[Paper Review] A Combinatorial Bit Bang Leading to Quaternions

Michael Manthey|ArXiv.org|Sep 11, 1998

Logic, programming, and type systems8 references3 citations

TL;DR

This paper proposes a combinatorial framework that derives discrete quaternions—representing 3D spatial structure—from primitive, abstract events through exclusion and co-occurrence rules, starting from a foundational 'Void'. Using Clifford algebras within a homology-cohomology structure, it emergently constructs quaternions and quark-like structures, offering a unified, hierarchical model compatible with quantum mechanics and relativity, with applications in AI and complex systems modeling.

ABSTRACT

This paper describes in detail how (discrete) quaternions - ie. the abstract structure of 3-D space - emerge from, first, the Void, and thence from primitive combinatorial structures, using only the exclusion and co-occurrence of otherwise unspecified events. We show how this computational view supplements and provides an interpretation for the mathematical structures, and derive quark structure. The build-up is emergently hierarchical, compatible with both quantum mechanics and relativity, and can be extended upwards to the macroscopic. The mathematics is that of Clifford algebras emplaced in the homology-cohomology structure pioneered by Kron. Interestingly, the ideas presented here were originally developed by the author to resolve fundamental limitations of existing AI paradigms. As such, the approach can be used for learning, planning, vision, NLP, pattern recognition; and as well, for modelling, simulation, and implementation of complex systems, eg. biological.

Motivation & Objective

• To establish a foundational model of 3D space using only combinatorial rules of event exclusion and co-occurrence.
• To derive discrete quaternions as an emergent structure from a primitive 'Void' without assuming pre-existing geometry or algebra.
• To provide a computational interpretation of mathematical structures in quantum physics and relativity via emergent hierarchy.
• To extend the framework to model complex systems, including biological systems and AI paradigms such as learning, vision, and NLP.
• To demonstrate compatibility of the emergent structure with both quantum mechanics and relativistic principles.

Proposed method

• Starting from a 'Void' of unspecified events, the model applies combinatorial rules to define relationships through exclusion and co-occurrence.
• The framework uses the homology-cohomology structure pioneered by Kron to embed Clifford algebras into the combinatorial foundation.
• Discrete quaternions emerge as a result of hierarchical organization of event combinations, forming a non-continuous, discrete spatial representation.
• The model leverages the algebraic properties of Clifford algebras to generate the quaternionic structure from combinatorial event sets.
• The construction is inherently computational, allowing for simulation and implementation in algorithmic systems.
• The emergent structure naturally supports the formation of quark-like constituents through further combinatorial decomposition.

Experimental results

Research questions

• RQ1 How can 3D spatial structure, represented by quaternions, emerge from purely combinatorial rules of event exclusion and co-occurrence?
• RQ2 What is the role of Clifford algebras and homology-cohomology in grounding discrete geometric structures in a foundational event-based framework?
• RQ3 Can the emergent quaternionic structure be made compatible with both quantum mechanics and relativity?
• RQ4 How does this combinatorial model support higher-level cognitive and computational functions such as learning, vision, and NLP?
• RQ5 What is the relationship between the emergent quaternions and fundamental particle structures, such as quarks?

Key findings

• Discrete quaternions emerge as a natural outcome of combinatorial event structures governed by exclusion and co-occurrence rules.
• The framework embeds Clifford algebras within a homology-cohomology structure, providing a topological foundation for geometric emergence.
• The model produces a hierarchical, discrete representation of 3D space that is compatible with both quantum mechanics and relativity.
• The emergent structure includes quark-like constituents, suggesting a fundamental link between combinatorial event patterns and elementary particle physics.
• The approach provides a computational substrate suitable for modeling complex systems, including biological systems and AI tasks.
• The construction is self-consistent and extends naturally to macroscopic scales, supporting a unified view of physical and computational emergence.