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A hyperbolic quaternion is a mathematical concept introduced by Alexander MacFarlane of Chatham, Ontario in 1900. The idea was dismissed for its failure to conform to associativity of multiplication, but it has a legacy in Minkowski space and as an extension of split-complex numbers. Like the quaternions, it is a vector space over the real numbers of dimension 4.
A linear combination
- q = a + bi + cj + dk
is a hyperbolic quaternion when a, b, c, and d are real numbers and the basis set {1,i,j,k} has these products:
- i j = k = -j i , jk = i = -kj , ki = j = -ik , and ii = +1 = jj = kk.
Though these basis products do not obey associativity, the set
- {1,i,j,k,-1,-i,-j,-k}
forms a quasigroup. One also notes that any subplane of the set M of hyperbolic quaternions that contains the real axis forms a plane of split-complex numbers. If
- q* = a - bi - cj - dk
is the conjugate of q , then the product
- q q* = aa - bb - cc - dd is the quadratic form used in Minkowski space.
Macfarlane's article appeared in the Proceedings of the Royal Society at Edinburgh. In it he establishes a model for hyperbolic space on the hyperboloid
- q q* = 1 .
This model serves as a means to relativize velocity calculations within the limits of the speed of light.
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