What question did this study set out to answer?

The aim is to explore the five-dimensional structure arising from quaternion rotations and their geometrical implications.

April 3, 2026Open Access

The Five-Dimensional Geometry of Quaternion Rotation

Key Points

The aim is to explore the five-dimensional structure arising from quaternion rotations and their geometrical implications.
Constructed the manifold M from quaternion conjugation and Hopf fibration
Analyzed the action of unit quaternions on the imaginary part of quaternion algebra
Derived relationships between spatial dimensions and rotation structures
Demonstrated M as a canonical five-dimensional manifold with a 3 + 2 decomposition
Identified the separation of rotation outcomes and residual twists
Showed that quaternion algebra depicts geometric configurations in both four and five dimensions

Abstract

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.

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