In the previous chapter, **Dual Dimensions Seen Simultaneously**, we considered the metaphor of metamorphosis. In a single lifespan, a lepidopteran lives on Earth as a caterpillar and lives in the Heavens as a butterfly. Introduced was the idea that the egg, which brings the caterpillar to its domain, and the chrysalis, which brings the butterfly to its domain, could be perceived as portals between the dual dimensions of this interesting insect.

We return now to the strange dream-time formula,

**( ia +1b) (ia -1b)**

to discover how the concept of portals was put into play.

The formula contains 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 BOTH are recorded in the Vesica Piscis, the area of overlap in the Pragmatic Schematic (see graphic below).

*NEITHER of the ‘ i*

*‘s alone are knowable.*

**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*

*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 that is within it. The ‘

**s are Neither +1 nor -1.**

*i’*The ‘a’s and ‘b’s are differentiated, so they must go in the outer crescents.

The lepidopteran’s egg and pupa facilitate Transitions between the Heavenly domain of the butterfly and the Earthly domain of the caterpillar. Also shown in the previous chapter and compared to the egg and chrysalis: the waxing and waning phases of the moon, similarly Transitioning between the invisible new moon and the brightly-visible full moon.

Mustn’t there always be some sort of portal through a coterminous boundary that BOTH connects and divides contiguous domains?

Mathematically speaking, would there then be Transitions/portals between contiguous exponential dimensions?

If so, perhaps the variables ‘a’ and ‘b’ represent a pair of portals aiding in dual Transitions: one coming into the dimension of the* singled-out* (+1 and -1) and the other leaving that lower dimension to return to the higher dimension of the *mingled-in* (** i** and

**).**

*i*In contemplating this possibility, I remembered from middle-school math the following pair of facts:

- We add on a number line.

This is 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.

The number two, in an abstract way, serves as a portal between the first and second exponential dimensions. Two can be added or multiplied to itself to achieve Four in either case, whether the units count on a linear-line or square onto a planar-surface.

To continue with my anything-but-ordinary 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 in the attached **Aside: Exponents Made Easy**, how I came to understand exponents so that they are no longer scary. If you are comfortable with exponents, just skip the aside. Otherwise, release the tension in your neck, shoulders, and tummy; take a deep breath and click the link provided in the previous sentence. By the end of the brief aside, 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**.

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. With our understanding of 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.

Now, using ordinary ‘if-then’ reasoning, we wonder: IF the bold equation above discloses a numerical bridge between one-dimensionality and two-dimensionality, THEN would a similar equation with stepped-up exponents reveal the numerical value of a bridge, or portal, between the second and third dimensions/domains?

**n² + n = n³**

Because we live as beings who are BOTH conscious (**INformation**** is 2D**) and corporeal (OUTformation is 3D), the possibility of numerical portals between the second and third exponential dimensions evokes heightened curiosity. Will these portals, once determined, be reflected in our conscious corporeal existence?

Perhaps the dream-time formula’s variables, ‘a’ and ‘b’, represent portals between the imperceptible 2D metaphysical realm of Patterns and the perceptible 3D physical world of Matter.

This suspicion spring-boarded me to exclaim, “The portals between mathematical dimensions must have specific numerical values!”

What doors may open by solving for the numerical value of these portals?

Let’s find out!

#### The Portals are Defined Mathematically

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 two-dimensional square gives its three-dimensional cube?”

##### Part One: Establishing the Equation

##### Part Two: Solving the Equation

To fully comprehend the significance of these solutions (and as a special reward for tolerating the quadratic formula), our next chapter will provide a scenic detour.

Through numerous narrated slide shows we will explore some awesome properties of our two solutions. As it turns out, these numerical values are related to an aesthetically exquisite proportion, considered throughout the ages to be sacred and even called “golden” and “divine”.

Those relative few who have been privy to the proportion’s profundity have used it to achieve the quality of timelessness in their artistic masterpieces and architectural wonders.

Of course, Mother Nature uses it ubiquitously. Even so, we must have the eyes to see. The upcoming video-driven chapter promises to be an eye-opener! In it Quadernity’s Cone-Spiral model is introduced.

After enjoying the provocative presentations in the next chapter, we will return to the dream-time formula, with all of its values in place and revel in amazement at how the once unintelligible formula was all along giving a mathematical answer to the question about creation stories that initially sparked my morning visions.

Derived from this understanding comes Quadernity’s Nature-Based Number Line.

But first, our **Detour into Magical, Mystical Mythematics**!