Geometry of involutions of quasi-quaternions

This study consists of four chapters. In the first chapter, the historical development of quaternions is discussed, while the second chapter presents the fundamental definitions and algebraic properties of real and quasi-quaternions. In the third chapter, two distinct transformations are defined using quasi-quaternions, each satisfying both the involution and anti-involution axioms. The geometric interpretations of these transformations in the 3-dimensional Euclidean space R^3 are also provided. Additionally, the corresponding matrices of these transformations are derived. The final chapter includes the conclusions and suggestions obtained from the study. The most distinctive feature of quasi-quaternions is the commutativity of their multiplication operation. Owing to this property, quasi-quaternions form a commutative algebra over the field of real numbers, and the two defined transformations satisfy both the involution and anti-involution axioms simultaneously. In each of the two transformations defined on quasi-quaternions, one of the quasi-quaternions is arbitrary, and the other is a unit element. As a result of both transformations, the scalar parts of the arbitrary quasi-quaternions remain invariant in the 4-dimensional Euclidean space R^4, whereas the behavior of the vector parts differs depending on the transformation. In the first transformation, the vector part undergoes a reflection through the origin in R^3, while in the second transformation, it remains unchanged. Therefore, in the second case, the entire arbitrary quasi-quaternion remains invariant in R^4.