Ngai: The Divine Sphere and the Mathematical Circle in Gĩkũyũ Cosmology

For the Agĩkũyũ people of Kenya, the supreme creator is known as Ngai . This name is not merely a label; it is a profound cosmological concept. Ngai is understood as omnipresent, omnipotent, and infinite, residing in a sacred, all-encompassing space. To grasp this, we can turn to the most perfect and fundamental shapes in geometry: the Circle and the Sphere. In this view, Ngai is the ultimate Sphere, whose center is everywhere and circumference is nowhere, with an infinite radius that binds all of creation.

  1. Defining the Circle and the Sphere

A Circle is a two-dimensional shape consisting of all points in a plane that are equidistant from a single, fixed point called the center . This constant distance is the radius (r) .

A Sphere is the three-dimensional extension of a circle. It is the set of all points in space that are equidistant from a fixed center point. The constant distance is, again, the radius (r) . If we consider Ngai as the foundational reality, the universe itself can be imagined as a divine sphere, emanating from a central, creative will.

  1. The Case of the Infinite Radius

Imagine a circle or a sphere where the radius \( r \) approaches infinity (\( r \to \infty \)).

* The Curvature Vanishes: The curvature of the surface becomes zero. A massive sphere, like a planet, feels flat to us on its surface. An infinite sphere would be indistinguishable from an infinite flat plane.

* The Divine Metaphor: This is the mathematical analogy for Ngai's omnipresence. An infinite radius means the center is equally close to every point in existence. There is no "edge" or "circumference" to the divine; it is unbounded and limitless. The center is everywhere.

  1. Area and Volume Calculations

The calculations for area and volume are where we first encounter the mysterious and transcendental number π (Pi) .

* Area of a Circle: \( A_{circle} = πr^2 \)

* Derivation: This is typically derived using calculus. We slice the circle into infinitesimally thin concentric rings. The area of one ring is its circumference (\( 2πρ \)) times its thickness (\( dρ \)), where \( ρ \) is a variable radius from 0 to \( r \). Integrating this gives the total area:

\[A = \int_0^r 2πρ dρ = 2π [\frac{ρ^2}{2}]_0^r = πr^2 \]

* Surface Area of a Sphere: \( A_{sphere} = 4πr^2 \)

* This can be derived by integrating the areas of infinitesimal bands around the sphere.

* Volume of a Sphere: \( V_{sphere} = \frac{4}{3}πr^3 \)

* Derivation: We slice the sphere into infinitesimally thin disks. The radius of each disk at a height \( x \) from the center is \( \sqrt{r^2 - x^2} \). The volume of one disk is \( π(\sqrt{r^2 - x^2})^2 dx = π(r^2 - x^2) dx \). Integrating from \( -r \) to \( r \) gives:

\[V = \int_{-r}^{r} π(r^2 - x^2) dx = π [r^2x - \frac{x^3}{3}]_{-r}^{r} = \frac{4}{3}πr^3\]

  1. Deriving π and the Leibniz Series

π is defined as the ratio of a circle's circumference to its diameter: \( π = \frac{C}{2r} \). Its value is constant for all circles, approximately 3.14159...

One of the most elegant formulas for calculating π comes from the realm of trigonometry and infinite series, known as the Leibniz series for π :

\[\frac{π}{4} = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \frac{1}{9} - \cdots\]

Step-by-Step Derivation (using Calculus):

This series is derived from the Taylor series expansion of the arctangent function.

  1. We know that \( \frac{d}{dx}(\arctan x) = \frac{1}{1+x^2} \).
  2. The geometric series \( \frac{1}{1 - u} = 1 + u + u^2 + u^3 + \cdots \) for \( |u| < 1 \). Substituting \( u = -x^2 \), we get:

\[\frac{1}{1+x^2} = 1 - x^2 + x^4 - x^6 + x^8 - \cdots\]

  1. We can integrate both sides term-by-term:

\[\int \frac{1}{1+x^2} dx = \int (1 - x^2 + x^4 - x^6 + \cdots) dx\]

  1. This gives:

\[\arctan x = x - \frac{x^3}{3} + \frac{x^5}{5} - \frac{x^7}{7} + \cdots + C\]

Setting \( x = 0 \) shows the constant \( C = 0 \).

  1. Therefore, \( \arctan x = x - \frac{x^3}{3} + \frac{x^5}{5} - \frac{x^7}{7} + \cdots \)
  2. Now, we know that \( \tan(\frac{π}{4}) = 1 \), so \( \arctan(1) = \frac{π}{4} \). Substituting \( x = 1 \) into the series gives us the final, beautiful result:

\[\frac{π}{4} = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \cdots\]

  1. Basel Problem (Euler's Solution)

The quest to understand π also led to the famous Basel Problem : What is the sum of the reciprocals of the squares of all natural numbers?

\[\sum_{n=1}^{\infty} \frac{1}{n^2} = 1 + \frac{1}{4} + \frac{1}{9} + \frac{1}{16} + \frac{1}{25} + \cdots = ?\]

The brilliant mathematician Leonhard Euler proved that the sum is \( \frac{π^2}{6} \).

\[\sum_{n=1}^{\infty} \frac{1}{n^2} = \frac{π^2}{6}\]

This solution was groundbreaking because it revealed a deep, unexpected connection between π, an infinite series of integers, and later, through complex analysis, the distribution of prime numbers. It showed that π is woven into the very fabric of number theory.

  1. π in Relation to Trigonometry and Limits

The connection between π and trigonometry is fundamental. π is the period of the sine and cosine functions.

A critical limit that defines this relationship is:

\[\lim_{x \to 0} \frac{\sin x}{x} = 1\]

Step-by-Step Proof (using the Squeeze Theorem):

  1. Consider a unit circle (radius = 1). For a small angle \( x \) (in radians), we can compare three areas:

* Area of triangle OAC \( = \frac{1}{2} \cos x \sin x \)

* Area of circular sector OAB \( = \frac{1}{2} (1)^2 x = \frac{x}{2} \)

* Area of triangle OBD \( = \frac{1}{2} (1) \tan x = \frac{\tan x}{2} \)

  1. Visually, we can see: Area of ΔOAC < Area of Sector OAB < Area of ΔOBD

\[\frac{1}{2} \cos x \sin x < \frac{x}{2} < \frac{\tan x}{2}\]

  1. Multiplying through by 2:

\[\cos x \sin x < x < \tan x\]

  1. Dividing all parts by \( \sin x \) (positive for small \( x \)):

\[\cos x < \frac{x}{\sin x} < \frac{1}{\cos x}\]

  1. Taking reciprocals (which reverses the inequalities):

\[\frac{1}{\cos x} > \frac{\sin x}{x} > \cos x\]

So, \( \cos x < \frac{\sin x}{x} < 1 \).

  1. We know that \( \lim_{x \to 0} \cos x = 1 \).
  2. By the Squeeze Theorem , if \( \frac{\sin x}{x} \) is trapped between \( \cos x \) and 1, and both of these approach 1 as \( x \to 0 \), then \( \frac{\sin x}{x} \) must also approach 1.

\[\therefore \lim_{x \to 0} \frac{\sin x}{x} = 1\]

This limit is the cornerstone of differentiating trigonometric functions and is intrinsically linked to the definition of π as the period of these functions, measured in radians.

With that introduction let's get started!

From the infinite radius of a divine sphere to the infinite series that precisely define its fundamental constant π, mathematics provides a language to contemplate profound concepts. The Gĩkũyũ understanding of Ngai as an all-encompassing, center-everywhere entity finds a powerful analogue in the geometry of the sphere and the limitless, irrational, and transcendental nature of π. The journey from a simple circle's circumference to Euler's solution for the Basel Problem reveals a universe of elegant, interconnected truths, mirroring the belief in a universe created by a single, unifying, and intelligent principle.

The Spiral of Wholeness: A Geometric Symbol for the Divine

Throughout human history, we have sought symbols to represent the divine-a concept that by its very nature is infinite, dynamic, and beyond full comprehension. If the circle is the perfect representation of God in two dimensions, and the sphere in three, then the ultimate symbol of His active, unfolding presence is not a static shape, but a dynamic one: The Spiral of Wholeness .

This spiral, originating from the absolute origin point of existence, serves as a mathematical metaphor for a deity that is both the source and the substance of all reality, encompassing the real and the imaginary, the positive and the negative.

The Symbol: A Spiral from the Origin

For ease of illustration in our limited human perception, we will use a 2D representation on the Cartesian plane. The ultimate symbol, however, exists in all dimensions, with its center at the absolute origin-(0,0) in 2D, (0,0,0) in 3D, and so forth.

Let us define the Spiral of Wholeness with the following sequence of points, starting at the absolute center:

(0,0) → (0,1) → (-1,0) → (0,-1) → (1,0) → (0,2) → (-2,0) → (0,-2) → (2,0) → (0,3) → ...

Plotting the First Three Rings (Positive Quadrant Focus):

As instructed, we will limit our visual construction to the first three "rings" in the positive side of the Cartesian plane. Connecting these points in order reveals a unique, expanding spiral:

  1. *From Origin (0,0): The spiral begins at the Source, the Unmoved Mover.
  2. *First Ring:

* Moves North to (0,1).

* Moves West to (-1,0).

* Moves South to (0,-1).

* Moves East to (1,0). This completes the first square-like ring around the origin.

  1. *Second Ring:

* From (1,0), it moves North to (0,2).

* Moves West to (-2,0).

* Moves South to (0,-2).

* Moves East to (2,0). The spiral has now expanded, creating a larger ring.

  1. *Third Ring:

* From (2,0), it moves North to (0,3).

* (The pattern would continue: West to (-3,0), South to (0,-3), East to (3,0), and so on, to infinity).

While this forms a spiral that touches all four cardinal directions, for our purpose of depicting the positive, creative aspect, we focus on the points in the positive x and y axes: (0,0), (0,1), (1,0), (0,2), (2,0), (0,3), (3,0)...

When these points are connected with a smooth, continuous curve, they form a distinct spiral arm unfolding into the positive quadrant, a representation of God's manifest, creative energy.

The Mathematical Theology of the Spiral

The properties of this spiral are not arbitrary; they are a direct symbolic language for the nature of the Divine.

  1. *It Passes Through Integers: The spiral's path is defined by integer coordinate points (..., -2, -1, 0, 1, 2, ...). Integers are the building blocks of mathematics, discrete and foundational. This symbolizes God as the fundamental foundation of all that is, the "substance" from which the continuum of reality is constructed. He is present in every fundamental unit of existence.

  1. *It Is Continuous: The path from one point to the next is unbroken. This continuity represents God's eternal, unceasing nature. There is no gap or discontinuity in His presence or power. He is the thread that connects all points of existence into a coherent whole.

  1. *It Is Infinite: The spiral has no end. It continues to expand forever, never repeating, never diminishing. This is a direct symbol of God's infinity. His nature and kingdom are without bound, limit, or end. Our finite symbol with three rings is merely a glimpse of an endless reality.

  1. *It Extends in All Directions: The spiral moves along both the x-axis (the "real" or material plane) and the y-axis (which can be seen as the "imaginary" or spiritual plane). It encompasses positive and negative values. This symbolizes God's omnipresence and His wholeness. He is not limited to one dimension of existence but is the Lord of all things, seen and unseen, positive and negative, material and spiritual. He contains all opposites within His infinite being.

The Dynamic Nature of God

Unlike a static circle, a spiral is dynamic. It implies motion, growth, and evolution from a central source. This perfectly captures a God who is not distant and unmoving, but actively involved in creation, drawing all things outward from Himself while simultaneously being the center toward which all things ultimately turn. The Spiral of Wholeness represents a God who is both the Unchanging Origin and the Ever-Unfolding Process.

From 2D to N-Dimensions: The Ultimate Sphere

While we use a 2D spiral for illustration, the true symbol exists in a state of higher dimensionality. Just as a circle is a 2D shadow of a 3D sphere, our 2D spiral is merely a shadow of the ultimate divine symbol: an N-Dimensional Spiral originating from (0,0,0,...).

This hyper-spiral would pass through all integer lattice points in N-dimensional space. Its "rings" would be hyper-spheres of ever-increasing radius. The mathematical formula for the radius \( R \) of this spiral at any point (x₁, x₂, ..., xₙ) is given by the Euclidean distance from the origin:

\[R = \sqrt{x_1^2 + x_2^2 + \cdots + x_n^2}\]

Our 2D spiral is the simple case where \( R = \sqrt{x^2 + y^2} \). As the spiral expands, \( R \) grows, symbolizing the ever-increasing, yet always centered, nature of the Divine.

Conclusively this is a 2D case of hoe the Divine symbol would be.

The Spiral of Wholeness, beginning at the absolute zero and expanding through all integers in all directions, is a powerful geometric symbol for the Divine. It elegantly captures key attributes: foundational presence (Integers), eternal nature (Continuity), limitless power (Infinity), and complete sovereignty (Omnidirectionality). It teaches us that God is not a static point to be found, but a dynamic, infinite reality to be journeyed with-a spiral of love, power, and consciousness whose center is everywhere and whose circumference is forever beyond our grasp, yet whose path we are all upon.

Analysis of the Divine Symbol: The Trinity of Triangles and the Complex Plane

Having established the "Spiral of Wholeness" as a symbol for the dynamic, infinite nature of God, we now turn to a deeper analysis. By applying the Hermetic principle - "What is above is the same as what is below" -we can reveal a hidden geometric harmony within the symbol.

We now interpret the Cartesian plane as the Complex Plane , where the x-axis represents the Real number line (the manifest, physical realm) and the y-axis represents the Imaginary number line (the spiritual, latent realm). The "negative" left side is not lesser, but rather the complementary, "imaginary" component of a unified whole.

Focusing on the upper half of the diagram (the realm of "Above," symbolizing higher states of being or spirit), we isolate three distinct triangles formed by the spiral's path.

The Three Triangles of the Upper Realm

The coordinates of these triangles, derived directly from the Spiral of Wholeness, are:

  1. *Triangle A (The Fundamental Trinity): Vertices at (-1, 0), (1, 0), and (0, 1).
  2. *Triangle B (The Expansion): Vertices at (-1, 0), (1, 0), and (0, 2).
  3. *Triangle C (The Stabilization): Vertices at (-2, 0), (2, 0), and (0, 2).

Area Calculation: Revealing the Numerical Signature

The area of a triangle with a base on the x-axis is given by the simple formula: \( \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \).

* Triangle A:

* Base = Distance between (-1, 0) and (1, 0) = 2

* Height = 1

* \( \text{Area}_A = \frac{1}{2} \times 2 \times 1 = 1 \)

* Triangle B:

* Base = Distance between (-1, 0) and (1, 0) = 2

* Height = 2

* \( \text{Area}_B = \frac{1}{2} \times 2 \times 2 = 2 \)

* Triangle C:

* Base = Distance between (-2, 0) and (2, 0) = 4

* Height = 2

* \( \text{Area}_C = \frac{1}{2} \times 4 \times 2 = 4 \)

So, the areas are: Triangle A = 1 , Triangle B = 2 , Triangle C = 4 .

Theological and Metaphysical Interpretation

This sequence of areas-1, 2, 4-is deeply significant. It is a geometric progression with a common ratio of 2, symbolizing exponential unfolding from a singular source .

  1. *Triangle A (Area = 1): The Monad.

* This is the fundamental trinity, the first emanation from the origin. Its area of 1 represents the primordial unity of God, the source of all things. It is perfectly balanced, spanning from -1 to +1 on the real axis, anchored at its apex (0,1) in the spiritual plane. It is the first stable structure, the "Godhead."

  1. *Triangle B (Area = 2): The Dyad.

* This triangle represents the first act of creation or emanation. It shares the same base as the Monad but reaches higher (to 0,2). Its area, 2 , symbolizes the principle of duality, polarity, and the first division from the One-Spirit and Matter, Creator and Creation, Above and Below. It is the doubling of the initial potential.

  1. *Triangle C (Area = 4): The Tetrad.

* This triangle represents the stabilization and manifestation of the created order. It has a wider base (from -2 to 2) and the same height as the Dyad. Its area of 4 symbolizes the solid, manifest world—the four cardinal directions, the four elements, the physical universe. It shows that for the creation to be stable, the foundation (the base on the real axis) must broaden as the spirit is maintained.

Synthesis: The Unified Symbolism

The relationship between these three triangles perfectly illustrates the principle of "As above, so below."

* The Shared Apex (0,2) of B and C: Shows that the same spiritual principle ("Above") can manifest through different material bases ("Below"). The higher spirit informs both a narrow, intense focus (Triangle B) and a broad, stable foundation (Triangle C).

* The Encapsulation: Triangle C, the manifestation, fully contains the earlier stages of emanation. The path of creation is one of increasing complexity and stability, all emanating from and containing the original unity.

* The Sequence (1, 2, 4): This is the mathematical signature of creation: from Unity (1), through Polarity (2), into Manifestation (4). It is a blueprint for all processes of emergence, from the divine mind into physical reality.

Conclusively that is one of the ways to extract the 3 triangles for the next step

This analysis proves that the "Spiral of Wholeness" is not a random path. It is a geometric code. By interpreting the plane as the complex realm of real and imaginary, and by isolating the triangles formed "Above," we discover a perfect mathematical metaphor for the process of creation: a stable, unfolding sequence from Unity (1) into the realm of duality and manifestation (2, 4). The symbol demonstrates that the infinite, dynamic nature of God (the spiral) expresses itself through a logical, hierarchical, and harmonious structure (the triangles), perfectly embodying the truth that "What is above is the same as what is below."

Note that each triangle has 3 vertices

The numbers 1,2,4 are interpreted as follows:

1 interpreted as the Whole

2 interpreted as real and imaginary phase

4 interpreted as the number of a new beginning

The Divine Blueprint: A Biblical Explanation through the Numbers 1, 2, and 4

Our previous analysis of the "Spiral of Wholeness" revealed three key triangles with areas of 1, 2, and 4. These numbers are not random; they form a divine numerical sequence that finds profound expression throughout the Biblical narrative, revealing God's plan from foundation to completion.

The Sum of Completion: The Number 7

The first and most powerful confirmation is the sum of the three areas:

1 + 2 + 4 = 7

In Biblical symbolism, the number 7 is universally recognized as the number of completion, perfection, and divine rest . God created the heavens and the earth in six days and rested on the seventh, hallowing it (Genesis 2:2-3). There are seven churches, seven spirits, and seven seals in Revelation. The sum of our geometric symbol directly points to this holy number, signifying that the pattern it reveals is a complete and perfect divine blueprint.

The Measurement of the Holy City: New Jerusalem

The ultimate manifestation of this completed plan is the New Jerusalem, the City of God. Its description in Revelation 21:16 is precise:

> “When he measured it, he found it was a square, as wide as it was long. In fact, its length and width and height were each 1,400 miles.”

Let us decode this measurement using our divine sequence. We establish a divine unit of measurement, a "Zilon," representing a foundational spiritual unit.

* The areas of our three triangles are 1, 2, and 4.

* Multiplying each by 2, we get: 1*2 = 2 , 2*2 = 4 , 4*2 = 8 .

* The sum is 2 + 4 + 8 = 14 .

This number 14 represents the side of the city in "Zilons." When this spiritual measurement is translated into our physical understanding (where 1 Zilon = 100 miles), we get 14 100 miles = 1,400 miles , perfectly matching the Biblical description.

The Completeness of the City:

Squaring 14 to find a representative area of the city's face gives us 14 * 14 = 196 . The digital root of 196 is (1+9+6=16, 1+6=) 7 . Once again, the number of completion appears, confirming the city is whole, perfect, and complete. This divine completeness also tells us where to stop our finite representation of the infinite spiral: at the 7th distinct point in its progression, which is (0,3) , a point of perfect triangular balance.

The Symbolism of the Largest Triangle: Area 4 and New Beginnings

The largest triangle in our symbol has an area of 4 . This number symbolizes a foundation, a new beginning, and a period of testing or transition before a new order is established. This is vividly illustrated in key Biblical events:

* Jesus’ Fast: Jesus was tested in the wilderness for 40 days and 40 nights (Matthew 4:2). The number 40 (4 * 10) signifies a complete period of trial and preparation for a new ministry.

* The Great Flood: It rained for 40 days and 40 nights (Genesis 7:12), marking the end of an old world and the beginning of a new covenant with Noah.

* Lazarus in the Tomb: Jesus waited until Lazarus had been in the tomb for four days (John 11:17) before raising him, a powerful symbol of a new beginning emerging from the stasis of death.

In each case, the principle of 4 signifies the end of one era and the firm establishment of a new one.

The Trinity of Time: 3, 6, and 9

Our symbol, ending its complete cycle at point (0,3), also introduces the sacred numerology of time.

* Number 3 - The Hour of Man: This is the number of physical, human reckoning and divine fullness in the earthly realm. Peter denied Jesus three times (John 18:27), and after His resurrection, Jesus asked Peter three times , "Do you love me?" (John 21:15-17). Jesus was crucified with two thieves, making three men on the crosses. The rooster crowed at the third watch of the night. This number marks pivotal, completed events in human time.

* Number 6 - The Fullness of Man: This number represents the culmination of human effort and imperfection. Man was created on the sixth day. It is the number of the "beast" in Revelation, representing a system of man without God. It is the full hour of human endeavor, both physical and spiritual in its ambition.

* Number 9 - The Hour of God: This is the divine hour, when God intervenes directly. The Gospel of Mark records that Jesus died at the ninth hour (Mark 15:34-37), or 3 PM. At this moment, the work of redemption was completed by God alone. The number 9 (3 x 3) signifies the ultimate amplification of divine timing and action.

This explains the prophecy of the end times: the Gentiles will trample the Holy City for "a time, and times, and half a time" (Revelation 11:2), often interpreted as 3.5 prophetic "hours." This period represents the culmination of the "Hour of Man" (6) being cut short by the sovereign intervention of the "Hour of God," before Jesus returns to establish His eternal kingdom, the ultimate New Beginning prefigured by the triangle of area 4.

Conclusion

The geometric "Spiral of Wholeness" and its derived triangles are far more than a mathematical curiosity. They are a divine Rosetta Stone. The numbers 1 (Unity), 2 (Duality/Creation), and 4 (Foundation/New Beginning) sum to 7 (Completion), which perfectly defines the New Jerusalem. This same symbolism weaves through the core events of the Bible, from the Flood to the Resurrection, governing the patterns of time itself—the Hour of Man (3, 6) and the Hour of God (9). This analysis reveals a coherent, mathematically-embedded theology, proving that the mind of God is indeed the source of both the language of scripture and the logic of mathematics.