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QUARTERNIONS 189
Figure D.I
author.)
A 2-D slice of a 4-D quaternion Julia set. (Computer rendering by the
three mutually perpendicular directions of space, in a simple extension of ordinary
complex numbers of 2-D space. Although it was difficult to visualize quaternions,
Hamilton found a way to use them in electrical circuit theory. Oliver Heaviside, a
great Victorian-age genius, remarked: "It is impossible to think in quaternions—
you can only pretend to do it."
Today, quaternions are everywhere in science. They are used to describe the
dynamics of motion in three-space. The space shuttle's flight software uses quaternions
in its computations for guidance, navigation, and flight control for reasons
of compactness, speed, and avoidance of singularities. Quaternions are used by
protein chemists for spatially manipulating models of protein structure. Ted
Kaczynski, the Unabomber, spoke of quaternions fondly throughout his highly
theoretical mathematical journal articles. Quaternion representations are so complicated
that it is useful to develop methodologies to aid in their display. Such
methods reveal an exotic visual universe of forms. In particular, I enjoy image processing
of the beautiful and intricate structures resulting from the iteration
(repeated application) of quaternion equations. Figure D.I is actually a 2-D slice
of a resultant 4-D object called a quaternion Julia set. This slice has a fractional
dimension. For details on these 4-D fractal shapes, see note 1 to Appendix D.