes (Mp), Perfect Numbers (PN), Complement Rectangles (CR), Butterfly Fractal 1 (BF1), Butterfly Fractal Matrix and the Exponential Power of 2 is found in plain sight right on the Inverse Square Law-driven BBS-ISL Matrix (BIM).

Three very short stories weave together both the generative ideas that informed both the research and discovery, and, subsequent paths for presenting the scope of the findings. Now we are really getting in to it as we move on with the third short story: Oceans of Numbers (Part II)...

There are numerous images, animations, tables, simple text and simple math.

Work in progress - ongoing - see Updates below

BOOK IV: Not to be outdone-the Artist-Geometer: MPSnext & 10 Common Parameters.(BIM-MPS_4)

An overview of the relationship between the Mersenne Prime Squares and the Mersenne Primes -- Perfect Numbers -- Exponential Power of 2 -- Butterfly Fractal 1 and Butterfly Fractal Matrix on the BIM. Perspective, especially visible perspective, can do wonders to "getting it!" Don't worry, there is also pure math coming up later.

You may want to familiarize yourself with this Introductory Outline, Ground "Rules" and Overview of the 10 Part "BIMMPS."

BIMMPS stands for BIM-Mersenne Prime Square connection that this series is all about. Everything you wanted to know about the Mersenne Prime Squares (MPS), Mersenne Primes (Mp), Perfect Numbers (PN), Complement Rectangles (CR), Butterfly Fractal 1 (BF1), Butterfly Fractal Matrix and the Exponential Power of 2 is found in plain sight right on the Inverse Square Law-driven BBS-ISL Matrix (BIM).

Three very short stories weave together both the generative ideas that informed both the research and discovery, and, subsequent paths for presenting the scope of the findings. Having laid out the expansive scope that the three short stores touched on, it's time to look at some of the details from a new perspective on the BIM.

There are numerous images, animations, tables, simple text and simple math.

Work in progress - ongoing - see Updates below

BOOK V: PTOFBMPPN: A Mathematicians Table.(BIM-MPS_5)

An overview of the relationship between the Mersenne Prime Squares and the Mersenne Primes -- Perfect Numbers -- Exponential Power of 2 -- Butterfly Fractal 1 and Butterfly Fractal Matrix on the BIM -- and then there are "tables." Tables have that wonderful quality of summation. Stay tuned for -- IMHO -- one of the most fantastic tables is revealed here: the PTOFBMPPNT. The Periodic Table Of Fractal-Based Mersenne Prime-Perfect Numbers Table.

You may want to familiarize yourself with this Introductory Outline, Ground "Rules" and Overview of the 10 Part "BIMMPS."

BIMMPS stands for BIM-Mersenne Prime Square connection that this series is all about. Everything you wanted to know about the Mersenne Prime Squares (MPS), Mersenne Primes (Mp), Perfect Numbers (PN), Complement Rectangles (CR), Butterfly Fractal 1 (BF1), Butterfly Fractal Matrix and the Exponential Power of 2 is found in plain sight right on the Inverse Square Law-driven BBS-ISL Matrix (BIM).

Three very short stories weave together both the generative ideas that informed both the research and discovery, and, subsequent paths for presenting the scope of the findings. Having laid out the expansive scope that are three short stores touched on, it's time to look at some of the details from a new perspective first OFF the BIM, and then back ON the BIM. Blows the mind!

There are numerous images, animations, tables, simple text and simple math.

Work in progress - ongoing - see Updates below

BOOK VI: Summary-Conclusion-References-Appendix I.(BIM-MPS_6)

A Summary-Conclusion-References-Appendix I of the relationship between the Mersenne Prime Squares and the Mersenne Primes -- Perfect Numbers -- Exponential Power of 2 -- Butterfly Fractal 1 and Butterfly Fractal Matrix on the BIM.

You may want to familiarize yourself with this Introductory Outline, Ground "Rules" and Overview of the 10 Part "BIMMPS."

BIMMPS stands for BIM-Mersenne Prime Square connection that this series is all about. Everything you wanted to know about the Mersenne Prime Squares (MPS), Mersenne Primes (Mp), Perfect Numbers (PN), Complement Rectangles (CR), Butterfly Fractal 1 (BF1), Butterfly Fractal Matrix and the Exponential Power of 2 is found in plain sight right on the Inverse Square Law-driven BBS-ISL Matrix (BIM).

Summary-Conclusion-References-Appendix I.

Work in progress - ongoing - see Updates below

BOOK VII: Appendix II: A researchers delight: Tables+.(BIM-MPS_7)

An overview - via TABLES --of the relationship between the Mersenne Prime Squares and the Mersenne Primes -- Perfect Numbers -- Exponential Power of 2 -- Butterfly Fractal 1 and Butterfly Fractal Matrix on the BIM.

You may want to familiarize yourself with this Introductory Outline, Ground "Rules" and Overview of the 10 Part "BIMMPS."

BIMMPS stands for BIM-Mersenne Prime Square connection that this series is all about. Everything you wanted to know about the Mersenne Prime Squares (MPS), Mersenne Primes (Mp), Perfect Numbers (PN), Complement Rectangles (CR), Butterfly Fractal 1 (BF1), Butterfly Fractal Matrix and the Exponential Power of 2 is found in plain sight right on the Inverse Square Law-driven BBS-ISL Matrix (BIM).

There are numerous tables, tables and tables.

Work in progress - ongoing - see Updates below

BOOK VIII: Appendix III: Do the math: Equations.(BIM-MPS_8)

A pure math-equations-based overview of the relationship between the Mersenne Prime Squares and the Mersenne Primes -- Perfect Numbers -- Exponential Power of 2 -- Butterfly Fractal 1 and Butterfly Fractal Matrix on the BIM.

You may want to familiarize yourself with this Introductory Outline, Ground "Rules" and Overview of the 10 Part "BIMMPS."

BIMMPS stands for BIM-Mersenne Prime Square connection that this series is all about. Everything you wanted to know about the Mersenne Prime Squares (MPS), Mersenne Primes (Mp), Perfect Numbers (PN), Complement Rectangles (CR), Butterfly Fractal 1 (BF1), Butterfly Fractal Matrix and the Exponential Power of 2 is found in plain sight right on the Inverse Square Law-driven BBS-ISL Matrix (BIM).

There are not numerous images, animations, tables, or simple text. There are lots of equations! So let's get to it!

Work in progress - ongoing - see Updates below

BOOK IX: Appendix IV: .(BIM-MPS_9)

The Area of the Perfect Number, when the value -- divided by 4 -- becomes the perimeter of a square containing multiple Mersenne Prime Squares.

The Butterfly Fractal that enforms the initial PN now expands to inform an larger area contaning multiple MPS.

This geometry becomes clearly defined on the BIM.

Work in progress - ongoing - see Updates below

BOOK X: Appendix V: .(BIM-MPS_10)

Another summary, resulting from a detailed look at the factors and divisors of both the "containers" of the Exponential Power of 2, and, their very special subset of Mersenne Prime-Perfect Number pairings, brings us closer to a fuller understanding of these "gems" of numbers.

It was through the interplay of inductive-deductive research that the pattern -- so obvious now in retrospect -- issued forth.

The conclusion(s) drawn beg much larger questions.

Some of these are addressed. None of them are answered.

Aperiodic Monotiles: Geometry

Simple Proof of Concept