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The Kakeya Conjecture, Electromagnetic Waves, and the Nature of Consciousness: A Cross-Disciplinary Mathematical, Physical and Philosophical Inquiry

发布时间：

2026-08-03

The Kakeya Conjecture, Electromagnetic Waves, and the Nature of Consciousness: A Cross-Disciplinary Mathematical, Physical and Philosophical Inquiry

Sun Zuodong

Abstract

As a core puzzle in geometric measure theory, the Kakeya conjecture reveals a profound topological property: a set of directional rays can fully preserve complete spatial dimensionality even when their enclosing volume approaches zero. The three-dimensional Kakeya conjecture has now been rigorously proven, laying a new theoretical foundation for spatial topology and harmonic analysis. Electromagnetic waves, as physical fields that permeate all space and propagate with conjugated energy and information, exhibit profound mathematical isomorphism with the geometric model of Kakeya sets. Neurons generate and radiate electromagnetic waves outward, and consciousness is fundamentally electromagnetic waves. From a triple perspective of mathematical geometry, electromagnetic physics and neural life science, this paper draws analogical reasoning using the geometric characteristics of Kakeya sets to construct a deductive framework linking Kakeya space, electromagnetic field propagation and conscious information fields. It should be clarified that this paper does not take the proven conclusion of the three-dimensional Kakeya conjecture as its core argument. While strictly defining the boundaries of scientific reasoning, it sorts out primitive perceptions of intangible spatial field states in Eastern and Western traditional cultures, distinguishing their cognitive origins from modern physical models rather than equating them to physical entities. This work aims to offer insights for paradigm reconstruction in fundamental scientific research.

Reading Note

This paper involves analogical speculation across multiple disciplines. Mathematical models, physical field theories and humanistic cognition belong to distinct discourse systems. Readers are advised to read the full text rather than interpret fragmented excerpts out of context.

I. Mathematical Core of the Kakeya Conjecture: Infinite Directionality Within Zero Volume

The Kakeya conjecture originates from the "rotating needle problem" proposed by Japanese mathematician Sōichi Kakeya. Its essence in three-dimensional space can be summarized as follows: there exists a set capable of containing unit line segments pointing in all directions; the region’s volume can shrink infinitely close to zero yet still fully encode all directional information of three-dimensional space. To put it simply, when an ideal infinitely thin needle with no thickness rotates to sweep through every possible spatial orientation, the area it traces can be compressed into an extremely tiny region while still covering all spatial directions.

The three-dimensional Kakeya conjecture remained unsolved for a long time. Wang Hong and collaborators integrated cutting-edge tools from harmonic analysis, geometric measure theory and combinatorial mathematics to deliver a rigorous proof. This breakthrough bridges deep theoretical channels between geometric topology and harmonic analysis, greatly expanding humanity’s understanding of the laws governing directional distribution in space.

The most striking feature of this model is not its counterintuitive volumetric property, but its capacity for lossless compression and holographic representation of directional information. When electromagnetic waves propagate through vacuum or media, the distribution of their polarization and wave vector directions in phase space resembles the superposition of countless directional rays. Communication signals and cosmic background radiation interpenetrate physical space without occupying tangible volume — a trait highly consistent with the geometric configuration of Kakeya sets, which "encompass all manifestations within an empty void". The two form an illuminating mathematical mapping: Kakeya sets characterize the topological completeness of directional space, while electromagnetic waves realize the dynamic diffusion of energy and information across space.

II. Electromagnetic Fields: The Physical Substrate for Diffusion and Integration of Biological Information

As objectively real physical entities with no mass, no fixed volume and governed by the superposition principle, electromagnetic fields serve as an ideal medium for signal interaction within living organisms. The human brain contains approximately 86 billion neurons that sustain continuous electrical activity, radiating extremely low-frequency biological electromagnetic fields outward.

Neuronal signaling relies not only on chemical and electrical transmission across synapses; synchronized oscillations of neuronal clusters emit electromagnetic waves that radiate beyond the physical neurons, forming a globally distributed biological information field surrounding neural tissue. Biological electromagnetic waves can detach from neurons and diffuse outward to fill the surrounding space. This mechanism aligns closely with the core tenet of the Conscious Electromagnetic Information (CEMI) Field Theory: unified conscious experience arises from a global integrating endogenous electromagnetic field generated by neurons, which unifies massive distributed neural signals. Biological electromagnetic fields may diffuse away from their original neuronal clusters, analogous to cellular base station signals propagating through open space.

Many Eastern and Western traditional cultures contain descriptions such as "vital energy field" or "spiritual essence". Folk sayings like "divine presence lingers three feet above the ground" also exist. Such notions are empirical, intuitive imaginations formed by ancient peoples without modern observational tools for physics. They can only be regarded as early primitive human perceptions of diffuse intangible spatial fields, and possess no objective physical substantiality, nor can they be directly equated to modern biological electromagnetic fields.

III. Model Construction and Boundaries of Scientific Deduction on the Nature of Consciousness

Based on the geometric properties of the Kakeya conjecture and physical laws of electromagnetic fields, a progressive deductive chain is established as follows:

1. Physical driver: Sustained electrical activity of neuronal clusters radiates characteristic beams of biological electromagnetic waves.

2. Field integration: Massive electromagnetic beams superimpose and interfere within the cranial cavity and adjacent space, forming a localized information field carrying global signatures of biological activity. The spatial distribution and information integration rules of this field can be modeled using the coverage theory of Kakeya sets.

Three rigid boundaries for scientific reasoning must be observed:

1. Hierarchical distinction: The Kakeya conjecture is a purely mathematical construct, while electromagnetic waves are physical entities. They serve only as heuristic analogies and cannot be directly equated.

2. Empirical limits: Empirical evidence confirms neurons radiate outward electromagnetic waves, yet sufficient proof is lacking to confirm that consciousness-bearing electromagnetic fields can persist independently of biological substrates for extended periods. After an organism perishes, endogenous electromagnetic fields lose their energy source, rapidly attenuate and disperse, and cannot retain complete information long-term.

3. Paradigm separation: Concepts such as "soul" or "spiritual consciousness" in traditional culture belong to humanistic and spiritual narratives. This paper only conducts speculative reasoning at the level of mathematical analogy. It does not claim that folk traditional concepts correspond to tangible physical entities, nor equate them with electromagnetic waves of specific frequency bands. The two are merely parallel descriptions of the same cognitive object within different discourse systems.

IV. Implications for Research Paradigms and Tool Innovation from an Interdisciplinary Perspective

Fundamental scientific research often faces a fork in the road: patching existing paradigms or confronting foundational unsolved problems. The operating mechanism of ion channels and the fundamental nature of consciousness are two longstanding core unresolved puzzles in life sciences. Against this backdrop, research paths that dare to confront foundational questions and pursue original exploration are particularly valuable. Wang Hong’s approach of tackling a century-old mathematical conjecture and persisting in original innovation offers an inspirational scholarly model for fundamental researchers.

Introducing the Kakeya conjecture as a geometric analytical tool can advance theoretical research on the spatial distribution laws of neural electromagnetic fields and encoding mechanisms of biological information fields. It can also provide geometric references for spatial dynamics simulation of artificial bionic transmembrane channels, guiding design work in synthetic biology.

Conclusion

The extreme geometric abstraction of the Kakeya conjecture, paired with the tangible, interconnected physical reality of electromagnetic waves, together form a compelling frame of reference for decoding consciousness — the ultimate grand puzzle of life science. Neurons generate and radiate outward electromagnetic waves, and consciousness is fundamentally electromagnetic waves. The biological electromagnetic information field emitted by neurons constitutes the critical physical layer bridging microscale neuronal electrical activity and macroscale subjective conscious experience. The true value of interdisciplinary research lies not in superficial grafting of terminology, but in rigorous analogical reasoning with well-defined boundaries that inspire novel ways of posing questions about age-old problems. On the journey to uncover the essence of consciousness, only deep integration of multi-disciplinary logic, restrained and prudent theoretical deduction, and unwavering commitment to empirical testing can reliably guide humanity toward truth.