So progresses arithmetic subtlety,
the end of which, as is said, is as refined as it is useless.
The intricate fractal shapes we are aiming to draw are based on the algebra and geometry of complex numbers. Complex numbers are really not as complex as you might expect from their name, particularly if we think of them in terms of the underlying two dimensional geometry which they describe. Perhaps it would have been better to call them ‘nature's numbers’. Behind complex numbers is a wonderful synthesis between two dimensional geometry and an elegant arithmetic in which every polynomial equation has a solution. When complex numbers were first dreamed of in the Renaissance, they were treated as an esoteric, almost mystical, concept. This aura of mystery persisted well into the twentieth century – the senior author's aunt Margaret Silcock (née Mumford), who studied mathematics at Girton College Cambridge in 1916, liked to describe them as a ‘delightful fiction’. In fact we still use the term ‘imaginary numbers’ to this day. Modern scientists, however, take complex numbers for granted, FORTRAN makes them a predefined data type, and they are standard toolkit for any electronic engineer. Perhaps the most remarkable fact about complex numbers is that they are absolutely essential to modern physics. In the theory of quantum mechanics, not only can the universe exist probabilistically in two states at once, but the uncertain composite state is constructed by adding the two simple states together with complex coefficients, introducing a complex ‘phase’.
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