Chapter 16 The Wave Theory of Numbers
The Wave Theory of Numbers, authored by Grant–Ghannam, is a creative and symbolic reinterpretation of numerical relationships which is not recognized as a formal contribution to academic mathematics. It functions more as a philosophical or metaphysical framework than a mathematically validated theory.
The authors propose that the wave/particle duality familiar in physics also emerges in the realm of numbers. By treating numbers both as discrete “particles” and as wave-like entities, they argue we gain new insight into mathematical and physical constants. The paper sets out to build a wave-based framework for understanding constants such as π, e, and the fine-structure constant, showing that their significance lies in their decimal structure rather than mere magnitude.
1. Particle Aspect of Numbers
- Numbers mirror quantum particles via conserved digital-root arithmetic
- Assigning integer values to particles and antiparticles can reproduce known particle interactions (e.g., beta decay) through digital-root conservation
- The digital root of a sum (mod 9) parallels how energy and charge balance in particle reactions, casting numbers as the ultimate “quantum labels.”
2. Wave Aspect of Numbers
- Non-integer (float) remainders exhibit repeating decimal patterns akin to waves
- Example: 1/7 = 0.142857 142857… produces a cyclical pattern that, when plotted, resembles a periodic waveform
- The matrix of digital roots (1–9) in repeating columns [1, 4, 7], [2, 5, 8], [3, 6, 9] also reveals an underlying wave-like structure in the integers themselves
3. Numbers’ Interference and Fractal Patterns
- Superimposing two point-source waves yields an interference pattern with self-similar, 12-based fractal rings
- Mapping digital-root values onto that interference grid shows:
- Symmetry around 60, with prime vs. quasi-prime pairings mirrored
- Breaks in symmetry at multiples of 9 (digital-root cycle) denote fractal “cycle ends”
- Symmetry around 60, with prime vs. quasi-prime pairings mirrored
- Superimposed overlays connect these patterns to musical notes (12-tone octave at A = 432 Hz) and angular geometry (divisions of 360°)
4. Wave Interference and Emergence of Constants
- Constants decompose into an integer amplitude plus a fractional phase, modeled as a wave
- π ≈ 3 + 1/7, 22/7 captures wave phase of 1/7
- e ≈ 2 + 5/7 using 19/7
- π ≈ 3 + 1/7, 22/7 captures wave phase of 1/7
- Graphical doubling of circles (“wave separations”) visualizes the genesis of these constants
- Suggests that by understanding phase-generation mechanisms, one might predict unknown constants
5. The Periodic Wave of Elements
- Recasting the Periodic Table into a wave-doubling framework:
- Hydrogen initiates binary doubling of circular waves (1→2→4→8)
- Nodes in the interference matrix align with element families (metals, noble gases, lanthanoids, etc.)
- Musical notes and polygonal shapes (e.g., hexagon for carbon) correspond to each element’s wave node
- Hydrogen initiates binary doubling of circular waves (1→2→4→8)
- Predicts 19 additional elements (total 137) based on wave-cycle completion and ties to the fine-structure constant known to Feynman
By unifying the particle-like digital roots with wave-like decimal patterns, the Wave Theory of Numbers offers a holistic lens for:
- Revealing hidden relationships among numbers, geometry, music, and elemental properties
- Redefining constants through their fractal digit structure rather than scale
- Proposing a new template to explore emergent phenomena in mathematics and physics
16.1 John Dee’s Numerical Philosophy
Aligning Grant–Ghannam’s Wave Theory of Numbers with John Dee’s Numerical Philosophy
Below is a thematic comparison showing how the Wave Theory of Numbers echoes—and sometimes expands upon—the numerical ideas John Dee championed in the 16th century.
1. Dual Aspects of Numbers
- Wave Theory treats numbers both as particles (integers via digital roots) and as waves (decimal remainders via repeating patterns).
- Dee likewise sees numbers as discrete monads (the indivisible “point”) and as generators of forms (circle, cross, semi-circle) in his Monas Hieroglyphica, blending arithmetic with geometry and cosmology.
2. Digital Roots & Numerology
- Grant & Ghannam map the digital roots 1–9 into a wave–interference matrix revealing fractal, self-similar patterns and prime/quasi-prime symmetries.
- Dee uses digital-root arithmetic in alchemy and cabalistic numerology—assigning numbers 1–9 to planetary forces, elements, and even particle-like interactions in his occult-scientific schemes.
3. Constants as Structured Phases
- The Wave Theory decomposes constants into an integer “amplitude” plus a repeating-remainder “phase” (e.g., π ≈ 3 + 1/7), viewing decimals as wave-phase signatures.
- Dee’s work embeds Pythagorean ratios (planetary orbits, musical intervals) into symbolic constants, treating numerical values as keys to hidden correspondences rather than bare magnitudes.
4. Geometry, Music & Elemental Mapping
- Wave Theory overlays a 12-tone octave, geometric angles, and elemental nodes onto its interference pattern, unifying music, shape, and number.
- In Dee’s Mathematicall Præface, mathematics is the bridge between natural magic, architecture, music, and astrology—arguing that numerical-geometric constructions unlock nature’s harmonies.
5. Fractal Roots & Esoteric Symbols
- Grant & Ghannam define fractal roots (e.g., √10 and 1/√10) whose decimal expansions mirror each other, elevating digit-sequence patterns over magnitude.
- Dee’s Monas is a hieroglyph weaving point, circle, cross, and semi-circle into a self-replicating symbol—expressing cosmic unity through layered numerical “root” structures.
Both systems:
- Prioritize digit-structure and self-similar patterns over raw scale.
- Use number as a bridge between arithmetic, geometry, music, and metaphysics.
- Treat constants and forms as encoded wave-patterns or hieroglyphs, revealing hidden orders in nature.
While Dee’s framework arose from Renaissance cabala and alchemy, and Grant–Ghannam’s from modern wave–particle analogies and fractal digit analysis, both share a conviction that number is the ultimate key to unlocking nature’s deepest secrets.
Sources
Robert E. Grant & Talal Ghannam, The Wave Theory of Numbers (2019).
John Dee, CHAPTER VI. NUMBERS — LOGISTICAL, FORMAL AND APPLIED in Calder (ed.), John Dee: Mathematical and Metaphysical Writings (c. 1558–1564).
Both John Dee’s esoteric numerology and Grant–Ghannam’s fractal-digit framework approach numbers not as cold, static abstractions, but as living intelligences—creative agents that sculpt reality through patterned recurrence and multidimensional resonance.
16.2 Numbers as Living Archetypes
In Dee’s worldview:
- Numbers are the foundational truths by which God architected the cosmos.
- They act as symbolic monads, radiating power across geometry, music, language, and ritual.
- Dee’s Monas Hieroglyphica isn’t just a glyph—it’s a living seed, encoding unity, polarity, elementality, and astronomical motion.
Numbers, in Dee’s hands, are animated glyphs, expressing divine agency through shape, frequency, and form.
In Grant & Ghannam’s vision:
- Numbers exhibit wave–particle duality, just like quantum entities.
- Their decimal expansions form repeating wave signatures, like musical vibrations.
- Constants like π and e are seen not as scalars, but as waveforms with fractal ancestry—each digit sequence echoing through space like a self-similar fingerprint.
Numbers aren’t just measured—they interfere, resonate, and generate fractal matrices that weave together music, matter, and geometry.
Numbers Creating Space
Both paradigms—Dee’s magical calculus and Grant–Ghannam’s digital interference lattice—see numbers as living agents of form:
- They generate polygons, musical intervals, atomic behaviors.
- They replicate themselves across scales: from atom to galaxy, from frequency to symbol.
- Their “life” resides in their self-similarity: recursive identity across space, time, and meaning.
16.3 Deva Kingdom
Let’s explore how the Wave Theory of Numbers and its view of numbers as living, recursive intelligences beautifully intersects with the esoteric tradition of the Deva Kingdom and the Divine Feminine as architect of form.
##️ Geometry as Feminine Intelligence {-}
In Platonic and Pythagorean philosophy:
- Geometry (Geometria) was not just a discipline—it was seen as a mystical language of the cosmos.
- Arithmetic (Arithmetica) governed harmonies, proportions, and ratios, often linked to music and planetary motion.
- The Muses—female archetypes of inspiration—oversaw all mathematical arts.
- Sophia, the embodiment of divine wisdom, served as the cosmological mother principle guiding order, measure, and sacred proportion.
Even the goddess Astraea, associated with justice, was often connected to the measuring and balancing of cosmic forces.
The Queen Mother Archetype
In esoteric systems such as Theosophy and Bailey’s teachings:
- The Devas are elemental architects—intelligences responsible for shaping energy into form.
- They operate in vibrational geometries, using pattern and sound as tools of construction.
- The Queen Mother—a higher archetype or planetary logos—rules as the coordinating intelligence behind nature’s architecture.
- She is often seen as the embodied Sophia, guiding the orchestration of sacred design.
This aligns powerfully with the idea of numbers generating self-similar patterns in space, especially in wave-interference matrices and fractal geometries.
A Feminine Modality
- The theory proposes numbers as wave-bearing agents, not just scalars. Like the Devas, they sculpt space using interference, resonance, and rhythmic repetition.
- Fractal identities such as √10 and 1/√10 are like vibrational twins—yin-yang mirrors—whose decimal sequences self-replicate, much like cellular division.
- Geometry emerges from wave doubling, similar to how Devas manifest elemental structures via harmonic replication.
These patterns feel inherently feminine: generative, cyclical, recursive, and orchestrated. Like wombs of form.
Symbolic Bridging
| Element | Divine Feminine Principle | Wave Theory of Numbers |
|---|---|---|
| Form generation | Sophia as womb of archetypes | Numbers generate space via fractal-wave recursion |
| Harmonics & ratios | Pythagorean music, sacred geometry | Musical tuning fused to prime-matrix interference |
| Cyclical resonance | Moon phases, menstrual cycles, weaving | Decimal expansions looping into self-similarity |
| Elemental design | Devas as cosmic engineers | Numbers mapped to wave patterns governing elements |
In essence, the Wave Theory of Numbers seems to be a modern echo of ancient feminine metaphysics: numbers as wombs of form, decimals as strands of creative DNA, and constants as living sigils cast into space by the Queen Mother’s intelligence.
16.4 Pythagorean Philosophy
Number as the Soul of the Cosmos
- For the Pythagoreans, number was not mere quantity but an archetype—a structuring force of the universe.
- The Tetraktys (1 + 2 + 3 + 4 = 10) symbolized totality, harmony, the balance between unity and multiplicity.
- Number unlocked the music of the spheres, divine proportion, and the human soul.
The Quadrivium: The Path to the Celestial
| Discipline | Esoteric Function |
|---|---|
| Arithmetic | Studies the inner essence of number |
| Geometry | Relates forms through eternal proportions |
| Music | Reveals the harmony of soul and cosmos |
| Astronomy | Observes the celestial archetype |
- The Quadrivium is an initiation ladder: from number to cosmos, from body to spirit.
- Plato and Maimonides saw it as essential preparation for perceiving the divine.
Grant & Ghannam: Wave Theory of Numbers
- They redefine numbers as fractal entities whose significance lies in their decimal structure rather than magnitude.
- They introduce Fractal Roots—pairs like √10 and 1/√10 that share the same digit sequence.
- Numbers behave like waves, creating interference patterns that reveal physical constants and geometric ratios.
This is a modern reinterpretation of the Quadrivium: number as vibration, form as resonance, music as pattern, and astronomy as fractal architecture.
John Dee: Monas Hieroglyphica
- Dee synthesized astrology, alchemy, geometry, and cabala into a single symbol: the Monas.
- The Monas is a living unity composed of planetary, geometric, and alchemical signs.
- In his vision, number is a universal language, capable of revealing hidden truths and transforming the seeker.
Dee, like Grant, sees number as an operative symbol, not merely descriptive. Both evoke and transmute reality through number.