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 Page 2: PIN: Pattern in Number...from primes to DNA.
A brief
primer on visual patterns found in simple, well known number systems leading into the
remarkable number magic and esthetic beauty of the DNA double helix (axial view) found
in Page 3. GoDNA and continued in Page 4. SCoDNA below. [ TOP ]
 Page 2b: Butterfly Primes...let the beauty seep in.
The lure and
elusiveness of prime numbers, those very same numbers which all the other natural
whole numbers can be reduced to, is thousands of years in the making, captivating
the minds of anyone...perhaps everyone...who has ever contemplated their beauty
and mystery. Can this beauty reveal a visual and mathematical pattern? First online: 120905 [ TOP ]
 Page 2c: Butterfly Prime Directive...metamorphosis.
This paper provides
further insight into the foundation of the "Butterfly Primes" pattern
and in doing so further substantiates the proof of Riemann's Hypothesis. First online: 010506 [ TOP ]
 Page 2d: Butterfly Prime Determinant Number Array (DNA) ~conspicuous abstinence~.
The role of
the highly ordered positive space pattern of the odd nonprimes in revealing the
reverse, negative space pattern of the primes as a determined number array is revealed.
First online: 021506 [ TOP ]
 Page 3: GoDNA: the Geometry of DNA (axial view) revealed.
Which came first?
The self replicating underlying geometry or the form itself with its dependence on
enzymes, hydrogen bonding and a matching code of bases? [ TOP ]
 Page 4: SCoDNA: the Structure and Chemistry of DNA (axial view).
Extending
GoDNA into the realm of structural detail and chemistry forming the composite design. All DNA works online: 2001. [ TOP ]

 Page 5a: DarkDarkLight: Dark Matter = Dark Energy (the inverse of) .
Conservation of SpaceTime as moderated by the Conservation of Force and
Energy. The LUFE Matrix is introduced. The relationship between the Higgs,
photon and graviton gauge bosons as the gatekeepers between spacetime and
matter is examined. Date first published: 050803; link to LUFE Matrix added 053003. [ TOP ]
 Page 5b: The History of the Universe in Scalar Graphics .
(Simultaneous Birth of Black Holes and the Inflationary Universe at the onset of the Big Bang)
A mostly graphical supplement to "Dark Matter = Dark Energy (the inverse of): The Conservation of Spacetime by The Conservation of Force".
Date online: 112804 [ TOP ]
 Page 5c: The History of the Universe  update "The Big Void" .
A short supplement to "Dark Matter = Dark Energy (the inverse of): The Conservation of
Spacetime by The Conservation of Force" and "The History of the Universe in Scalar Graphics."
Date online: 090507 [ TOP ]
 Page 6: The LUFE Matrix.
Here is a fully digitized and interactive version of The LUFE Matrix The
LUFE Matrix: the distillation of SI units into more fundamentally base units of
space and time dimensions (Short Title: The Multidimensional LUFE Matrix).
Date first published: 1985 (LUFE); Date first published by itself: 1991. Date online: 053003.
PreviewReview Slide Show added 080503 [ TOP ]
 Page 7: The LUFE Matrix Supplement.
The supplement, first published along with the The LUFE Matrix in 1991, contains
over 200 examples and proofs covering a wide range of classical and modern physics
and cosmology. The paper version is 97 pages long. The digital version is now completed.
Date first published: 1985 (LUFE); Date first published by itself: 1991. Date online: partial 063003. Completed:072703 [ TOP ]
 Page 8: The LUFE Matrix: Infinite Dimensions.
~a room with a
view...within a room with a view...within a room with a view~
A new, speculative journey to account for the multiple dimensions that have informed
The LUFE Matrix since its introduction in 1985. Date online: partial 031004. Completed: 031404 [ TOP ]
 Page 9: The LUFE Matrix: E=mc^{2}
~and then some~
The focus is on energy. From Einstein's initial insight into the transformation
between mass and energy, we now have a large number of similar transformations between
energy and the other fundamental parameters of matter...i.e., charge and spin...and
their activities within their spacetime environments such as velocity and acceleration, frequency
and temperature, momentum, pressure and force, potential difference, heat and work, etc.
The original PreviewReview Slide Show "Can you say energy 10 different ways?" (see above),
included here, is now further enhanced with a showcase of select LUFE Matrix Supplement
examples all focused on the various forms that energy can take.
Date online: partial 080503. Completed: 101705 [ TOP ]
 Page10: Quantum Gravity..by the book
~merging of Quantum Mechanics with Gravity ~
How do the mechanics of a written language explain quantum mechanics and general relativity (gravity)...the merging of the very small and the very large. Could it be that the same evolutionary neural development that allowed us the creation and development of our language...and more than just mimicking the results thereof...actually was developed in the same nursery of elemental spacetime physics that form our understanding of the cosmos. Simple elemental combinations produce the greatest richness and diversity within a harmonious whole.
Date online: 041008. Completed (Table added): 042908 [ TOP ]
 Page11: Conservation of SpaceTime
~The role of the Higgs, graviton and photon bosons in defining Dark Matter = Dark Energy (the inverse of) ~
In this paper, we will bring together important concepts laid out in more detail in earlier works.
Specifically, we will be providing some simple visual portraits of the elementary particles. These diagrammatic illustrations will allow us to "see" how the concepts might in fact play out. Of course, what we are doing is illustrating the invisible… namely spacetime (SpaceTime=ST) itself and the matter and fields which are to populate it. As it will hopefully be clear, ST, and the matter and fields, are integrally interrelated... one defining the other and it is in the ultimate conservation of ST that results in and from the individual conservations of energy, force, and momentum. Any successful merging of the very large (general relativity) and the very small (quantum mechanics) into a theory of Quantum Gravity will have to address these fundamentals of ST. Date online: 123108.
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 Page 12a: LUFE: The Layman's Unified Field Expose`
~IntroductionContentsList of Figures~
A textural, visual and mathematical work that presents a clear vision of our Universe(s). More relevant today, than ever.
The SpaceTime (ST) continuum ...the matter and fields that are intimately derived from ...singularities, the Planck Event Horizon,
pulse generation ...the expression of mass, charge and spin and other fundamental constants ...the role of symmetry and asymmetry
in the expression of ST units like the fermions and the forcemediating vector bosons ...is explored from the visible to the hidden
through the interface of the photon, graviton and Higgs bosons ...the source of the fractal Universe.
Date first published by itself: 1985. Date online: 112608.
 Page 12b: LUFE: The Layman's Unified Field Expose`
~Chapter I: Symmetry of Design (Fields)
~sonata form~~
Date first published by itself: 1985. Date online: 112608.
 Page 12c: LUFE: The Layman's Unified Field Expose`
~Chapter II: Introduction to the Geometry
Of Interactions
~rhythms and melodies~~
Date first published by itself: 1985. Date online: 112608.
 Page 12d: LUFE: The Layman's Unified Field Expose`
~Chapter III: The Geometry of Elemental Forms
~scales~~
Date first published by itself: 1985. Date online: 112608.
 Page 12e: LUFE: The Layman's Unified Field Expose`
~Chapter IV: The Geometry of Universal Constants
~obstinate bass~~
Date first published by itself: 1985. Date online: 120808.
 Page 12f: LUFE: The Layman's Unified Field Expose`
~Chapter V: The Geometry of Intermediate Forms
~chords~~
Date first published by itself: 1985. Date online: 120808.
 Page 12g: LUFE: The Layman's Unified Field Expose`
~Chapter VI: The Geometry of Change
~polyrhythms and polyphony~~
Date first published by itself: 1985. Date online: 120808.
 Page12h: LUFE: The Layman's Unified Field Expose`
~Chapter VII: The Geometry of Paradox
~counterpoint~~
Date first published by itself: 1985. Date online: 120808.
 Page12i: LUFE: The Layman's Unified Field Expose`
~Chapter VIII: The Geometry of Large Forms
~harmony~~
Date first published by itself: 1985. Date online: 120808.
 Page12j: LUFE: The Layman's Unified Field Expose`
~Appendix Set I:~
Date first published by itself: 1985. Date online: 120808.
 Page12k: LUFE: The Layman's Unified Field Expose`
~Appendix Set II:~Fundamental Constants
These five followup supplemental papers, collectively called "Fundamental Constants,"
are very important works that present key themes which run through LUFE. For some readers, reading these shorter, selfcontained works first,
in reverse order, will provide the benefit of a spotlight on those key themes that will then more fully resonate with the complete LUFE read.
The works are now included, in the original unaltered state from 198507, as part of LUFE under this Appendix Set II. Date first published by itself: 198587. Date online: 120808.
 Page12k: LUFE: The Layman's Unified Field Expose`
~Appendix Set II:~Fundamental Constant ~1. Mass
Mass expression.
Date first published by itself: 198587. Date online: 120808.
 Page12l: LUFE: The Layman's Unified Field Expose`
~Appendix Set II:~Fundamental Constant ~2. Ratios
Ratios of the Electron and the Proton.
Date first published by itself: 198587. Date online: 120808.
 Page12m: LUFE: The Layman's Unified Field Expose`
~Appendix Set II:~Fundamental Constant ~3. Charge
Charge expression. There are particles (ST units) with mass and no charge, but never with charge without mass!
Date first published by itself: 198587. Date online: 120808.
 Page12n: LUFE: The Layman's Unified Field Expose`
~Appendix Set II:~Fundamental Constant ~4. The LUFE Matrix
Common units and The LUFE Matrix.
Date first published by itself: 198587. Date online: 120808.
 Page12o: LUFE: The Layman's Unified Field Expose`
~Appendix Set II:~Fundamental Constant ~5. The Atom
Mass variation in the electron by photon absorption, and, The Inverse Square Law.
Date first published by itself: 198587. Date online: 120808.
 Page12p: LUFE: The Layman's Unified Field Expose`
~Acknowledgements and Bibliography~
Date first published by itself: 1985. Date online: 120808.
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 Page 13: GoMAS: The Geometry of Music, Art and Structure ...linking science, art and esthetics
That simple geometric principles can be responsible for the
manifestation of the multitudes of beauty, diversity and richness of Nature is
further supported here in this paper. The InverseSquareLaw (ISL), a simple
geometric principle, is shown to be the common link between physics, chemistry,
biology, and the science and esthetics of music and art. Nature has
incorporated the ISL as the major game rule to building the cosmos. Our
perception of pleasing form, color and sound is prophetically built into our
sensory being, our biologic evolution, and our structural past. Key to our
understanding of the geometry of the ISL is the interference of complementary
waveforms. Date first published by itself: 1987, 1996 (updated), and 1998 (condensed). Date online (new, expanded): 030909. New Graphic link added: 102809 [ TOP ]

 Page 14: GoMAC: The Geometry of Music And Color, Parts II and III. How the natural, geometric divisions of the musical monochord and the color spectrum are inherently derived from the same universal law that informs how light, sound, electromagnetism, gravity...indeed, "The Architecture of SpaceTime" itself...behaves. That law is the Inverse Square Law (ISL) and it has a specific relationship to all the quantitative, whole integer numbers as shown in the Brooks (Base) Square Matrix (BBS).
Part II of the threepart Geometry of Music And Art (GoMAC) series focuses on a visual journey of how the musical spectrum...and specifically the chromatic and diatonic scales...is based on the ISL. A visual walkthrough and summary of key concepts relating the ISLderived Brooks (Base) Square, BBS, Matrix with a geometrically derived division of both the musical monochord...resulting in the chromatic and diatonic major scale...and the color spectrum (as seen in GoMAC part III). Modulations around the Circle of Fifths and the Golden Rectangle are...key.
Based on "The Geometry of Music, Art and Structure...linking science, art and esthetics" white paper, "Geometry of Music And Color, I" new media net.art project, and the BBS Matrix. Original soundtracks (A and B).
Part III focuses on a visual journey of how the color spectrum...and specifically the complementary colors of each hue, is based on the ISL. A visual walkthrough and summary of key concepts relating the ISLderived Brooks (Base) Square, BBS, Matrix with a geometrically derived division of both the musical monochord...resulting in the chromatic and diatonic major scale (part II)...and the color spectrum, in which color complements inside and outside of the 400750 nm visible spectrum dictate the colors that we perceive.
Based on "The Geometry of Music, Art and Structure...linking science, art and esthetics" white paper, "Geometry of Music And Color, I" new media net.art project, and the BBS Matrix. Original soundtracks (C and D).Date on line: 052512 [ TOP ]

 Page 15: Brooks (Base) Square (BS): The Architecture of SpaceTime (TAOST) and The Conspicuous Absence of Primes (TCAOP) the complete work (Rules 1177)
The table of numbers that, while excluding the *prime numbers ... and only the *prime numbers ... reveals the number patterns that are primary ... primordial ... to the structure of the Universe.
This primordial matrix defines the Inverse Square Law (ISL) that informs gravity, light (electromagnetism) and atomic structure ... the underpinnings of that which we see as the observable Universe.
It is the architecture of spacetime itself ... that which ultimately informs ... that is, reveals its disposition that in turn presents the physical laws, and their expression, of light, gravity and the structure of matter ... that fundamentally defines the true structure of the Universe ... seen and unseen ... encompassing all matter (light and dark) and all energy (light and dark).
The ISL is a key relationship describing the interplay of space and time.
It is postulated here that the true underpinnings of spacetime itself are inherently defined by numerical relationships ... numbers ... the numbers of Brooks (Base) Square (BS). Date online (Intro) : 123109; Date on line (full version): 050910. Appendix added: 060110. Appendix updated (Rules 178181) 081010. Appendix B added 11711. Appendix B updated (Rules 182200) 010812. [ TOP ]

 Page 16: Brooks (Base) Square interactive (BBSi) Matrix: Part I: BASICS
This "BASIC" introduction to the matrix grid and the simplest, most fundamental pattern of the numbers that are formed from the main, Prime Diagonal are presented in a step by step fashion in five different interactive, multimedia formats. The Prime Diagonal is none other than square of the axial numbers (1,2,3,...) forming the core of the Inverse Square Law sequence (1,4, 9,...).
All the numbers...and their many exquisite patterns...are derived directly from the Prime Diagonal numbers. This is Part I. Part II will follow and will present some of the more interesting patterns.
The Inverse Square Law defines how a force or influence (like energy or light) falls off in strength as the distance is increased. It is one of the fundamental relationship laws of the universe effecting everything from gravity to electromagnetism. It can not but help to also be part of the spacetime fabric. It's part of "The Architecture Of SpaceTime" (TAOST). Date online (Intro): 110511. Appendix B added: 110711. Hands0n Interactive BBSISL grids added 101413. Six more instructional MOVIES added 121813. [ TOP ]

 Page 17: The Architecture Of SpaceTime (TAOST) as defined by the Brooks (Base) Square Matrix and the Inverse Square Law (ISL)
"The single most important concept to take from here is that not only is ST not rigid and static...indeed it is more like a fluid made of light, each quanta of which flashes on and off...pulsepropagates into and out of existence...at a frequency (and wavelength) according to its energy...and this fluid light ST unit resonates off a wave of energy at the peak of each pulseflash...that wave, like a pond ripple, fans out to infinity (or the edge of the Universe...which ever comes first) where it is absorbed back into the subPlanck event horizon (Singularity) to fuel the next round...communicating...passing on information...as to the location source and energy density...the amount and degree of its presence...its curvature...to every other ST unit throughout the Universe(s), BUT that the ST formation and curvature information propagation is informed, dictated and quantified by the matrix of BBS and the ISL...the bean counters par excellence!" Date online: 121011. [ TOP ]

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 Page 19: AFPOP: A Fresh Piece of Pi(e)...and the √2, too...Fractal  Fractal  Fractal
"Circles and squares...as well as other geometric forms...have always been interrelated. Inherent in their geometry is the relationship of their linear elements to their areas...forming the basis of the Inverse Square Law (ISL), as formally presented in the Brooks Base Square (BBS) Matrix. At their core, these geometric forms are based on three irrational, infinitybased numbers of proportion...π, √2 and ϕ...pi, the square root of two, and the golden mean. These core proportionbased relationships suggest that they may be acting as fractals in the formation of The Architecture Of SpaceTime (TAOST). Fractals of infinity.!" Date online: 090112. [ TOP ]

 Pages 2027: LightspeedST: 0. The Ultimate Question
The Ultimate Question Intro to this 5+part series examining the fundamental nature of SpaceTime (ST) and the role the constant "the speed of light" plays. Is there a connection? Date online: 032313. [ TOP ]

 Page 20: LightspeedST: 1. 20 Questions
Part 1 of the 5part series. Twenty plus question examining the fundamental nature of SpaceTime (ST) and the role the constant "the speed of light" plays. Is there a connection? Date online: 121312. [ TOP ]

 Page 21: LightspeedST: 2. 20 Answers
Part 2 of the 5part series. Twenty plus answers to the 20+ questions, framed in bitesized chunks...the better to swallow and digest these tender new morsels. Lightspeed=ST suggests a new view of our Universe(s). Date online: 121312. [ TOP ]

 Page 22: LightspeedST: 3. The Mechanics of SpaceTime (ST)
Part 3 of the 5part series. Actually, this "Portrait of SpaceTime" was written first. (Actually...really...this is all based on L.U.F.E., the Layman's Unified Field Expose´ from 1985. See Links.) The goal: in the most concise and succinct manner...and always with an eye on simplification...lay out the case for tying together the quantum nature of light with the gravitational dependence of spacetime. LightspeedST Parts 1,2,4,and 5 quickly followed as the antidote. Date online: 121312. [ TOP ]

 Page 23: LightspeedST: 4. The Movie Script
Part 4 of the 5part series. All text and no play doesn't mean anything more than the notion that every one's individua imagination will play out differently...because that is a good thing! And yet, there is something especially apropos about seeing the original intent...the roadmap to infinity. Don't miss this one! Date text online: 121312. Movies: 022713. [ TOP ]

 Page 24: LightspeedST: 5. The Movie
Part 5 of the 5part series. As a visual journey, "LightspeedST: 5. The Movie" will quickly swoosh you...through moving images and text...up and over, down and through, the highlights of LightspeedST...leapfrogging @ the speed of light. Date text online: 121312. Movies: 022713. [ TOP ]

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 Page 26: LightspeedST: 7. Quark Portraits
A quick reference to the I and Lquarks and the composite particles that they make. Includes reference charts, a quick look at L.U.F.E. and The LUFE Matrix, and a whole section on Spinor Rotation. Date online: 012313. [ TOP ]

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 Page 28: LightspeedST: The eBook
This interactive book aims to be useful and valuable to both those who know little about the grand description of our physical world and those who know a lot.
For the new student, let this guide be like a story — really, more like a storyboard — for making a movie about the physical laws of Nature.
You start with the ultimate question: Why is the speed of light constant?
Wow! That's a big question! You break it down to a series of 20 or so smaller questions and answers. How might these go together?
With that previewoverview in mind, you now tell your story with an imaginative mix of words and hundreds of images, with some 50 short videos, all organized around The Players, The Rules, The Subplot, The Plot, and The Conclusion. This is followed up with The Movie (review summary), a substantial Appendix, Links and References, and interactive sketching and note taking widgets embedded throughout.
Without memorizing a single fact, you can experience and gain insight into the workings of Nature.
For the seasoned veteran student immersed in the Standard Model of Quantum Mechanics — matter, fields and exchange quanta; gluons, photons, gravitons, Higgs Bosons, W⁺W⁻Z°; fermions, bosons, leptons, baryons, hadrons; charge, mass and spin vectors; the strong, residual nuclear, weak, electromagnetic and gravitational interactions; the Planck Event Horizon, spinor rotation and much more — and searching to reconcile with General Relativity, this book is exactly what you want to bring fresh insights to that Ultimate Question: Why is the speed of light constant?
This question is so important because it goes directly to the heart of the matter — all matter, all energy — including Dark Matter and Dark Energy.
A unified and harmonious vision of the cosmos awaits you. Date online: 060513. [ TOP ]

 Page 29: MathspeedST: The eBook
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) SquareInverse Square Law, or BBSISL Matrix for short, under the title MathspeedST. It includes hundreds of images, a number of videos and lots of testyourself, interactive widgets in presenting the derivation, layout, rules, patterns and ISLderived mathematical basis of the matrix. Ages 810 and up.
A new paradigm in the visualization of basic mathematics is presented!. Date online: 112613. [ TOP ]

 Page 30: MathspeedST: The Ubiquitous Information of Numbers (or, How Fractals Exceed the Speed Limit of Light)
The General Ubiquity Rule (GUR) in effect, asserts that if you have a fixed numerical pattern that is ubiquitously present across all space and time (spacetime, ST), that information is present across the same, passing any need to actually travel or transports said information at or above the speed of light.
The Ubiquitous Information of Numbers new media net.art project presents a summary of this idea in a somewhat playful form. The gist of the argument is as stated in the GUR above. A more local example of a checkered tablecloth is used to demonstrate the point. No matter how folded (or not), of whatever size, once you know the pattern of the cloth in any one grid square you know the the pattern of grids at any other spot on the cloth. Substituting the BBSISL (Brooks Base SquareInverse Square Law) Matrix, with its fixed numerical pattern that expands across any and all sizes, spaces, and times of distribution, gives us exactly that: The Ubiquitous Information of Numbers.
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) SquareInverse Square Law, or BBSISL Matrix for short, under the title MathspeedST. It includes hundreds of images, a number of videos and lots of testyourself, interactive widgets in presenting the derivation, layout, rules, patterns and ISLderived mathematical basis of the matrix. Ages 810 and up.
A new paradigm in the visualization of basic mathematics is presented! Date online: 031314. [ TOP ]

 Page 31: TPISC: The Pythagorean — Inverse Square Connection ~~~A MathspeedST Supplement~~~
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 (BBSISL Matrix) is an infinitely expandable grid that reveals ALL Pythagorean Triples — both Primitive Triples (PPT) and their nonPrimitive multiples (nPTT). An extremely simple geometric AREA proof of the Pythagorean Theorem — c^{2}=a^{2}+b^{2} — is built into the BBSISL 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 BBSISL Matrix grid is composed of Pythagorean Triples crisscrossing over much of the entire grid. An intimate interconnection 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 BBSISL 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) SquareInverse Square Law, or BBSISL Matrix for short, under the title MathspeedST. It includes hundreds of images, a number of videos and lots of testyourself, interactive widgets in presenting the derivation, layout, rules, patterns and ISLderived mathematical basis of the matrix. Ages 810 and up.
A new paradigm in the visualization of basic mathematics is presented! Date online: 071714. NOTE: See INTRO below (page 32). [ TOP ]

[ TOP ]

 Page 33: TPISC I: Basics ~The ebook
The Pythagorean — Inverse Square Connection ~~~A MathspeedST Supplement~~~
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 (BBSISL) Matrix) is an infinitely expandable grid that reveals ALL Pythagorean Triples — both Primitive Triples (PPT) and their nonPrimitive multiples (nPTT). An extremely simple geometric AREA proof of the Pythagorean Theorem — c^{2}=a^{2}+b^{2} — is built into the BBSISL 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 BBSISL Matrix grid is composed of Pythagorean Triples crisscrossing over much of the entire grid. An intimate interconnection 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 BBSISL 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 BBSISL 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 testyourself, interactive widgets in presenting the derivation, layout, proofs, and patterns connecting the two geometries. Ages 810 and up.
A new paradigm in the visualization of basic mathematics is presented! Date online: 092915. NOTE: See INTRO above (page 32). [ TOP ]

 Page 34: TPISC II: Advanced ~The ebook
The Pythagorean — Inverse Square Connection ~~~A MathspeedST Supplement~~~
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 BBSISL Matrix and the Pythagorean Triples are related. In fact, the two are so interconnected 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 (BBSISL) Matrix is an infinitely expandable grid that reveals ALL Pythagorean Triples — both Primitive Triples (PPT) and their nonPrimitive multiples (nPTT). An extremely simple geometric AREA proof of the Pythagorean Theorem — c^{2}=a^{2}+b^{2} — is built into the BBSISL 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 BBSISL Matrix grid is composed of Pythagorean Triples crisscrossing over much of the entire grid. An intimate interconnection 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 BBSISL 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 BBSISL 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 testyourself, interactive widgets in presenting the derivation, layout, proofs, and extended fractallike patterns connecting the two geometries. Ages 810 and up.
A new paradigm in the visualization of basic mathematics is presented! Date online: 092915. NOTE: See INTRO above (page 32).

 Page 35: TPISC I & II: INTRODUCTION
TPISC (The Pythagorean — Inverse Square Connection) on the BBSISL Matrix ~~~A NEW Simplified Introduction~~~
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 BBSISL Matrix and the Pythagorean Triples are related. In fact, the two are so interconnected 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!
Date online: 101815. NOTE: See the more advanced INTRO above (page 32) also. [ TOP ]

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 Page 37: Time and Structure: The role of the Inverse Square Law.
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/rafaelsorkin):
"order + number = geometry."
And why is that?
What does it mean to say that the progression of interconnected 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 interconnectedness? 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 BBSISL 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!
Date online: 08052016. [ TOP ]

 Page 38: Simple Visual Guide to making a BBSISL Matrix!.
Simplification
Take 5 or 10 minutes (more or less) and learn to make the 10x10 — or any other size —BBSISL Matrix!.
Once you do, you will open the door, get over any fears, and fly right on through to some amazing mathematics!
The BBSISL Matrix! is based on simple, whole integer numbers: 1 2 3 ,... and their squares: 1 4 9,....
Simple AdditionSubtraction & MultiplicationDivision will define the Geometry.
The BBSISL Matrix! combines the simplest of algebraic and geometric concepts into easy visualizations!
The BBSISL 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 BBSISL Matrix!
Let's get started!
Date online: 09202016. See above Pages 1519, 2936 for more. [ TOP ]

 Page 39: TPISC III: Clarity—Simplified: ToPPT ~Tree of Primitive Pythagorean Triples.
Simplification
Take the BBSISL Matrix (BIM)for short,
and reveal the Pythagorean Triples (PTs) embedded ubiquitously throughout.
The BBSISL Matrix not only reveals every possible PT — both Primitive PTs (PPTs) and their nonPrimitive 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 345 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 BBSISL 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!
Date online: 07222017. Updated: 11112017. More to come. [ TOP ]

 Page 40: BIMtreeToPPT_AREAS: 12 Easy Pieces.
TPISC IIIV
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 interconnected by AREAS.
Date online: 11042017. [ TOP ]

 Page 41: TPISC III: Clarity & Simplification: Tree of Primitive Pythagorean Triples (ToPPT) (~ A MathspeedST Supplement~) eBook.
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 fractalnodes 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 BBSISL 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 nonisosceles 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 — BBSISL Matrix — is the geometry defining SpaceTime!”
MathspeedST: Leapfrogging Lightspeed FASTER than The Speed of Light
The MathspeedST Supplements:
 TPISC I: Basics
 TPISC II: Advanced
 TPISC III: Clarity & Simplification (ToPPT)
 TPISC IV: Details
 TPISC V: Exponentials (coming)
This is a multitouch, interactive ebook made with iBooks Author and published in the Apple iBooks Store.
Excerpt From: Reginald Brooks. “TPISC_IIIClarity....” iBooks.
Date published: 11092017. [ TOP ]

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 Page 45: TPISC IV: Details: A closer look on #2 below: A special connection between the BBSISL Matrix + Pythagorean Triples + axial DNA decagon (double pentagonal) geometry. Here is the supplemental page to Page 44 above: "Phi (ϕ), Fibonacci, Pentagon Connections to the PTs (~ A MathspeedST Supplement~) .".
NEW! Connections between:
 1. BIM (BBSISL Matrix) + PTs (Pythagorean Triples);
 2. PT + Pentagon & Decagon (double pentagon of DNA);
 3. BIM + PT + DNA;
 4. BIM + PT + DNA + Zika Virus (cryoem imagery).
Here we continue with TPISC IV: The Pythagorean  Inverse Square Connection: Details. A special connection between the BBSISL Matrix + Pythagorean Triples + axial DNA decagon (double pentagonal) geometry was the topic of the previous white paper (Page 44). The bottom of that page linked to the resource links and comments page on "Phi (ϕ), Fibonacci, Pentagon Connections to the PTs" — a page so full of information that it has been referenced here as its own white paper entry (Page 45).
Referenced here is the long history relating phi, the golden ratio, the Fibonacci numbers, Kepler Triangle, Pythagorean Theorem, and the pentagonal geometry of the dodecahedron and icosahedron of the Platonic Solids. Because the Pythagorean Triples have a strong tie in with the Fibonacci numbers, and thus phi and the pentagons, they, too, are part of this interconnection between the transcendental irrational numbers of the universal constants and the natural, whole integer numbers of the Pythagorean Triples and the Inverse Square Law. Much to see here!
Date published: 04102018. Updated: 08212018 [ TOP ]

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 Page 47: TPISC IV: Details: DSEQEC (DoubleSlit Experiment – Quantum Entanglement Conjecture (Part I of II) TPISC III: Clarity & Simplification and TPISC IV: Details: Tree of Primitive Pythagorean Triples (ToPPT) (~ A MathspeedST Supplement~) Interactive Graphics on BIM & BIMtree AREAS..
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 pulsepropagation 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 Lawbased 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 fractallike 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 pulsepropagation, and we we now have a fractalbased, holographic quantum universe(s)!
Such profound harmony of form naturally begs the question: to what effect? Several conjectures come to mind:  Would not the fractalnodes 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 nonSquare Rectangles holding these PTs?
 Would not these same fractalnodes of the PTs be internal, integral and identifying structural parameters (Quantum State Numbers) of any waveparticle ST unit formed?
 Would not the immensely interconnecting linkage of the PTs, between the PTs and between the PTs and the BIM itself, provide the very basis of the waveparticle duality and entanglement phenomenon we see in the *DoubleSlit 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:  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 builtin velocity of light, c;
 ST itself, and thus its ST progeny, must be conserved;
 ST itself, and thus it ST progeny, must be fully accountable for all intricacies of the DoubleSlit Experiment.
SEE Page 49 (below) for Part II of II CaCoST  DSEQEC. SEE Page 50 (below) for spreadsheets: BIM: How to Make.
Date published: 08102018. Updated: 12122018, links 05222019 [ TOP ]

 Page 48: TPISC IV: Details: PPT Profiles; PPT and PRIMES DistributionActive Rows_24; and, DSEQEC (DoubleSlit Experiment – Quantum Entanglement Conjecture TPISC III: Clarity & Simplification and TPISC IV: Details: Tree of Primitive Pythagorean Triples (ToPPT) (~ A MathspeedST Supplement~) Interactive Graphics on BIM & BIMtree AREAS..
Here is the working draft of the TPISC IV: Details: PPT Profiles; PPT and PRIMES DistributionActive Rows_24; and, DSEQEC (DoubleSlit Experiment – Quantum Entanglement Conjecture ebook. This includes the above Pages 4248. 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 DoubleSlit 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 overfocusing 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~). This is an interactive graphics page on the BIM and BIMtree_AREAS — bridging TPISC I, II, and III with TPISC IV: Details. Each canvas may be interactively deconstructed to see how the geometric components are related. Simply refresh the page to start over. A great learning and teaching resource.
Please keep this in mind: The quantum universe begs the question: what is the quantum? Perhaps we can say the basic pulsepropagation 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 Lawbased 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 fractallike 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 pulsepropagation, and we we now have a fractalbased, holographic quantum universe(s)!
SEE Page 50 (below) for spreadsheets: BIM: How to Make.
SEE Page 51 (below) for Big OUTLINE: Big OUTLINE.
Date published: 08212018. Updated: ongoing (last: 03162019, 04132019, 04212019, 04302019, 05222019, 10052019, 11132019, 12072019, 03152021) [ TOP ]

 Page 49: TPISC IV: Details: CaCoST (Creation and Conservation of SpaceTime) — DSEQEC (DoubleSlit Experiment – Quantum Entanglement Conjecture (Part I of II) TPISC III: Clarity & Simplification and TPISC IV: Details: Tree of Primitive Pythagorean Triples (ToPPT) (~ A MathspeedST Supplement~) Interactive Graphics on BIM & BIMtree AREAS..
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 pulsepropagation 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 Lawbased 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 fractallike 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 pulsepropagation, and we we now have a fractalbased, 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:  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 builtin velocity of light, c;
 ST itself, and thus its ST progeny, must be conserved;
 ST itself, and thus it ST progeny, must be fully accountable for all intricacies of the DoubleSlit Experiment.
In Part II, that very Creation and Conservation of SpaceTime will form the context in which the DSEQEC comes to life!
SEE Page 47 (above) for Part I of II CaCoST  DSEQEC. SEE Page 50 (below) for spreadsheets: BIM: How to Make.
Date published: 10052018. Updated: 12122018, links 05222019 [ TOP ]

 Page 50: TPISC IV: Details: Make any size BIM easily on your spreadsheet app.
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 spreadsheetBIM 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: 12112018. Updated: 12282018, 05222019 [ TOP ]

 Page 51: TPISC IV: Details: Big OUTLINE.
Big OUTLINE for TPISC IV: Details
A simple, linked outline for the contents of TPISC IV: Details (page 48 above) of PPT Profiles; PPTs and PRIMES DistributionActive Rows_24; and, DSEQEC (DoubleSlit Experiment – Quantum Entanglement Conjecture)
Date published: 02182019. Updated:


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 Page 54: TPISC Artworks.
Artworks specifically in reference to MathspeedST  TPISC IIV series:
 DRAWING & PRINTS:
 PAINTING:
Date published: 04132019. Updated: 04212019, 04302019, 10052019

 Page 55: PRIMES vs NOPRIMES: Two definitive methods for revealing all PRIME numbers) (~ A MathspeedST Supplement~) eBook.
eBook
Statement
Gather the NOPRIMES
6yx+y
Remains are the PRIMES
How can a deceptively simple matrix grid square give us:
1. Inverse Square Law (ISL);
2. Pythagorean Triples (PTs);
3. Primes (P) vs. NoPrimes (NP)?
The BIM (BBSISL Matrix) does that!
What started as the invention/discovery of the ISL matrix — a simple grid of natural Whole Integer Numbers (WIN) — has led to an unforeseen connection between the ISL — PPTs — PRIMES.
The BIM is an infinitely expanding matrix where every cell within the grid is uniquely occupied by a WIN that is itself simply the difference between its horizontal and vertical intersection values of the main, Prime Diagonal (PD).
The PD mirror divides the grid into symmetrical halves. The PD is key as it consists solely of the squares of the Axis numbers (1,2,3,...) as A^{2} = PD.
In fact, drop vertically down from any PD squared number until you intersect with its squared complement and you will have landed on a Primitive Pythagorean Triple (PPT) row, with a^{2} + b^{2} culminating with c^{2} where that row now intercepts horizontally with the PD, giving a^{2} + b^{2} = c^{2}.
Repeating this will identify all the PPTs.
But wait! Now divide every Inner Grid (IG) cell value by 24 and you will generate a crisscrossing diamond grid overlay pattern with almost magical properties!
The BIM÷24 grid will break out the overall matrix into two (2) types of ODD Axis rows:
• those ÷3 that are never PPT, never PRIMES = NonActive Rows (NONAR);
• those NOT ÷3 that PPTs and PRIMES are
exclusively found on = Active Row Sets (ARS) or Active Rows (AR) for short. NOTE: a necessary, but not sufficient, condition as some AR have neither.
The reason the ARS is plural is because there are two ODD Axis rows — AR — in each ARS.
These two (2) ODD Axis AR alternate with an ODD Axis NONAR giving the (NONAR)—AR—AR—(NONAR)—AR—AR—(NONAR)... pattern. Basically for every ÷3 row, one has two NOT ÷3 rows.
This becomes important as the ÷3/NOT ÷3 pattern is key to identifying the Number Pattern Sequence (NPS) of ALL the NOPRIMES (NP) and that NPS ultimately defines the elusive pattern of the PRIMES as: ALL ODD WIN (≧3)  NP = P.
This has culminated — at least at this point in this exciting mathematical journey — in two (2) simple methods for generating ALL the NP:
1. Algebraic: *NP = 6yx ± y let y = ODD WIN, x = 1,2,3,... (*one must add the exponentials of 3, 3x, if one does not first eliminate all ÷3 WIN).
2. Algebraic Geometry: where it all began on the BIM by subtracting (eliminating) ALL the **NP that are shapelocated directly on the BIM, one can reveal the remaining PRIMES (**here, one must include the ODD Axis^{2} values).
Both methods nicely dovetail directly on the BIM.
*NP = 6yx ± y is the simple, algebraic equation that is derived from and summarizes the visualization approach we see directly on the BIM. This dovetailing of the pure algebra approach with the pure geometry of the BIM gives us the algebraic geometry method. Visual thinkers like to "see" the algebra and algebraic geometry at work — play!
The regular BIM by itself contains all the same information for either method. The NPS of the algebraic and algebraic geometry methods pops out more readily when we do a minor alteration to the BIM — squaring the Axis numbers. Since we are only working with the ODD numbers, this means simply squaring the ODD Axis numbers. And to completely correlate the *NP = 6yx ± y algebraic method to the BIM, we need to manually add the exponentials of 3 (3x) to the listings (including those in spreadsheet table format) of the NP numbers generated by the equation. That's it!
Looking at the BIM's lower symmetry — after eliminating all the EVEN numbers and squaring the ODD Axis numbers — a clear NPS of ALL the NP is revealed. A Lshapeddoublewidediagonalpath pattern that corresponds systematically, stepbystep with algebraic solutions of *NP = 6yx ± y and the pre geometryshape visualization of the algebraic geometry method. Both methods give the same identical NP results because each is simply the same logical process only coming at it from two different approaches.
Now the really interesting part of all of this is all the NPS revealed in the exposure of ALL of the NP is itself simply the converse — inverse — of ALL the PRIMES.
One — take your pick — is simply the negative space of the other. If we call the PRIMES the positive space, then their NPS is revealed by the negative space NOPRIMES!
When visualizing the PRIMES vs. NOPRIMES on the BIM, it is much easier to "see" the NP as the positive space with all the NPS and the PRIMES as the remaining negative space — a space completely defined by that of its opposite (inverse) NOPRIMES space as: NP + P = ALL ODD numbers (≧3).
Yes, a deceptively simple matrix grid square (BIM) can give us:
1. Inverse Square Law (ISL);
2. Pythagorean Triples (PTs);
3. Primes (P) vs. NoPrimes (NP)!
~~~
This is an additional supplement to the MathspeedST ebook. Other titles include The PythagoreanInverse Square Connection (TPISC):
TPISC I: Basics
TPISC II: Advanced
TPISC III: Clarity and the Tree of Primitive Pythagorean Triples
TPISC IV: Details, DSEQEC and PRIMES.
This is a multitouch, interactive ebook made with iBooks Author and published in the Apple iBooks Store.
Excerpt From: Reginald Brooks. “PRIMES vs NOPRIMES....” iBook.
Date published: 05202019. [ TOP ]

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 Page 57: Periodic Table of Primes (PTOP): The Goldbach Conjecture.
Details of the PTOP and the patterns proving the Goldbach Conjecture (strong form):
 Periodic Table of Primes (PTOP)
 Goldbach Conjecture (Euler's "strong" form)
Date published: 10182019. Updated: 11132019, 1272019 [ TOP ]

 Page 58: PTOP (Periodic Table Of PRIMES) and the Goldbach Conjecture A NEW Pattern reveals how every EVEN is made of 2 PRIMES
~A MathspeedST Supplement~ eBook.
eBook
Statement
Gather the EVENS
Steps are P2A
A=half E
Here’s the thing. Amongst a myriad of other connections, there exists an intimate connection between three number systems on the BBSISL Matrix (BIM):  The Inverse Square Law (ISL) as laid out in the BIM;
 PTs — and most especially PPTs — as laid out on the BIM;
 The PRIME numbers — PRIMES — as laid out on the BIM.
The BIM is the FIXED GRID numerical array of the ISL. PTs are the Pythagorean Triples and PPTs are the Primitive Pythagorean Triples. They have been extensively covered in the TPISC (The PythagoreanInverse Square Connection) series as: TPISC I Basics, TPISC II Advanced, TPISC III Clarity, TPISC IV Details. (See links at bottom.) The PRIMES vs NOPRIMES (2019) was covered earlier. The original MathspeedST (200913) work, Brooks (Base) Square** **(200911), that started this journey, was divided in to two sections: I. TAOST (The Architecture Of SpaceTime); II. TCAOP (The Conspicuous Absence Of PRIMES). You see, other than the natural whole numbers that form the BIM Axis’ and the standard 1st Parallel Diagonal (containing ALL the ODDs (≥3), there are NO PRIMES on the BIM. This is the basis of the PRIMES vs NOPRIMES work. Yet, convert that same 1st Parallel Diagonal to ALL EVENS (≥4, by adding 1 to each former ODD), and now the BIM reveals the stealthily hidden PRIMES relationship in forming symmetrical pairs of PRIMES that are ALWAYS equal distance (STEPS) from ANY given EVEN # divided by two. All this occurs on the BIM Axis. The apex of the 90° Rightangled, isosceles triangle so formed lies on a straight line path from the EVEN/2 to the given EVEN # located on that converted 1st Parallel Diagonal. This relationship is geometrically true and easily seen on the BIM. Extracting those PRIMEPair sets (PPsets) for each given EVEN forms the basis for the PTOP — Periodic Table Of PRIMES. While “hidden” on the BIM, it clearly forms a definitive pattern on the PTOP: for EVERY 3+PRIME PPset that forms the 2nd column on the PTOP — acting like a bifurcation point — a “Trail” of PPsets forms a zigzagging diagonal pointing down and to the right. These are ALWAYS — much like a fractal — added with the SAME PRIME Sequence (3,5,7,11,13,17,19,23,29,31,37,...). When you read a given horizontal line from left to right across the PTOP, you see that each given PPset that was contributed by a PPset Trail, adds up — composes — its respective EVEN. The sum (∑) number of PPsets that form its EVEN is totaled in the last column. The fractallike addition of one new, additional PRIME Sequence PPset to each subsequent “Trail” formed results in the overlapping trails growing at a rate that far, far exceeds the growth and incidence of the PRIME Gaps. This ensures that there will always be at least one PPset that will be present to compose ANY EVEN. The Goldbach Conjecture has been satisfied and a new PRIME pattern has been found. TPISC Media Resource Center https://vimeo.com/372243091 Yes, a deceptively simple matrix grid square (BIM) can give us: 1. Inverse Square Law (ISL); 2. Pythagorean Triples (PTs); 3. PRIMES (P) Goldbach Conjecture! ~~~ This is an additional supplement to the MathspeedST ebook. Other titles include The PythagoreanInverse Square Connection (TPISC): TPISC I: Basics TPISC II: Advanced TPISC III: Clarity and the Tree of Primitive Pythagorean Triples TPISC IV: Details, DSEQEC and PRIMES. PRIMES vs NOPRIMES
Date published: 12112019. [ TOP ]

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 Page 60: Make the PTOP with Fractals
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 atomicbase 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! PSFractals.
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 PSFractal method of making the PTOP showcases the repetitive, reiterate, selfsimilar 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.
Date published: 01182020. Updated: 01202020 [ TOP ]

 Page 61: PRIMES 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 = BBSISL 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 (P_{1}, P_{2}), can be found and individually profiled DIRECTLY on the BIM as seen in BIM: Part II.
Plotting all the Lower Diagonal P_{2} PRIMES on a table and then replotting those results back onto the BIM opened up a new vista. By substituting the AXIS values for each of those P_{2 }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 IIII, 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 PSFractal 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 IIV. 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, mindblowing 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.
Date published: 04302020. Updated: 05072020 [ TOP ]

 Page 62: PRIMES on the BIM ebook
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 = BBSISL 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 (P_{1}, P_{2}), can be found and individually profiled DIRECTLY on the BIM as seen in BIM: Part II.
Plotting all the Lower Diagonal P_{2} PRIMES on a table and then replotting those results back onto the BIM opened up a new vista. By substituting the AXIS values for each of those P_{2 }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 IIII, 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 PSFractal 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 IIV. 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, mindblowing 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.
Date published: 05052020. Updated: 05102020 [ TOP ]

 Page 63: DSEQEC (DoubleSlit Experiment  Quantum Entanglement Conjecture) and CaCoST (Creation and Construction of SpaceTime) ebook
DSEQEC (DoubleSlit Experiment  Quantum Entanglement Conjecture) and CaCoST (Creation and Construction of SpaceTime) eBook ebook
ABSTRACT
DSEQEC & CaCoST
.
DSEQEC: The DoubleSlit 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 DoubleSlit 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 waveparticle ST units exhibit unusual and nonintuitive 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 MatterAntiMatter alternating "current" of pp ST units — here it is the photon, but it holds for all ST units — are far and wide. The troubling and poorly understood DoubleSlit Experiments were trying to tell us something all along. Our onesided 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, twosided alternating "reality."
Where did all the antimatter go? No where! Its "hiding" right where it has been since the inflationary, BigBang!
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 pp, 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 pp into and out of the OVMatter and AVAntiMatter 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 (BBSISL 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 (BBSISL 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 DoubleSlit 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, interconnected and harmonious “system” by means of symmetry, repetition, reiteration 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 bipolar.
Our “Universe” and all that “matter” is but half of the full Continuum, the “other” half being all that “antimatter,” reverse charge and spin “Universe” that coexists 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!
Date published: 05252020. [ TOP ]

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 Page 65: Math—Art—Physics (MAP) Index
Math—Art—Physics (MAP)
Interactive Slideshow Links of White Papers and eBooks
Date published: 06132020. Updated: 07102020 [ TOP ]

 Page 66: BIMGoldbach Conjecture, I
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!
Date published: 12292020. Updated: 02072021 [ TOP ]

 Page 67: BIMGoldbach Conjecture, II
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!
Date published: 02072021. Updated: [ TOP ]

 Page 68: BIMGoldbach Conjecture, III
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.
Date published: 03152021. Updated: [ TOP ]

 Page 69: Mersenne Prime Squares
Mersenne Prime Squares: The Geometry of the Mersenne PrimePerfect 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 BIMGoldbach_Conjecture series described above (Pages 6668).
Every Mersenne Prime has a 1to1 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: 04052021. Updated: 06202021 [ TOP ]

 Page 70: Mersenne Prime Squares
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:
 Basics
 Intermediate
 Advanced
Every Mersenne Prime has a 1to1 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: 06102021. Updated: 06202021 [ TOP ]

 Page 71: Mersenne Prime Squares: Advanced
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, stepbystep analysis of that visualization will be presented. Basically, we will be taking the invisible EuclidEuler Theorem and transforming it into the highly visible geometric components (lines, areas and shapes) that make up the Mersenne Prime Square.
 MPS divides into:
 PN + OC subdivide into:
 PNS + CR + CR + OCS
Every Mersenne Prime has a 1to1 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: 06152021. Updated: 06202021 [ TOP ]

 Page 72: The MPS Project (Mersenne Prime Squares Project)
The MPS Project
In July 116, 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 EuclidEuler 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 barrelvault space. The three overlapping areas (below) will be stacked directly on top of each other.
 MPS divides into:
 PN + OC subdivide into:
 PNS + CR + CR + OCS
Every Mersenne Prime has a 1to1 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.
Date published: 06152021. Updated: 07102021 [ TOP ]

 Page 73: The Butterfly Fractal 1: the basis of the Mersenne Primes  Perfect Numbers and the Mersenne Prime Squares (MPS)
The Butterfly Fractal 1
Every Mersenne Prime has a 1to1 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 EuclidEuler 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 xx^{2}yy^{2}zz^{2}xyxzxy.
 MPS divides into:
 PN + OC subdivide into:
 PNS + CR + CR + OCS
As the Mersenne Prime (Mp = z) is based on the exponential power of 2  1, it follows that xx^{2}yy^{2}zz^{2}xyxzxy can be shown to be simple additive combinations  better known as the running sums (∑) of 2^{n} 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 371531.... 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 124816... repeats in a redundant, reiterative, selfsimilar 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.
Date published: 06062022. Updated: [ TOP ]

 Page 74: 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.
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: 07282022. Updated: [ TOP ]


 


