(Basic Geometry of Cognitive Activity)
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Lookiloos as of August 4th, 2025
Let me momentarily digress to supply some references to the point- line- plane of geometry. Note in the following illutration that one, two, three points are progressively used as part of the explanation which itself needs some referencing because of the attempt to display the three concepts as basic, fundamental, pristine points of cognitive origination to be equated with unequivocal truth, and yet the use of a progressive serialization of point quantity typically accompanies many such efforts:
Internet query parameter "what are the three undefined terms in geometry?" yields the following:
The three undefined terms in geometry are point, line, and plane. These terms are considered undefined because they are the foundational building blocks upon which all other geometric concepts are built and cannot be defined using simpler geometric terms, according to multiple math websites. Here's a bit more about each:
- Point:
- A point represents a specific location in space. It has no dimension (no length, width, or height).
- Line:
- A line is a one-dimensional figure that extends infinitely in both directions. It has no thickness.
- Plane:
- A plane is a two-dimensional flat surface that extends infinitely in all directions. It has no thickness.
While the ideas are simple, they have not been looked at from the perspective of developmental cognitive activity taking place in a similar fashion in multiple subject areas. And they do not stress the progressive lineage towards some "three" or "third" position of accomplishment. After the one (or first) comes the two (or second) and after the two comes the three (or third). It is a cognitive value notation that I explore in the concept After Every Third, and how it relates to the survivability of biological life under the conditions of an incrementally deteriorating Triplex (Sun-Moon-Earth) planetary system. Multiple ideas reference the situation but you need to understand the triplet code being used by Nature in terms of developmental cognitive behavior over time.
As a further, but related aside comment, we can catalogue "three undefined ideas" in religion and philosophy as well, though I am sure the reader might be able to see such a lineup in other contexts, as well as realize that several concepts overlap with other subjects, including a written assignment involving Body- Title- Conclusion:
- Religion:
- What or who is God, if any?
- What is sacred?
- What is belief and faith?
- Philosophy:
- What is life?
- What is truth?
- What is real?
- Metaphysics:
- What is mind, body, and soul?
Here some examples of specific philosophical orientations using a tripartite theme (Source: BTR 5):
|
Thesis ~ Antithesis ~ Synthesis Indulgence ~ "Middle Way" ~ Ascetism Major Premise ~ Minor Premise ~ Conclusion Contradiction ~ Excluded Middle ~ Identity Principal |
||
| "God-ology": Omnipresent Omnipotent Omniscient |
"Metaphysics-ology": What is real How change comes What is mind |
Marxian "Dialectology": Unity of opposites Quantity & quality Negation of negation |
| Epistemology: How we know What is truth What is mind |
Axiology: Nature of good Nature of beautiful Nature of religious |
Ontology: Quality (1st-ness) Relation (2nd-ness) Representation (3rd-ness) |
Let me now digress into displaying various ideas rendered in a tripartite mathematical form: (Source: BTR 3)
| C = pi x D | (Circumference of a Circle) | E = H x V | (Planck's Constant) |
| A = L x W | (Area of a Rectangle) | P = I x E | (Watt's "Law") |
| A = 1/2B x H | (Area of a Triangle) | E = I x R | (Ohm's "Law") |
| A = pi x R2 | (Area of a Circle) | E = V1 + V2 | (Kirchhoffs Voltage Law) |
| E = M x C2 | (Einstein's formula) | P = R x B | (Percentage formulas) |
| Z = Eg ÷ Ig | (Thevenin impedance formula) | A2 + B2 = C2 | (Pythagorean theorem) |
Now let's go a step further along this aside path and query the internet with "famous three part equations", resulting in:
Here are some examples of famous and influential equations with three parts:
- Pythagorean theorem:
- Equation: a2 + b2 = c2
- Description: This fundamental theorem in geometry relates the sides of a right triangle: the square of the hypotenuse (the longest side, denoted by 'c') equals the sum of the squares of the other two sides ('a' and 'b').
- Significance: Widely used in various fields like engineering, construction, and navigation.
- Sine, cosine, and tangent are (the three) trigonometric functions that relate the angles and side lengths of right triangles, while the Pythagorean theorem specifically relates the squares of the side lengths of a right triangle. The Pythagorean theorem (a2 + b2 = c2) helps find a missing side length when you know the other two. Trigonometric functions (SOH CAH TOA) are used when you know one side and one angle (or two sides and need to find an angle) to find other side lengths.
- Newton's Second (of three) Law(s) of Motion:
- Equation: F = ma
- Description: States that the force (F) acting on an object is equal to its mass (m) multiplied by its acceleration (a).
- Significance: Foundational to classical mechanics and crucial for understanding and predicting the motion of objects in the physical world.
- Einstein's Mass-Energy Equivalence:
- Equation: E = mc2
- Description: Relates energy (E) and mass (m) as interchangeable forms of the same thing, with the speed of light (c) being the constant factor connecting them.
- Significance: A cornerstone of modern physics, it led to the development of nuclear energy and the atomic bomb.
- 4. Equations of Motion (Kinematics)
- Equations:
- v = u + at
- s = ut + ½2
- v2 = u2 + 2as
- Description: These three equations relate velocity, displacement, acceleration, and time when an object moves with constant acceleration in a straight line.
- Significance: Essential for studying linear motion and understanding how objects move in response to forces.
- Ohm's Law:
- Equation: V = IR
- Description: Describes the relationship between voltage (V), current (I), and resistance (R) in an electrical circuit.
- Significance: A fundamental law in electrical engineering, it's vital for designing and analyzing electrical circuits.
It's important to note that the term "three-part equation" can sometimes refer to systems of equations with three variables or to equations with three terms on one side. However, the examples above represent distinct mathematical and scientific relationships commonly recognized for their three key components.
- It doesn't take a genius to see that "A" (one) Rock is just "A" (one) large Point.
- Nor does it take a lot of imagination to see that the plural word "scissors" means the attachment point of two separate "scissor" item-s is analogous to a geometric line reference having a value of two, typically assigned with at least one 'point', but often two points distintionn.
- And it is rather troubling if you can't see a piece of paper as a plane upon which are often described the placement of 3 points to create a coordinate system, and that each provides a type of directional covering.
While these three (plus the eating utensils example) are easily identified as correlations, some of the later shown examples may be difficult for some to visualize and localize within their personal systems of logic, because they are not familiar with the tool of trichotomization being applied over a large area of different subjects.. No doubt some readers will want to move the items into different slots because they feel how they define a given idea provides a better fit, but this is to be expected and does not diminish what I am trying to convey about the existence of a basic cognitive theme being expressed, but has largely gone unrecognized, much less catalogued over time to determine if, when, and how human cognition is changing in this respect... which, if my suppositions are correct, there is an ongoing fusion of the elements that is presently seen in the form of an overlap, as is the standard rule in the rock- scissors- paper game... that suggests the human model of present day conceptualization (for the most part) is in a state of solidification to be followed by (or overlap with) a state of liquidity that will then be followed by a stage of gaseous models of conceptualization, if we can use the analogy of ice, water, steam as an appropriate 3-part/3-stage analogy.
From a solid patterning of... one, to a liquid/flow patterning of... two, to a gaseous triple point-like existence mistakenly defined as a sought for state of future harmony, peace, enlightenment that is not to be measured by time, but a loss of that which keeps the state from being sustained. The many fluctuations of heat, pressure, and vacuum in the present reality are not designed for a sustained triple point equilibrium, as noted by the difference in proportionality of the supposed 3 Universe "stable(?)" constants: Matter (5%), Dark Matter (27%), Dark Energy (68%), which can lead into contemplations involving the presence or absence of constants in dimensionless- dual-dimension- triple dimensionality, when known constants may have differing proportionalities.
This also brings to the surface the Einsteinian concepts of special and let us say common (general) Relativity, though others worked along these lines as well. The internet provided the following definitions:
- Special relativity deals with the relationship between space and time for observers in uniform motion.
- General relativity extends this to include gravity and accelerated motion.
We can add a "c" or 3rd type of relativity which deals with perception relative to the rate in which biological processes are adapting (adjusting) to the incremental deteriorations of the triplex planetary system which are likewise subject to outlying changes in the galaxy and elsewhere; which brings to mind the consideration that the proposed constants we claim to see and abide by, may themselves be subject to incremental changes we make "tweaking" adjustments for and claim it is due better "sciencing" efforts (equipment, experimentation). On the other hand, if the constants are long enduring, we still need to make relative adjustments to changes in our peceptibiities in relation to alterations required by adaptations necessary to maintain some semblance of equalibrium with the triplex deteriorational effects on our biology. No doubt there is some level of rationalization imported to account for what may be viewed as small changes that are dismissed by way of an accepted plus/minus attitude awarded to a perception of human imperfection. Hence, we might cite this as a "Super Special" or "Super General" Relativity concept:
Brief internet definitions:
- A "super general concept" refers to an idea or principle that is extremely broad, abstract, and applicable across a wide range of situations, subjects, or domains.
- "Super Special" can imply beyond the norm.
Regardless of which one might use, if not something different, the point of the matter is that our present relativity concepts do not take into consideration that perceived constants which actually change, may well be rationalized (tweaked) to fit within the parameter of normalcy we are incrementally forced to adapt to as a survival mechanism which seeks a measured stability of equilibrium.
The reason for using Trichotomization is that Nature expects us to as a survival mechanism under the present circumstances of an incrementally deteriorating planetary environmental system, as described by life on the 3rd planet within a triplex system (moon- earth- sun), a tripartite particle physics, a tripartite biological development system (triplet code of cellular energy sources (AMP- ADP- ATP), triplet code of DNA and a triplet code of 3 germ layers, to name a few select items.
Very many of our ideas are based on faith. This "faith" is then displayed with words and/or symbols meant to establish a desire for establishing proof and thus some purported reality which may use common references of everyday perception as supportive evidence. Let us take for example the case for geometry. As many a student has come to realize, what they are being taught rightly gives the impression that ideas being offered by an instructor are little more than an acceptance of rules, even if those rules are based on imagination being presented with words labeled as logic and truth which are to be interpreted as a good enough proof. Yes, they are a proof so long as everyone accepts them as a proof, just as everyone in the village decided to go along with the view that the Empower/King was wearing a find suit of clothes even though he was stark naked and rightly identified as such by a child.
However, if enough people believe in the imaginatively created proofing system, then this becomes the standard by which everyone is required to accept, adopt, and mimic as truth in a society which advances whatever nonsense it wants and creates a working social system with such a doctrine. To tell you that all systems of government and education are made up in the imaginations of people may be hard for some to accept as a basic reality to the extent they begin to look more closely at religion, at mathematics, at government, at the stock market, at education, at sports, etc., and realize they are participating in a collection of fairy tales. That which was accepted as truth in the past had eventually become recognized as idiocy, just as many of the truths today will likewise be seen as nonsense in the future. And for those who see such today, your mindset is geared more-so for some future age which may make living in the present collection of nonsense quite hard to cope with and cause you to indulge in self-defeating activities.
Let us look at how a series of ideas which begin as suppositions are narratively orchestrated towards a profile of presenting a system of logic based on this stepwise increase of assertions that eventually become labeled as a rigorous system of proofing by way of rationalization. I used the internet query "the proof of geometry is often taken on faith", which receives a reply of representing a half-truth based on the view of an applied setting:
The statement "the proof of geometry is often taken on faith" implies that geometric principles are sometimes accepted as true without rigorous demonstration. This perspective isn't entirely accurate, (and can be misleading) though it can reflect how geometry is taught and understood in some educational settings.
- Axiomatic Foundations: Geometry, particularly Euclidean geometry, is built upon a system of axioms or postulates – fundamental statements assumed to be true without proof. These serve as the basis for deriving other theorems through logical reasoning. For example, one of Euclid's postulates states that "A straight line segment can be drawn joining any two points".
- Theorems and Proofs: Theorems are derived from these axioms and previously proven theorems through a process of rigorous logical deduction. This process is crucial in establishing the validity and reliability of mathematical results.
- Mathematical Rigor: Mathematical rigor, the precision and logical soundness of arguments, is highly valued in geometry and other areas of mathematics. Rigorous proofs ensure that conclusions are correct and that the reasoning can withstand scrutiny.
- Challenges in Education: Some research suggests that geometry instruction can sometimes focus on rote memorization or procedural skills rather than on the underlying logical reasoning and proof-building process. This can lead students to perceive geometric principles as truths to be accepted on faith rather than as logical deductions from established axioms.
- Misconceptions: Students may struggle with proofs due to difficulties with geometric vocabulary, recognizing different components of problems, or even retaining prerequisite algebra skills.
In essence, geometry is built on a foundation of axioms and a system of logical deduction and proof. While the educational approach can sometimes lead to an impression of "taking things on faith," the core of geometry lies in rigorous proofs derived from fundamental assumptions.
Page initially created: Sunday, July 20th, 2025... 4:44 AM
Initial posting: Monday, Aug. 4th, 2025... 8:06 AM