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Quaternions, the first example of a non-commutative algebra, arose as a by-product of Hamilton's failed attempt to construct an "algebra of triples". Hamilton envisaged the quaternions as the "new language" of science and technology, but their place was usurped by vector analysis, an algebraically crude and overtly pragmatic subset of the quaternion algebra. A simple quaternion expression automatically generates Pythagorean quartuples of polynomials, thus yielding an elegant rotation invariant characterization of Pythagorean hodographs in R^3. Quaternions provide compact and intuitive descriptions for rotations in R^3, a fact that has lead to a renewed interest in them for robotics, computer graphics, animation, and related fields. Quaternions also describe rotations in R^4, whose strange properties provide a cautionary tale against extrapolating our geometric intuition from R^2 and R^3 to Euclidean spaces of higher dimension.