The Written Version

This work is not a scientific theory. It is a symbolic cosmology — a personal language built from geometry, numbers, cycles, and ancient motifs. It explores meaning, not physics.

Numbers

Digital Roots

Cycles, couples, and the numerical structure behind the fabric of reality.

Numbers and Digits

“A digit is a written symbol used alone or in combination to represent whole numbers. In a positional numeral system such as the decimal system, a small set of digits is enough to express any value. The number of digits in the system is its base. The decimal system, the most common numeral system, contains ten digits representing the numbers from zero to nine.”

“A number is a mathematical concept used to evaluate and compare quantities or ratios of magnitudes, and also to order elements by indicating their rank. Often written using one or more digits, numbers interact through operations summarized by calculation rules. The properties of these relationships are studied first in integer arithmetic, then more broadly in number theory.”

Digits and Numbers: Tools for Counting and Calculating

Digits are like the letters of an alphabet — but for numbers. They are used to write numbers. In our usual numeral system (the decimal system), we use ten digits:

0, 1, 2, 3, 4, 5, 6, 7, 8, 9

By combining these digits, we can write any number, no matter how large.

Numbers are the concepts represented by digits. They allow us to count, measure, and compare quantities. For example, the number “5” represents the idea of five objects.

In summary:

Numbers are used to count, measure, and compare.

Digits are symbols; numbers are ideas.

Digits are used to write numbers.

Digital Roots

“The digital root of a natural number is obtained by adding all its digits (in base 10), then adding the digits of the result, and repeating until a single digit remains.” — Wikipedia

The Digital Root: A Deeper Insight

The digital root — also called repeated digit sum or numerical residue — is a simple yet powerful arithmetic operation.
It consists of adding the digits of a number until only one digit remains.

Mathematical Definition

Let n be a natural number. The digital root of n, noted dr(n), is defined as:

• If n is a single digit (0–9), then dr(n) = n.
• Otherwise, dr(n) = dr(s(n)), where s(n) is the sum of the digits of n.

Example: n = 12345 1 + 2 + 3 + 4 + 5 = 15 1 + 5 = 6 → dr(12345) = 6

Interesting Properties

• Congruence mod 9: The digital root of a number is congruent to that number modulo 9. This is the basis of the “proof by 9” method.
• Addition invariance: dr(a + b) = dr(dr(a) + dr(b))
• Multiplication invariance: dr(a × b) = dr(dr(a) × dr(b))

Applications

• Checking calculations (proof by 9)
• Numerology
• Algorithms (sorting, hashing)

In summary

The digital root is a simple but powerful tool revealing hidden structures in integers.

The Number 9

9 is neutral in digital root arithmetic and is considered equivalent to 0.

Addition / Subtraction

1 + 8 = 9 2 + 7 = 9 3 + 6 = 9 4 + 5 = 9

1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 = 36 → 3 + 6 = 9 (neutron) 1 + 2 + … + 9 = 45 → 4 + 5 = 9 (proton)

1 − 1 = 0 → Union of Wadjet and Nekhbet creating the cosmic egg.

1 − 1 + 1 − 1 + 1 − 1 + 1 − 1 = 0 → The eight primordial Heh, arranged in four pairs, creating the cosmic egg.

Multiplication / Division

Multiples of 1 (and inversely multiples of 8)

1, 2, 3, 4, 5, 6, 7, 8 … (cycle repeats)

Multiples of 2 (and inversely multiples of 7)

1×2 = 2 2×2 = 4 3×2 = 6 4×2 = 8 5×2 = 10 → 1 6×2 = 12 → 3 7×2 = 14 → 5 8×2 = 16 → 7

Cycle: 2, 4, 6, 8, 1, 3, 5, 7 Sum = 36 → 3 + 6 = 9

Multiples of 3 (and inversely multiples of 6)

1×3 = 3 2×3 = 6 3×3 = 9 4×3 = 12 → 3 5×3 = 15 → 6 6×3 = 18 → 9

Cycle: 3, 6, 9

Multiples of 4 (and inversely multiples of 5)

1×4 = 4 2×4 = 8 3×4 = 12 → 3 4×4 = 16 → 7 5×4 = 20 → 2 6×4 = 24 → 6 7×4 = 28 → 1 8×4 = 32 → 5

Cycle: 4, 8, 3, 7, 2, 6, 1, 5

We obtain 8 sequences arranged in 4 pairs.

Exponents

Sequence 1: 1 Sequence 2: 1, 2, 4, 8, 7, 5 … Sequence 3: 3, 9, 9, 9 … Sequence 4: 4, 7, 1 … Sequence 5: 5, 7, 8, 4, 2, 1 … Sequence 6: 6, 9, 9, 9 … Sequence 7: 7, 4, 1 … Sequence 8: 8, 1 …

Again: 8 sequences in 4 pairs

(8, 1)

(2, 5)

(4, 7)

(3, 6)

Divisions

9 ÷ 2 = 4.5 → 4 + 5 = 9

4.5 ÷ 2 = 2.25 → 2 + 2 + 5 = 9

2.25 ÷ 2 = 1.125 → 1 + 1 + 2 + 5 = 9

And so on, infinitely.

The Number 32

“In the depths of the Chamber of Life, a flower of light began to grow and gain strength to push back the night. At its heart emanated a ray of great power, giving life, light, and strength to all who approached it. Around this flower, they formed a circle with 32 thrones where the Children of Light could immerse themselves in its radiance and fill themselves with eternal light.” — The XV Tablets of Thoth

32 reduces and divides:

32 → 3 + 2 = 5

32 ÷ 2 = 16 → 1 + 6 = 7

16 ÷ 2 = 8

8 ÷ 2 = 4

4 ÷ 2 = 2

2 ÷ 2 = 1

1 ÷ 2 = 0.5 → 0 + 5 = 5

0.5 ÷ 2 = 0.25 → 0 + 2 + 5 = 7

0.25 ÷ 2 = 0.125 → 0 + 1 + 2 + 5 = 8

Cycle: 5, 7, 8, 4, 2, 1 … → This corresponds exactly to Sequence 5 in the exponent cycles.

Irrational Numbers — The Signature of Complexity

Some numbers cannot be expressed as fractions. They escape perfect representation. They are irrational.

π is one of them — irrational and transcendent. Its infinite, non‑repeating decimals reveal something fundamental:

The universe cannot be fully captured by rational structures.

π appears everywhere :

• in circles
• in waves
• in oscillations
• in quantum fields
• in the geometry of space

Its irrationality is not a flaw. It is a signature — the mathematical imprint of the universe’s complexity.

Square Roots — Geometry, Proportion, and Universal Structure

The square root is the inverse of squaring. Simple in appearance, but foundational in geometry, physics, and vibration.

Geometric Proportions

• A square of side 1 has a diagonal of √2.
• A cube of side 1 has an internal diagonal of √3.
• A golden rectangle has a diagonal of φ√2.
• Kepler’s triangle links √2, φ, and √3 in a perfect progression.

These proportions appear in:

• crystals
• molecules
• biological structures
• cosmic patterns

They are universal axes of symmetry.

Square Roots in Physics and Vibrations

Square roots govern:

• oscillations
• frequencies
• wave equations
• quantum fluctuations
• zero‑point energy

At the Planck scale, the vacuum vibrates. Its fluctuations are modeled by square roots — confirming that vibration is the foundation of reality.

Numerical Cycles and Harmonic Resonances

Even the digital roots of square roots form cycles. These cycles resemble harmonic frequencies, suggesting that numbers themselves may encode the resonances of the quantum fields.

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