3D & 4D Quaternion Groups: Exploring Reflections, Rotations, Integrals

This paper explores the properties of quaternions in three and four dimensions, focusing on their application in rotations and reflections. It examines the formation of chiral and achiral groups in 3D, further divided into polyhedral and dihedral structures. The analysis extends to 4D quaternions, addressing their complexity by utilizing complex notations and the Hurwitz integral. The report details the structural formation from mathematical expressions, the outcomes of rotations and reflections, and the application of theorems such as the Fundamental Theorem of Algebra and spherical excess theorems. The study uses figures to illustrate triangulation and areas on a sphere, offering a comprehensive overview of quaternion behavior in various dimensions.