A Geometric Construction of Quaternion Algebra: Deriving Anti-commutativity from a Rotating Skew Axis and Cross Product

中文摘要: 标准四元数教材通常以哈密顿公理 i^2=j^2=k^2=ijk=-1 为起点,再指出四元数乘法与向量叉积的关系 uv = -u· v + u× v 。本文反转了这一认知路径:从一条在三维空间中连续运动的斜轴 T(θ) 出发,定义 i = T(θ) ,并通过叉积 j = n× i 、 k = i× j 生成正交三元组。在该构造下,借助标准四元数乘法公式 uv = -u· v + u× v 作为计算工具,反交换性由叉积的反对称性保证,而基元本身由几何运动生成,因此哈密顿公理成为整套几何构造的后验推论。本文给出了严格的正交性证明和反交换性推导,并用 Python 在参数 θ ∈ [0^∘, 330^∘] 的全范围内( θ 每隔 30^∘ 取 12 个采样点,结合 α∈\{0,0.5,1.0\} 三组参数,共 36 个状态)数值验证了所有代数规则。本工作为四元数教学提供了一种几何直观的替代路径。 英文摘要: Standard quaternion textbooks typically start from Hamilton's axioms i^2=j^2=k^2=ijk=-1 and then note the relationship between quaternion multiplication and the vector cross product uv = -u· v + u× v . This paper reverses that cognitive path: starting from a continuously rotating skew axis T(θ) in 3D space, we define i = T(θ) and generate an orthogonal triple via the cross products j = n× i and k = i× j . Under this construction, using the standard quaternion multiplication formula uv = -u· v + u× v as a computational tool, anti-commutativity is guaranteed by the antisymmetry of the cross product, while the basis elements are generated by geometric motion—thus Hamilton's axioms become emergent corollaries of the geometric construction. We provide rigorous proofs of orthogonality and the derivation of anti-commutativity, and numerically verify all algebraic rules across the full range θ ∈ [0^∘, 330^∘] (12 sampling points at 30^∘ intervals, combined with three parameter sets α∈\{0,0.5,1.0\} , totaling 36 states) using Python. This work offers a geometrically intuitive alternative pathway for teaching quaternions. 中文关键词: 四元数;反交换性;叉积;几何构造;教学法 英文关键词: quaternion; anti-commutativity; cross product; geometric construction; pedagogy