Exploring Quaternions and their Natural Structure.

This Master's thesis (2015) investigates the algebraic and geometric foundations of Quaternions (H), proposing a non-standard framework for Relativistic Kinematics. The work is structured around three main axes: Chapter 1 - Topological Foundations: Starting from the classification of Division Algebras endowed with a non-trivial topology, the work highlights the uniqueness of Quaternions as the most general Connected Locally Compact Division Algebra. It provides a formal construction of H based on these properties rather than the traditional i,j,k definitions. Chapter 2 - Differential Algebra Perspective: Algebraic and geometric properties (projections, rotations) are explored through the lens of Differential Algebra. By treating commutators as algebraic derivations satisfying the Leibniz rule, the work interprets rotation groups as "Differential Galois Groups", providing a novel angle on standard quaternionic automorphisms. Chapter 3 - Quaternionic Möbius Transformations: The core of the thesis examines the Quaternionic Möbius Group. Three fundamental subgroups are identified, which together generate the full group. A specific subgroup isomorphic to the Lorentz Group is studied in detail; these transformations preserve the set of square roots of −1 ("light-like velocities") and exhibit conformal action. The chapter provides "ruler-and-compass" constructions for these transformations and concludes with a relativistic reformulation, with Unit Quaternions rather than the traditional "Four-Velocities", consequently allowing for a trivially intuitive visualization of the "Twin Paradox". An elegant velocity addition formula is provided, expressed in two different ways, and a Compass-and-Straightedge construction is naturally associated with the latter: one can visualize the "Relativistic Velocity Addition" by just "drawing one circle unit Hypersphere and two lines". Errata & Technical Note: Velocity Addition: The "single-piece" formula provided in the text is correct. However, a typo exists in the "alternative formulation" (introduced in the summary/intro at the end of page 7, and again on page 161): the parameter b should be (v−1)/(v+1) instead of (1+v)/(1−v). These are mutually inverse transformations, and the error resulted from a late-stage clerical slip. Typos: As this document is the original, unedited version submitted to the University of Mons in 2015, other minor typos may be present. For instance, a typo was identified on page 30 in the proof of a minor result, though the validity of the conclusion remains unaffected. .Structure & Scope: The original project was intended to have a more specific title and include a "Chapter 4" dedicated to in-depth relativistic implications. Due to time constraints, the work was finalized with three chapters, offering only a "glimpse" of the physics applications in Chapter 3. These topics will be addressed more in detail in the author's forthcoming (2026) monograph. Context: This 2015 thesis builds upon the author's preliminary research (2013-2014) which first introduced the quaternionic velocity addition formula. Together, these works serve as the historical and mathematical foundation for the author’s forthcoming 2026 monograph.