When the order of grouping matters.
Pentarnion research investigates five-dimensional nonassociative algebraic structures and the differential equations they generate. Unlike associative multiplication, changing the parentheses in a product can change its value—and, in an algebra-valued differential equation, the resulting dynamics.
With Dr Susanne Pumplün at the University of Nottingham, I am developing a mathematical programme connecting algebraic structure, geometry, elliptic integrability and nonlinear dynamics. Three research preprints are publicly available and submitted for publication.
From Hodgkin–Huxley generalisation to algebra.
The programme arose from investigations of algebra-valued formulations of polynomial differential equations motivated by generalising Hodgkin–Huxley modelling towards GTG. Those questions developed into an independent mathematical research direction.
The connection is one of scientific origin. Pentarnion algebra is not presented here as an implemented component of GTG’s governing equations.
Structure, geometry and dynamics.
The work includes exact algebraic and dynamical results alongside computational investigations. In the bracketing-dependent examples, exact identities describe the associator contribution while numerical diagnostics support the reported chaotic behaviour; those are different kinds of evidence.
The three preprints.
From Quaternions to Pentarnions I: Algebra, geometry, and a spin model
arXiv:2610.31485 · Preprint · Submitted for publication
Nonassociative algebra, geometry and a classical spin-model application. Public preprint; submitted for publication.
From Quaternions to Pentarnions II: quadratic differential equations
arXiv:2610.34201 · Preprint · Submitted for publication
An investigation of quadratic differential equations in the quaternion-to-Pentarnion research programme.
A Five-Dimensional Nonassociative Algebra: Elliptic Integrability and Bracketing-Dependent Dynamics
arXiv:2610.34202 · Preprint · Submitted for publication
Five-dimensional nonassociative algebra, exact elliptic solutions and the effect of parenthesisation on nonlinear dynamics.
Pentarnion Studio.
I developed Pentarnion Studio to make algebraic investigations computationally accessible. The research environment combines a MATLAB-style command window, script editing, native numerical modules and dockable scientific figures, retaining operand order and parentheses in nonassociative multiplication.
The repository includes source, examples, language specifications and build instructions. Supported algebra contexts and numerical operations have explicit limits described in the documentation. The software implements research methods; its availability is not itself proof of a mathematical result.