Chapter 30. Quaternions and the Four Division Algebras

It is clear from the stories of William Rowan Hamilton that quaternions were discovered because he persisted in asking the question “What generalizes the complex numbers.” Therefore, it is appropriate to continue our study of quaternions by asking the question “What do we discover when we attempt to generalize quaternions?” The answers fall into several categories. Some features, such as the algebraic properties of quaternions, have very limited generalizability to other dimensions. Other features, such as the constructability of double-valued parameterizations of Euclidean rotations (the spin representations) generalize to all dimensions. In this chapter we look at the properties that are ...

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