When I was in high school, my mother told me “girls do not need college.” She went on to explain how, regardless of my grades and honors, I would end up being a secretary, so I needed to focus on vocational education classes to prepare me for this preordained path.
Only two math classes were required of me. I took pre-Algebra, during which I missed a month of school from having a bad spell of asthmatic bronchitis. Never quite catching up, and knowing I would probably never need these arcane formulae, I skimmed through this course without much of it establishing a file in my long-term memory.
The other math class was an introduction to Geometry, which I totally loved. For my classmate, Denise, who believed herself to be the smartest girl in school, the logic of geometry was impenetrable. Her bewilderment reached full-out exasperation as she detected my enthusiasm for it. That the shy, skinny, sickly kid could grok what she found utterly incomprehensible frosted her cake with a thick layer of cognitive dissonance. When I would answer a question correctly, she would turn around and look at me agape, as if I were an alien. (To the best of my recall, this was the first of countless times that I have gotten this look. )
After my obligatory classes were finished, I never used geometry or algebra for anything other than to calculate the area I wished to cover with paint and to determine whether a 15% discount was enough of an incentive to buy something I would not otherwise buy at full price.
In my prep for becoming a secretary, I did take a bookkeeping class; however, basic arithmetic of adding and subtracting is all that is used in bookkeeping. The skill there comes from knowing into which column to put the numbers, as either credits or debits.
I share this story with you so that you will realize how weird it was when, in a morning meditation, I see this thing sparkling at me:
(ia +1b) (ia -1b)
At that time, I had become used to daily visions of abstract patterns, whose meanings I somehow intuited. None of my visions, however, had ever been made so starkly clear as this one.
The relatively rare times when a single, adamant directive has come to me, it has been audible, in a deep male voice. I would know without question what to do.
This time a clear communication came without any instructions. I would not have even known the string of symbols was mathematical if it weren’t for the plus and minus signs!
I simply wrote it down, remaining in wonderment until I eventually hired a math tutor to help me interpret the symbols and determine its meaning.
The narrated video from the mast-head landing page for Mathematical Models is offered again here. It is optional viewing for readers who would like a brief explanation of what I learned from the tutor. (Keep your cursor off the video screen for unobstructed viewing.)
In summary, here are the three main points:
- The letter ‘i’ stood for the word, imaginary. One can only imagine the √-1, he explained. Squaring either positive or negative numbers always produces positive numbers, never negative numbers. There is, in fact, no real number that, when squared, equals -1.
- The FOIL method provided the correct sequence (First, Outer, Inside, Last) for multiplying the terms inside one set of parentheses by the terms in the other set. The solution to the formula turned out to be: -1(a² + b²).
- The letters ‘a’ and ‘b’ are called variables. As the formula stood alone, no numbers were assigned to, or represented by, these variables, so the mystery continued.
This chapter reveals the process of determining the values of the variables, ‘a’ and ‘b’.
An aside is offered to refresh our memories of some previously discussed conceptss, and to offer a few new tidbits that will help prepare us for the logic and method used to arrive at the values of these variables. Conceptual Preparation for Obtaining Values for the Variables.
Evaluating the Variables from the Dream-Time Formula:
With clues from the attached aside, we are ready to reconsider the dream-time formula:
(ia +1b) (ia -1b)
In the formula are six letters: two ‘i‘s, two ‘a’s, two ‘b’s, and two numbers: a +1 and a -1.
BOTH +1 and -1 are singled-out as known solutions for √+1. Together they go in the Vesica Piscis, the area of overlap in the Pragmatic Schematic (see below).
NEITHER of the ‘i‘s alone are knowable. i stands for an imaginary number. We do not know what i actually is, but we do know that when i is squared, it produces a real value, √-1. In other words, the ‘i’s must be mingled-in to be realized.
The Female and Male mingle in only in the Private Nighttime Relationship, which is unobserved, invisible, and occurs in the metaphysical domain. Therefore, we will put the ‘i’s in the transcendent realm above and beyond the Pragmatic Schematic below. The ‘i’s are Neither +1 nor -1.
A Subject’s first act is always to PULL. The Female Pulls from the metaphysical domain and delivers Her Object down into the physical domain. The Male Pulls from the physical domain and delivers His Object up to the metaphysical domain.
Although the results of my process may be borne out by specialists, the process itself is largely artistic. Recalling from the Geometry of Universal Mother and Father that, symbolically speaking, the Female is round and the Male is straight. So, taking artistic license, I placed the round numbers, 0/0, in the metaphysical domain transcending the PS, as seen below. The final reason that convinced me to do so is this: the √-1 cannot be known, and, similarly, 0/0 is a symbol of indeterminateness.
The 1/1, the straight numbers, would then go into the contrary physical domain, represented by the almond-shaped ‘mandorla’ within the PS.
The ‘a’s and ‘b’s are differentiated, so they must go in the outer crescents. They represent the Transitions coming into distinct physical being (singled-out), and then leaving it in favor of integration with a greater whole; in other words, to be mingled-in and undifferentiated.
The lepidopteran’s egg and pupa facilitate opposite Transitions between the different domains of the butterfly and caterpillar. The waxing and waning of the moon similarly reflect Transitions between the new moon and full moon phases.
To go between dimensions there must be a connector, or portal of sorts. The thought came to mind that the variables, ‘a’ and ‘b’, might represent these Transitions, or portals.
This thought spring-boarded me to a wild idea: Would the portals between dimensions have specific values, mathematically speaking?
In contemplating the above-stated question, the following ideas came to mind:
- We add on a number line, a linear process with a one-dimensional result. For example if we want to know the combined length of two sticks, each 2 feet long, we would line them up and measure them together. 2 + 2 = 4 linear units of distance.
- We multiply to get a two dimensional area. If we want to know the area covered by a square with side lengths of two linear units, we would multiply two by two to get four square units. 2 x 2 = 4 square units of area.
To continue with my reasoning, we will need to use exponents. Using exponents used to intimidate me, and they might be foreign to you. Just in case, I will explain below how I came to understand exponents so that they are no longer scary. If you are comfortable with exponents, just skip past the indented tutorial.
Otherwise, release the tension in your neck, shoulders, and tummy and take a deep breath. By the end of the few following paragraphs, you will probably laugh at how easy my little tricks are and how confident you will be from now on when you encounter a number written in scientific notation.
We will use the example 3². Three is the base number (written on the base line) and two is the exponent (the little superscript number). This is commonly read, 3 to the power of 2.
Even in this simple little example, there are some hidden assumptions. Positive numbers are not preceded by a + sign, though negative values are preceded by a – sign.
We do not generally use the exponent ¹, because it does not change the base number (number on the base line) into a new value. (Any number multiplied by one remains the same, e.g., 2¹ = 2.) On the other hand, an exponent of -1 would change the value, so it must be used when applicable. (2¯¹ = 1/2.)
It has served me well to think that there is an implied +1 multiplied by any number I see standing alone. You will soon see why this little trick is useful.
Here is how I would think of 3²: +1 (+3 x +3). An exponent of two tells me to multiply positive one by the base number, in this case positive three, two times.
If we see the number 3 all by itself, it means 3¹. Multiply positive one (implied) by positive three (the base number) only one time (says the exponent of 1).
If we see the number 3°, it means multiply +1 by 3 ZERO times. In other words, any number with a zero exponent means you do nothing to the one. This is why it helps to imagine a +1 before all base numbers.
If you see a negative exponent, it is telling us to DIVIDE our initial +1 by the base number given, however many times the exponent indicates.
3-² means +1 divided by (+3 x +3), or +1/9. By convention, we would drop the + sign and state our answer as 1/9.
Summary: the exponent (positive or negative) tells you how many times you multiply or divide +1 by the base number.
Okay, back to my anything-but-ordinary reasoning, which is leading us to find the values for our variables, ‘a’ and ‘b’.
Because we know that 2 + 2 = 2 x 2, we know there is a number (two), that when added to itself produces the square of itself. Now that we understand exponents, we can report this fact as n¹ + n = n² so that other mathematicians can solve for n and discover that 2 is the only value possible for n, given that n + n = n x n.
Using normal ‘if-then’ reasoning, we can safely assume that IF the bold equation above discloses a numerical bridge between one-dimensionality and two-dimensionality, THEN a similar equation with stepped-up exponents should reveal the numerical value of a bridge, or portal, between the second and third dimensions.
Because we live as beings that are of BOTH of Consciousness (INformation is 2D) and of corporeality (OUTformation is 3D), the possibility of a numerical portal between our dual domains should evoke a heightened curiosity as to whether we are also beings of mathematical proportionality!
Perhaps the formula’s variables represent portals between the imperceptible 2D metaphysical realm of Patterns and the perceptible 3D physical world of Matter. What will we learn from discovering the numerical value of our portals? What doors will this open? Let’s find out!
The Value of Portals
In the following two short step-by-step videos, we establish and solve the equation, n² + n = n³. In English, this statement asks, What number when added to its 2D square gives its 3D cube?
Part One: Establishing the Equation
Part Two: Solving the Equation
We will return again to the dream-time formula with all of its values in place.
In preparation to fully comprehend the significance of the solution (and as a special reward for tolerating the quadratic formula), our next chapter will provide a scenic detour.
We will explore some magical, mystical mathematical properties, and learn of a proportion so aesthetically stunning that it has been called “golden” and “divine”!