Points [vectors] in the plane can be represented using complex numbers, and then can be “multiplied”, i.e. can be given the structure of an
-algebra. Conversely, complex numbers can be viewed as geometric objects, and operations on them given geometric meaning, by the same construction.
Is there an analogue of this for points in 3-space? Hamilton tried to find such an analogue, and was led to the quaternions. There is a (possibly apocryphal) story of how his son asked him every morning: “Well, Papa, can you multiply triplets?” and always got the same answer: “No, I can only add and subtract them”, with a sad shake of the head. What Hamilton realized was that this is indeed not quite possible … unless we embed the triplets in a 4-dimensional algebra (the quaternions)—this is what Hamilton realized that morning and inscribed on Brougham Bridge
“… quaternions appear to exude an air of nineteenth century decay, as a rather unsuccessful species in the struggle-for-life of mathematical ideas. Mathematicians, admittedly, still keep a warm place in their hearts for the remarkable algebraic properties of quaternions but, alas, such enthusiasm means little to the harder-headed physical scientist.” (1986.)
- Wikipedia’s “History of quaternions” and “Quaternion“
- (see also) John Conway and Derek Smith, On Quaternions and Octonions: Their Geometry, Arithmetic, and Symmetry