clifford_a-_pickover_surfing_through_hyperspacebookfi-org
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appendix d<br />
quarternions<br />
The invention of quaternions must be regarded as a most remarkable<br />
feat of human ingenuity.<br />
—Oliver Heaviside<br />
It is as unfair to call a vector a quaternion as to call a man a quadruped.<br />
—Oliver Heaviside<br />
Some of you may be familiar with the concept of "complex numbers" that have a<br />
real and imaginary part of the form a + hi, where z = v — 1. (If you've never heard<br />
of complex numbers, feel free to skip this section and simply enjoy the pretty fractal<br />
image.) When these 2-D numbers were invented, many people were not sure of<br />
their validity. What real-world significance could such imaginary numbers have?<br />
However, it didn't take long for scientists to discover many applications for these<br />
numbers—from hydrodynamics to electricity.<br />
Quaternions are similar to complex numbers but of the form a + hi + cj.+ dk<br />
with one real and three imaginary parts. 1 The addition of these 4-D numbers is<br />
fairly easy, but the multiplication is more complicated. How could such numbers<br />
have practical application? It turns out that quaternions can be used to describe the<br />
orbits of pairs of pendulums and to specify rotations in computer graphics.<br />
Quaternions are an extension of the complex plane and were discovered in<br />
1843 by William Hamilton while attempting to define 3-D multiplications.<br />
Hamilton was a brilliant Irish mathematician whose genius for languages was evident<br />
at an early age. He could read at three—by four he had started on Greek,<br />
Latin, and Hebrew—and by ten had become familiar with Sanskrit. By age seventeen,<br />
his mathematical prowess became evident.<br />
In 1843, during a flash of inspiration while walking with his wife, Hamilton<br />
realized that it took four (not three) numbers to accomplish a 3-D transformation<br />
of one vector into another. In that instant, Hamilton saw that one number was<br />
needed to adjust the length, another to specify the amount of rotation, and two<br />
more to specify the plane in which rotation takes place. This physical insight led<br />
Hamilton to study hypercomplex numbers (or quaternions) with four components,<br />
sometimes written with the form: Q = a 0 + a l i + a-j + a 3 k where the as are<br />
ordinary real numbers, and i,j, and k are each an imaginary unit vector pointing in<br />
188