clifford_a-_pickover_surfing_through_hyperspacebookfi-org

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