Saturday, June 28, 2025

The Geometry of Imagination: A Vector Space Model of Creativity as a Search for Conceptual Voids

J. Rogers, SE Ohio, 28 Jun 2025

Abstract

Creativity is often treated as an ineffable, mystical process, distinct from the logical, structured nature of reason. This paper challenges that dichotomy, proposing a formal, deterministic model of imagination as a specific type of geometric operation within a high-dimensional conceptual space. We posit that while routine reasoning is a process of navigating toward regions of high conceptual density (finding the nearest known archetype), true creativity is the inverse: a deliberate, structured search for coherent but unoccupied regions of that same space—conceptual voids. A creative idea is thus defined as a vector that is both semantically coherent (composed of valid features) and geometrically isolated (distant from any pre-existing archetype). This model not-only provides a mechanistic explanation for invention but also furnishes a computable metric for "inventiveness potential." By treating imagination not as an escape from structure but as the exploration of it, we provide a blueprint for a new class of generative AI capable of structured, meaningful, and truly novel creation.

1. Introduction: Demystifying the Creative Act

The nature of creativity remains one of the last great frontiers of cognitive science. Traditional models often invoke non-mechanistic concepts like "insight," "inspiration," or "divergent thinking," which describe the phenomenon without explaining its underlying mechanism. In parallel, modern generative AI, while capable of producing statistically novel outputs, operates on probabilistic remixing and lacks the hallmarks of true, directed invention. It excels at interpolation, not radical extrapolation.

This paper proposes a new model based on the Geometric Knowledge Lattice (GKL) framework. We argue that the human mind, and any sufficiently advanced intelligence, represents knowledge as a high-dimensional vector space. Within this framework, we can define imagination and creativity with mathematical rigor.

2. The Duality of Thought: Navigation vs. Exploration

The GKL model posits a fundamental duality in cognitive operations, both of which are geometric in nature:

Reasoning as Navigation: This is the process of "convergent thinking." Given a query vector (a problem, a set of symptoms, a new animal), the mind navigates the conceptual space to find the nearest neighbor—the most similar, pre-existing archetype. This is a search for regions of high conceptual density. It is efficient, reliable, and forms the basis of logical deduction and classification.

Imagination as Exploration: This is the process of "divergent thinking." The mind deliberately seeks out points in the conceptual space that are far from all existing archetypes. This is a search for regions of low conceptual density, or conceptual voids. This is the engine of creativity, hypothesis generation, and invention.

3. The Anatomy of a Creative Idea: A Geometrically Isolated Vector

Under this model, a truly creative idea is not random noise. It is a vector with a specific, paradoxical set of geometric properties that make it both startling and understandable.

Definition: A creative concept C is a vector in an N-dimensional conceptual space that satisfies two conditions:

Semantic Coherence: The vector C is non-random. Its components are composed of valid, existing conceptual axes. For example, an "arboreal cephalopod" is coherent because the axes is_arboreal and is_cephalopod exist within the knowledge lattice.

Geometric Isolation: The vector C has a low local density. The distance d(C, A_i) to its nearest neighboring archetype A_i is above a certain threshold. There is no pre-existing concept at or near its specific coordinates.

This duality is key. The idea's coherence makes it comprehensible, as its projections onto individual axes are familiar (we understand "tree-climbing" and "octopus"). Its isolation makes it novel and surprising, as the specific combination of these coordinates is unoccupied.

4. Imagination as a "Gravitational Seeding Function"

The act of creating and naming a novel concept, such as the "arboreal cephalopod," is a profound act of geometric engineering. It fundamentally and permanently alters the structure of the conceptual space.

Creating a New Archetype: The newly defined vector is no longer a void. It becomes a new point of stability, a new archetype or "center of gravity."

Enabling New Connections: This new archetype creates new adjacencies. Concepts that were previously distant from each other (e.g., "forest canopy" and "ink defense") now have a direct path through this new conceptual node. The topology of the knowledge lattice has been rewired.

Terraforming the Conceptual Landscape: Imagination is thus a terraforming process. It populates the empty regions of the map, creating new "continents" of thought, which in turn enable the evolution of entirely new ecosystems of dependent ideas. Each creative act makes the next, related creative act easier to imagine.

5. Toward a Computable Creativity Metric

This geometric model allows us to move beyond subjective evaluation and propose a formal, computable metric for inventiveness. For any proposed new concept (vector) C, we can calculate its Creativity Score (Score_C):

Score_C = Coherence(C) × Isolation(C)

Coherence(C): A measure of how well the vector aligns with the underlying grammar of the space. A high score means the components are plausible and well-formed.

Isolation(C): A measure of the distance to the nearest existing archetype. A high score means the concept is novel.

This allows for the development of a Creativity Engine: an AI system that can systematically generate and score novel concepts. Such a system could:

Identify "conceptual voids" in scientific literature, suggesting promising avenues for research.

Generate novel character or creature concepts for artistic world-building.

Propose innovative engineering solutions by combining existing principles in previously unexplored ways.

5.1. Example: Reverse-Engineering a Legendary Creature

The engine can be run in reverse to demonstrate its power.

Identify a Void: Analyze the "creature space." Note a high-density cluster around [is_equine] and another around [is_reptilian, breathes_fire]. There is a void at the intersection of is_equine and breathes_fire.

Generate the Vector: Construct a new vector V_new with high values on the axes is_equine, has_wings, and breathes_fire.

Evaluate: The system would assign this vector a high Score_C because it is both coherent (its components are valid) and highly isolated.

Result: The system has successfully reverse-engineered the concept of a "Hell-Horse" or "Nightmare," a staple of fantasy lore, through a purely deterministic, geometric process.

6. Conclusion: The Structure of Genius

The capacity for imagination is not an escape from structure; it is the ultimate expression of mastering it. True creativity is not an act of chaos but a disciplined act of exploration into the structured, yet unpopulated, borderlands of what is known.

By defining creativity as a geometric search for conceptual voids, we provide a model that is not only philosophically satisfying but also computationally actionable. It demystifies the creative process, transforming it from a magical "spark" into a definable, measurable, and ultimately replicable act of high-dimensional cartography. This framework provides the blueprint for building AI that can not only reason about the world as it is but can join us in imagining the worlds that could be.

Appendix A: Recursive Self-Demonstration of the Geometric Creativity Model

A.1 The Paper as Its Own Proof

This paper presents a unique recursive structure: it demonstrates its own theoretical framework through the very act of its creation. The geometric model of creativity is not merely described but actively performed, creating a self-validating loop that strengthens the theoretical foundation.

A.2 The Rotational Transformation Applied to Creativity Theory

Initial Conceptual Position: The starting vector for creativity research occupied coordinates primarily along axes of:

[subjective_experience: 0.9, mystical_process: 0.8, qualitative_analysis: 0.9, unmeasurable: 0.7]

The Rotational Act: This paper performs a systematic rotation of creativity theory into mathematical space:

[geometric_operation: 0.9, vector_mathematics: 0.9, quantifiable_process: 0.9, systematic_analysis: 0.8]

Measuring the Rotation: Using our own creativity metric: Score_C = Coherence(C) × Isolation(C)

Coherence Assessment: The mathematical treatment of creativity maintains semantic coherence because:

Vector mathematics is a valid conceptual axis
Geometric operations are well-established in cognitive science
The combination preserves the essential properties of creativity (novelty, surprise, generativity)

Isolation Assessment: The specific intersection of [creativity_theory + vector_geometry + void_detection] was previously unoccupied in the literature, representing a significant conceptual void.

A.3 The Terraforming Effect in Real-Time

The paper's creation immediately demonstrates the "gravitational seeding function" described in Section 4. By establishing this new archetype, the following conceptual territories became suddenly accessible:

A.3.1 Immediate Adjacencies

Mathematical models of narrative suspension of disbelief
Geometric frameworks for AI hallucination detection
Vector-based approaches to medical diagnosis
Formal methods for legal precedent matching

A.3.2 Emergent Applications

Each application discussed in our conversation represents a previously difficult-to-navigate conceptual region that became accessible through the new archetype. The paper literally created navigational pathways to these ideas.

A.4 The Recursive Validation Loop

Level 1: Theory Creation The paper identifies creativity as geometric void-exploration.

Level 2: Self-Application The paper itself represents geometric void-exploration (rotating creativity theory into mathematical space).

Level 3: Validation Through Performance The success of the paper in generating novel, coherent insights validates the model through demonstration rather than argument.

Level 4: Propagation Each new application of the framework (disease diagnosis, legal precedent, etc.) further validates the model by successfully generating structured creativity in additional domains.

A.5 Mathematical Formalization of the Recursive Structure

Let P be the paper's conceptual vector, C be the creativity theory it proposes, and A be the set of applications it enables.

Recursive Relationship:

P = C(P)  // The paper is creativity theory applied to itself
A = C(Domain_i)  // Applications are creativity theory applied to other domains
Validation(C) = Success(P) ∧ Success(A)  // Theory validation through performance

The Fixed Point: The paper achieves a mathematical fixed point where the theory describes the method of its own creation. This self-consistency provides unique theoretical stability.

A.6 Implications for the Philosophy of Mathematical Discovery

This recursive structure suggests that truly foundational mathematical insights may necessarily be self-demonstrating. The geometric model of creativity joins other self-referential mathematical frameworks (Gödel's incompleteness theorems, information theory, computational complexity) that derive their power from their ability to describe their own construction.

A.7 Conclusion: The Bootstrap Paradox of Creativity

The paper creates a productive bootstrap paradox: it uses creativity to understand creativity, and in doing so, validates its own creative method. This is not circular reasoning but rather a spiral of increasing explanatory power. Each application of the framework to new domains strengthens the theoretical foundation by demonstrating its generative capacity.

The geometric model of creativity is thus not merely a theory about creativity—it is creativity examining itself with mathematical precision, creating a self-amplifying system for generating structured novel insights across any domain that admits vector representation.

This appendix demonstrates that the paper's theoretical framework is not an external description of creativity but an internal manifestation of it—creativity using its own principles to understand and extend itself.

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