Threesology Research Journal: Static versus Dynamic Equations
"Infinitesimal versus Accordian Calculus"
(Static -vrs- Dynamic)

~ The Study of Threes ~

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Assisting Mathematic's to breathe more deeply by way of Accordian Calculus

Very few take an extended look at the world-view expressed of those labeled a child, a pet, an indentured servant or slave. Much less a homeless person, an ex-convict, drug or alcohol abuser or someone wearing front and back placards inscribed with a Doomsday scenario. If anyone of them asked for an indulgence to view an idea concerning one or more patterns of behavior they have witnessed and derived some meaning from, each one of the listed examples... themselves customarily viewed in terms of exhibiting a given behavior associated with some level of intelligence typically assigned a value less than those who automatically discredit their appeal; most people would walk away. If no one listens to their tale, their options in the current social environment leaves them with the choice of making a youttube and/or creating a webpage such as this one. Yet, even these avenues may be overlooked far beyond the lifetime of the individual. While a more radical approach would be to hold numerous people hostage in a given situation so as to elicit news-media coverage, there is no guarantee that one's message will be heard or even listened to because of the anti-social effort used in attempting to establish a pro-social idea. Such a catch-22 option is not very appealing.

If however the person or persons advancing an idea do approximate some agreed upon means of expression to be sustained in the public domain, they might being with what amounts to be a plea for the consideration of others to take an extended look at some basic patterns of behavior which reveal an interesting cognitive kaleidoscope of patterning from which the revelation of a standard cognitive formula can be ascertained and used to provide humanity with a fortuitous path for grasping a reality that is long overdue for being appreciated as a prosperous trail if we would only collectively indulge ourselves a moment's hesitation from routines of thought, and examine the repeating patterns of (metaphorically described) bread crumbs, broken branches, and trampled brush suggesting someone(s) or something(s) have past this way for some reason(s) and those reasons might well be worth investigating if we would only allow ourselves the chance by labeling it some sort of game like a scavenger hunt, a supposed treasure hunt, or even some mystery developed by a caressed display of intriguing "maybes"... unless of course you are afraid of peering too closely to ideas which have been formed and framed as ghostly apparitions you stay away from.

Yet, that's enough of this type of artistically-inclined self-indulgent monologuing. Let's move forward in the primary intent and effort of this present essay:

It is of need at this moment to point out the presence of a philosophical debate concerning the ability of humans to learn language, then counting and then some form of mathematics. Indeed, the standard Nature versus Nurture or Nativist versus social/cultural indoctrination is a long-standing issue. Yet my interest in bringing it up is not to voice an opinion one way or the other at this moment in time. I simply want sot point out that it is a dichotomy, or otherwise noted as a duality, contrast, and whichever word you prefer to use when describing the discursive event. So too must I point out those who prefer some middle-ground variation typically described with the comment that both situations effect behavioral outcomes. Hence, we have a 1 versus 1, unless we illustrate it as 'one or the other', and in some sense arrive at a two-variation idea, or instead choose a third option. Overall what we see being displayed is an elaborate model of a basic enumeration of preferring 1 idea, the 2nd idea, or a composite 3rd view. Like any primitive counting system, philosophical debates routinely migrate to embellishing what appear to be a practice in a simple counting theme of 1- 2- 3. Despite all the vocabulary, all the phrasing and asserted justification of one or another point, the underlying cognitive pattern being displayed is a simply counting scheme sometimes philosophically expressed as the monad- dyad- triad, with the triad sometimes replaced by a term related to "many" such as polyad in philosophy and polynomial in mathematics.

In addition to consciously acknowledging the presence of a duality is to recognize its multi-variant forms such as word-to-number relationships and symbol-to-quantity associations. Some dualities may appear oppositional while others look like complements to one another. But the fact remains we are dealing with "twoness" and not "oneness, threeness or some value of many-ness, heap-ness, several-ness, much-ness, more-ness, a lot-ness, whole bunch-ness", etc.

According to this Wikipedia article on Equation, we find the idea that there are only two types of equation, but I shall insert a short reference in the excerpt which describes three:

In mathematics, an equation is a statement that asserts the equality of two expressions, which are connected by the equals sign "=". The word equation and its cognates in other languages may have subtly different meanings; for example, in French an équation is defined as containing one or more variables, while in English, any equality is an equation.

Solving an equation containing variables consists of determining which values of the variables make the equality true. The variables for which the equation has to be solved are also called unknowns, and the values of the unknowns that satisfy the equality are called solutions of the equation. There are two kinds of equations:

  1. Identity equations An identity is true for all values of the variable.
  2. Conditional equations A conditional equation is only true for particular values of the variables.

Insert: three basic types of equations College Algebra by Lial Hornsby Schneider (I rearranged the definitions to provide continuity in the statements.):

  1. Identity: An equation satisfied by every number that is a meaningful replacement for the variable is called an identity, such as 3(X 1) 3X + 3.
  2. Conditional: An equation satisfied by some numbers but not others is called a conditional, such as 2x = 4.
  3. Contradiction: An equation that has no solution is called a contradiction, such as x = x + 1.

Let's continue with the wikipedia article:

An equation is written as two expressions, connected by an equals sign ("="). The expressions on the two sides of the equals sign are called the "left-hand side" and "right-hand side" of the equation. Very often the right-hand side of an equation is assumed to be zero. Assuming this does not reduce the generality, as this can be realized by subtracting the right-hand side from both sides.

One author says two basic types of equation and another says three types. This "2 versus 3" is an important reference which I will get into later one, because it represents a developmental change in cognitive activity easily recognized when we take up the issue of the development of a number concept in primitive humans; since it portrays a transition that is not always consciously acknowledged but plays an important role in describing perceptual differences. These differences are noted by the dual interpretive representation of numbers referred to as Cardinal and Ordinal. Cardinal referring to the situation when one counts such as 1- 2- 3-..., etc., and Ordinal referring to the time, position and speed by which a particular number or numbers occurs in a given sequence, whether it be a regular from beginning to end Cardinal sequence of the 1- 2- 3-... regimentation, a non-regular Cardinal sequence which reverses the order, or some configured set of numbers such as 1- 3- 5-..., etc..., or 2- 4- 6- etc., or whatever compilation one is involved with for a given moment and task. The entirety of the compiled transition may be that which is viewed as an order and/or some segment there of such as extracting a series of both odd and even numbers juxtaposed like the black and white keys on a piano (with up to three pedals), piano accordion (with bass buttons), or the red/black betting spots on a Roulette wheel (though there also exists a green spot[s] betting option).

Let me now approach the topic of basic types of equations from a different perspective:

Despite the various names (and 2 divisions of pure mathematics) of Mathematical branches, there are... for the present discussion... only three types of Basic mathematical equations as seen from a two-dimensional frame of referencing:

  1. Linear equations. [1 + 2 = 3, or: A2 + B2 = C2, etc...]
  2. Circular equations (not to be confused with circle equations).

For example, we would most likely end up using a linear equation such as:

  • [1 + 1 + 1 + 1 = 4,
  • or while standing, one points in four directions (front, back, left, right), or six directions (front, back, left, right, up, down); presenting us with a four-point series noted as 4, 5, 6, 7] that have physically determined finite distances around which we encapsulate with a spherical projection.

Here are two illustrations of these ideas about the Human Body Axis in the environment, without emphasizing the person as the fifth and seventh points:

Four directions concept Six directions concept

We can take both the four and six directions concepts and encircle the conserved points of reference:

Encircling the tips of the four directions 
concept. Encircling the tips of the six directions 

One problem with our modern enforcement of an inscribed encirclement of the exterior points of the conceptualized directions is that they do not effectively point out the probability of perception which once existed involving the perceptions of the ground and horizon as limits, while the perspective above looking outwards towards the stars (we can imagine) was viewed as being further out. Hence, prior to the ideas of Celestial Spheres and the frequent present day usage of box-like portrayals, we would have had considerations of half-a-sphere, even if these were neither commonly articulated nor illustrated. The chance that many common others... than those depicted by historians... may well have conceptualized a similar image when given a chance to muse upon life; is not something we should overlook. This is similar to the notion that many everyday common people who are not mathematicians may well have conceptualizations which are just as intricate as are the equations being used in pure and applied mathematics.

Half a sphere review 1 Half a sphere review 2.

What the foregoing illustrations were trying to describe in a pictographic way, is that there are no actual circular equations. While we have many "encircling" ideas, all of them are expressed in terms of linear equations or "portraits" as the foregoing images illustrate. Such an idea... as one thinking that a linear equation can actually be a substitute for a "circular equations" is an undefined manifestation of a mathematical embellishment of our reality and an attempted idealization beyond our physical limitation. It is like a primitive human saying "ugh" to reference a concept beyond the value of "one", yet neither they nor anyone else has any grasp of such a realization available to a consciousness that needs a few millennia before such a perspective is tenable. We use linear equations to describe circles. Typically, let us note the usage of quadrilaterals/quaternions as conjectures to circular equations which are undeniably two-dimensional.

Triangular equations. Ancient Egyptians crudely attempted to create this mathematical entity by compiling four triangular-faced objects into a single form called the pyramid. While they could imagine three-dimensions, they thought that by compounding a 1 with 3 they would create what we of today describe as the 3 dimensions plus time. They may not have use the same language nor symbology, but their efforts are starkly clear, despite all the attendant anthropologically defined associations to religiousity and after-life ceremonialisms. In short, like circles, many have tried to use linear Algebraic equations in an attempt to illustrate this phenomenal type of triangle or triad, thus providing a mathematical expression for the 3 dimensions and time. However, because the desired triangle/triad exists in three dimension, linear equations which are functionally appropriate for addressing a 2-dimensional reality fall short of being able to create an accurate portrait of three dimensions. Mathematicians such as Sir William Roland Howell will always search in vain for creating Algebraic triads.

Succinctly put, we have linear equations, but no circular nor triangular ones. We use linear equations as an attempted substitute for the realization of an idea that is as yet poorly formed in the consciousness of humanity, yet humanity, because it straddles a 2-dimensional and 3-dimensional frame of referencing, has perceived an echo, a sublimation, an image, a shadow, a ghost, a spectre, a caricature, a silhouette of these other two types of equations. However, unless we alter the environment in which such conceptualizations are most favorably attuned with, we will be forever lost in a limbo... trying to use compounded embellishments and other mathematical stage-crafts to portray that which we get a whiff of, but can't get past the small taste on the tip of our tongue in order to take a full drink or large bite of such ideas to the point of being able to recreate them as might a chemist or cook.

Let me explain myself. In this Sciencealert article: A Physicist Has Calculated That Life Really Could Exist in a 2D Universe(by David Nield, 27 June, 2019), the physicist James Scargill argues for the possibility that life could indeed exist in two dimensions. So here once again we are confronted by the "2 versus 3" idea in that we are viewing ideas about 2 versus 3 dimensions. I want to suggest that our propensity for using two-patterned based ideas (yin/yang, binary code, dichotomies, dualities, oppositionals, contradictions, etc...) are representative of our human existence in both two and three dimensions. Like an ancient counting scheme in which an early human had reached the pinnacle of counting up to the value of "2" before transitioning into the conceptual range for recognizing the existence of the value 3; our multiple diverse usage of the two and three by way of variously named identities (biads/triads, dualities/trichotomies, yin-yang/yin-unity-yang, binary/ternary, etc.,), is describing our existence in both two-dimensions and three-dimensions. In short, we have not fully matured, as a species, into a full 3-dimensional frame of reference.

Because Mathematics and other subjects rely heavily on "persistent" dichotomies as part of their basic formulation which frequently end up with tripartite configurations, we must all ourselves the benefit of the doubt for taking the possibility of existing in a transitional state of life. Whereas we typically think we live in a single frame of existential referencing, we may in fact be living in a transitional state of two-dimensions. Such a realization would then provide a clue as to the very many different opinions being realized from perceptions that gather referencing material from a 2-dimensional preference, and/or a 3-dimensional preference, and/or an inter-mediary range of perception and collation of materials being used as intellectually (mentally) configured substances for the purposes of establishing some relative equilibrium... an equilibrium that is being subjected to an enforced conservation as an adaptive measure related to incrementally deteriorating environmental conditions with regard to the Earth, the Moon and the Solar system, though some readers may wish to include the galaxy and the entire Universe as well in their personal equations.

Let us take a look at a simple lineup of geometrically portrayed dimensional scenarios, though exceptions can be noted such as a circle referencing 1 or 2-dimensions and a sphere referencing 3-dimensions. Every single one of the following examples can be viewed as a linear equation, though the expressions do not use customarily assigned numbers or symbols. Ask someone to create an image, and their brain will routinely engage in a silent form of calculating what to use for the writing instrument (pen, pencil, paintbrush, nail, file, finger, etc..), the medium upon which the structure is to be composed (paper, chalk board, sand, etc...), and how large/small, wide/thin, position on the medium, and when they will begin. While some brains calculate fast, others are slower and some people will seek later confirmation that they "did it right" as expected.

geomentric illustrations of different dimensions Word explanations of different diimensions

As I said, one can point out exceptions. For example, is a point the end of a line? How long is a line? At what size does a point become a plane and thereby enter into the 1-dimensional domain? At what width or length does a 1-dimensional line become a 2-dimensional plane? Is a circle actually a sphere but labeled a circle because it is placed in two dimensions? Or are we permitted to consider fractal dimension properties and therefore refer to a circle as being 1-dimensional? (1-dimensional sphere.

Let us also consider the circle as an n-sphere):

The dimension of n-sphere is n, and must not be confused with the dimension (n + 1) of the Euclidean space in which it is naturally embedded. An n-sphere is the surface or boundary of an (n + 1)-dimensional ball.

In particular:

  • The pair of points at the ends of a (one-dimensional) line segment is a 0-sphere.
  • A circle, which is the one-dimensional circumference of a (two-dimensional) disk, is a 1-sphere.
  • The two-dimensional surface of a three-dimensional ball is a 2-sphere, often simply called a sphere.
  • The three-dimensional boundary of a (four-dimensional) 4-ball is a 3-sphere.
  • The n – 1 dimensional boundary of a (n-dimensional) n-ball is an (n – 1)-sphere.

Can a triangle be a combination of three lines seen from different 2-dimensional vantage points and therefore defined as a plane, yet called a "triangular prism" when a triangle is portrayed in a side-view highlighted by two or three different colors to denote a 3-dimensional object?

Let's take a look at some simply 2D and 3D objects:

Examples of 2-dimensional and 3-dimensional objects which can alternatively be viewed as projected thoughts.

Again the question pops up as to what constitutes a supposed 2-sided figure:

An idea representing the view of a 2-sided figure

Now we get to the point where some may ask how many dimensions are there and how many different types of objects can one devise to portray the different dimensions? Since some readers may claim that the quantity is "infinite", whether or not an adequate model can be readily portrayed, I submit their usage of the word "infinite" is comparable to the words "Much, Many, Heap, Unlimited, etc...". In other words, cognitively speaking, they are resorting to the same type of expression that primitive peoples used when they were developing a sense of number-quantity, where in historical development we find that there were stops and starts (much like the dots and dashes of Morse code); where beyond an initial quantity such as one, or two, or three and perhaps four as well, the quantity beyond this valuation was considered to be "Many". In other words, their imagination did not contemplate an exact figure. It is much the same case for the usage of numbers which stop at "9-numbers" and the English alphabet stops at "26-letters". Our imaginations have reached an end... unless we consider the alternative that this is an expressed conservation forced upon the human species due to being subjected to an incrementally deteriorating environment where such conservation activities is required to provide us with the appearance of having achieved an equilibrium as a survival mechanism, which in such a case conservation of mental activity is fortuitous to go along with dwindling resources that can not renew as fast as humanity is consuming them, even if the Earth, the Moon, and the Solar system were stable.

Let me switch gears here and turn the radio to a different channel. The station we are on for the moment is one of biology. Because of the ongoing concerns with the covid-19 (and variants), let me briefly note that viruses can appear with linear, circular, and triangular forms, involving symmetry. It would appear that a way to address the issues of viruses would be to disrupt the symmetry by infecting viruses with a false-imaging mirror, so to speak, whereby they would become confused like a person in a fun house of distorted mirrors. Anyway, the point of biology I want to bring up in the present context since I have inserted commentary about different dimensions, is that we have one and two-stranded RNA and DNA forms, with there being a Triple-stranded DNA form but that the existence of a quadruple form of DNA is rare. One might consider that after billions of years, we would have RNA and DNA functionalities with more than the 3-dimensional varieties we are seeing. Can we not have many multiples of strandedness inside and outside 3-dimensions? Or is the case of such a paucity of strandedness another indication of a prevailing conservation effect being imposed on biology by a degrading environment, and we simply rationalize away any alternative suggestions because rationalization is part of the conservation being used to maintain some semblance of equilibrium?

When we (mathematically) speak of the 3 + 1 dimensions, what we are unknowingly ascribing ourselves to is a scalar factor... simply stated as ratio. By placing both RNA and DNA in a linear-like equation such as is described by a simple addition problem as:

3 to 1 ratio correlations

Needless to say, we need a new mathematics. This is only possible if we create an origin and succession of new assumptions and beliefs from which will be derived a set of axioms appropriate to this endeavor. Yet, there is a decide stumbling block and this is mathematicians themselves. Like politicians who immerse themselves in a given culture where they carve out a comfortable niche', mathematicians, like so many professionals in the past, are those whose challenges are sometimes quite valuable, but at other times offer little but obstinacy like those professionals who were at one time confronted by a new idea that later proved to be a value by which later standards of their profession were built on; such as the Theory of Evolution, Germ Theory, propriety of vaccination, theory of continental drift, heliocentric theory, usage of the plane by armed forces, 1- 2- 3- shot rules in basketball, standardized baseballs, basketballs, tennis balls, steel-plated ships, environmental protections, tobacco hazards, exercise needs for health, child labor laws, vote for women, Equal Rights Amendment, mental health advances, helmets for football players, kindergarten, rite-of-passage graduation ceremonies, courtship (as opposed to hitting a woman over the head with a club and dragging them back to your cave), etc...

With any mathematic's equation we necessarily devise a means of proofing what at first is an hypothesis. For example, even the simple linear equation of 1 + 1 = 2 demands a provable explanation. The explanation for which was established long ago, but at the moment of inception, the belief in it needed to be reconciled with multiple vantage points of perception and interpretation, despite the presumed simplicity of its symbolism which was an effort to achieve a level of basic language expression there would be no room for ambiguity. While most of us today take such an equation for granted and claim that there are many representative examples which help to assert the validity of such a (once philosophical) expression, problems arise as mathematical expressions transgress into areas of pure mathematics where we can encounter suppositions whose proof(s) may require a rigor of mathematics that tends to incline towards the usage of explanations which are as unconventional as the new-found equation(s) themselves. Hence, with respect to the present thesis, let me state:

Conventional mathematics cannot be used to provide the proof(s) needed to substantiate the validity and reality of a mathematical foundation beyond present mathematical conventions. You cannot use the standard (system of) Assumptions and Beliefs from which axioms are produced (derived) and used for equations which are equivalent to the primitive concept of "1"; when your equation(s) are describing the concepts of "2" and "3" that are equivalent to a new type of mathematics existing beyond the Calculus and enters into the realm of pure mathematics from the vantage point of a new realm of perception which grasps those infinitesimals of consideration being presently overlooked in terms of collated correlationals akin to a sense of numerousity which preceded the development of what we today label as advanced mathematics.

In the development of a new mathematics to step beyond the static disposition of present Calculus in order to venture into a domain of a Dynamical formula I am referring to as Accordian Calculus, we need to subject the subject of Mathematics to an infinitesimal gradient itself. In other words, mathematics itself must be analyzed infinitesimally. This requires a wide-spectrum approach from many disciplines since the ideas in different subjects is where the assumptions and beliefs to produce axioms for later proofing arise from. It is actually a very simple concept which entails looking at many different ideas from a different perspective that affect mathematical thinking. For example, let us place a pencil (or pen, chalk, marker, etc...) point on a page and refer to this as a zero dimension. (Despite any readership argument to the contrary.), and then extend the point (as if it were elastic or a piece of gum or stretchable balloon from which animal characters are made, taffy, etc...), so that we now have a one-dimensional characteristic. From here let's turn at a 90° angle so that where we started from is just another straight line and not a continuation of the first line, though they are connected. In other words, we started at one end of the first line in one dimension and have created another instance of a single dimension that presents us with the simple equation of 1 + 1 = 2 dimensions. (Pretty simple stuff.) From this point let us momentarily mention (as a related digression) the use of triangles used in linear equations (but not routinely noted as a three-patterned ensemble such as 3:4:5) as opposed to supposed dynamic equations (which reveal the use of a dichotomy labeled as intouch/extouch triangles): Dynamic Triangle Geometry: Families of Lines with Equal Intercepts by Paul Yiu. Because it is untypical to think in terms of analyzing mathematices with mathematics with respect to outlinging basic cognitive patterns, the fact that dichotomies presently play a large role in the study of triangles disguises the presence of a recurring primitivity of mental exercising.

What I am describing is a scene of long ago where algebra nor geometry existed and for most people was not even a fragment of consideration... though there were a few whose mental functions were using rudimentary expressions thereof in their day-to-day activities which may not have anything to do with the language of mathematical expressions. In their own way, in their own requisite day-to-day concerns, they expressed algebraic and geometric orientations of thought; but only those versed in the rudimentary forms of mathematical expressions became privy to the development of that which would be later labeled as Algebra, Geometry, Trigonometry, Calculus, etc... If one is not accustomed to using the language of mathematics to express concepts defined as mathematical equations, then that with which they are accustomed with will have to suffice, regardless of its primivity due to the context in which it occurs, because so many areas of activity external to mathematics are relatively conserved, constrained and otherwise ill-developed to express deep philosophical issues that mathematics has been practicing at for generations. Likewise, so has music in its own way, but this is not the case in other subjects areas where much human industry of activity has been applied. Because far too many human activities have not been developed in a rigor as has mathematics, those individuals whose mental activity functions at more complex levels, must therefore either turn towards a means by which to suppress or distract their complex thinking patterns or find themselves indulging in multiple areas of interest as a compensatory activity. While some turn to drugs, alcohol, sports, fighting, sexuality, and other activities which help to distract and/or suppress a mind that indulges in complex patterns for which a given individual may not have a favorable social or other environment in which to adequately develop themselves, others may submerge themselves in multiple or elaborate forms of music, art, acting, sports, entrepreneurship, cooking, baking, sewing, hunting, fishing, criminal activity, political stratagizing, academic study/self-teaching, etc., or otherwise formulas of mixing and matching the depths of a particular subject to a high level of knowledge and proficiency. The characteristic colloquialism concerning a person who "makes a science" out of a particular activity that is not typically viewed as a science is no doubt well known to some readers.

Date of Origination: 26 February 2021... 6:59 AM
Initial Posting: June 5th, 2021... 7:39 AM