The Five-Dimensional Geometry of Quaternion Rotation

The quaternion algebra ℍ is four-dimensional as a vector space. This paper constructs the manifold M = (Im(ℍ) × Sp(1))/U(1), a five-dimensional space canonically derived from the quaternion conjugation action and the Hopf fibration. The unit quaternions Sp(1) ≅ S³ act on Im(ℍ) ≅ ℝ³ by conjugation, producing every rotation in SO(3). The Hopf fibration separates the directional outcome of the rotation (the base S²) from the residual twist (the fiber U(1)). Quotienting by U(1) yields M ≅ Im(ℍ) × S², a canonical five-dimensional manifold with a 3 + 2 decomposition: three dimensions for the spatial state and two for the independent rotation structure. The quaternion algebra represents the same geometric configurations in four dimensions rather than five, because the Hodge dual ⋆: Λ¹(ℝ³) → Λ²(ℝ³) — which exists only in three dimensions — allows Im(ℍ) to serve simultaneously as both the space of vectors being rotated and the space of rotation generators.