Fractals: A Very Short Introduction (Very Short Introductions) by Falconer Kenneth

Author:Falconer, Kenneth [Falconer, Kenneth]
Language: eng
Format: mobi, epub
Publisher: Oxford University Press, USA
Published: 2013-08-05T16:00:00+00:00

Chapter 4

Julia sets and the Mandelbrot set

Julia sets and the Mandelbrot set are amongst the most frequently pictured fractals, combining both aesthetic and actual beauty. They have been used as a basis for modern art, as scenery in science fiction films, and simply as mystical symbols. The complexity of the Mandelbrot set is mind-boggling and its mathematical properties are far from fully understood, yet its definition is simple and it can be realized on a computer screen with just a few lines of computer code. Like some of the other fractals we have encountered it is determined by the behaviour of itineraries around the plane—once again repeating a simple step over and over again leads to extraordinarily complicated objects.

It is possible to think of Julia sets and the Mandelbrot set purely in terms of iteration of functions expressed in coordinate form. However, a little knowledge of complex numbers highlights the simple elegance of these functions.

Complex numbers

By definition, the square root of a number is a number which, when squared, gives that number. For example, since 32 = (–3)2 = 9, the square roots of 9 are 3 and –3. A quirk of the usual or real number system is that, whilst positive numbers have square roots, negative numbers do not. As the square of every number is non-negative there is no real number whose square is –9. Hence –9, –100, –¼, and all other negative numbers do not have a square root. This asymmetrical situation unsettled mathematicians for thousands of years and it was not until the 16th century that the matter was resolved by inventing a new entity, namely the square root of the simplest negative number, –1.

Thus, we bring in to the usual number system a ‘number’ which we think of as , the square root of –1. It is denoted by i and has the defining property that its square is –1, so i2 = i × i = –1. This enables us to express the square roots of all negative numbers in terms of i. For example, , since (3i)2 = 32 × i2 = 9 × (–1) = –9. Similarly and This leads to an enlarged number system in which we can add, subtract, multiply, and divide in a consistent way.

A ‘number’ of the form x + yi is called a complex number where x and y are ordinary real numbers and i is thought of as the square root of –1. We call x and y the real part and the imaginary part of x + yi respectively. For example,

1 + 3i, 2 + i, 3 – 5i, 0 – 3i, ½ + i, – 1.5 + 2.8i

are complex numbers, and the number 2 + 3i has real part 2 and imaginary part 3. It is often convenient to think of complex numbers as a single entity, so we may write z = x + yi and refer to this as ‘the complex number z’.

The basic arithmetic operations of addition, subtraction, and multiplication of

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