23. The effect on length, area, and d-dimensional measurement of enlargement by a factor 2

In fact, there are difficulties in taking the multiplier c in a power law as a ‘d-dimensional measurement’; to begin with it assumes that there is actually a constant multiplier, whereas for many fractals it ‘wobbles about a bit’. However, this heuristic argument is a first stab at generalizing the familiar concepts of length, area, and volume to fractional dimension in a natural way. This idea leads to what is known as d-dimensional Hausdorff measure, named after the German mathematician Felix Hausdorff (1868–1942), which is widely used by mathematicians.

Properties of dimension

For dimension to be a worthwhile concept, it should have properties that are both natural and useful. The first thing that box-counting dimension achieves is to extend the classical notion of dimension to a much wider setting. The box dimension of smooth curves such as circles or ellipses is 1 and of surfaces or plane regions such as solid squares is 2, consistent with the traditional concept of dimension as a whole number. However, box-dimension applies to essentially any geometric objects, not just curves and surfaces. This approach of taking a familiar idea and finding a way to generalize it to a wider setting is regularly seen in mathematics and has led to many significant developments. the product and power rules for, 41

Dimension has several properties without which it would be of little use. Firstly, ‘the larger an object, the larger its dimension’. For example, erasing any part of the Sierpiński triangle gives a smaller set whose dimension therefore can be no larger than 1.585, the dimension of the original Sierpiński triangle.

If a set comprises just a finite number of individual points scattered around the plane, its dimension is 0. On the other hand the dimension of a set in the plane cannot be more than 2, the dimension of the plane itself. Similarly, if we consider fractals in 3-dimensional space, where box-counting dimension involves counting cubes, then dimensions can be anything up to 3, the dimension of space itself. (Strange spaces are encountered in some areas of mathematics or physics where dimensions can be larger, but these need not concern us here.)

A continuous curve in the plane always has dimension at least 1. Conversely, if a fractal has dimension less than 1 then it must be ‘dust-like’ or ‘totally disconnected’, that is, the fractal is so sparse that it is impossible to travel between any two points without going outside the fractal. For if it were possible to trace a continuous path inside the fractal then the fractal’s dimension would be at least that of the path, that is at least 1.

Given two fractals in the plane, for example two fractal ferns, we might wish to lump them together and regard them as a single object, with its dimension given by box-counting across the entire picture. Then the dimension of the combined object is (normally) the larger of the two individual dimensions—the fractal with the larger dimension ‘dominates’ the other. Thus a von Koch curve with a straight line adjoined to one end has dimension 1.26, the larger of 1.26 and 1.

Dimension is a stable quantity—distorting a fractal in a smooth manner does not change its dimension. If a fractal is drawn on a thin sheet of rubber which is then stretched or deformed smoothly, without introducing any folds, corners, or kinks, its box-counting dimension is unchanged. In particular transforming a fractal by a similarity does not affect the dimension (though it will change any d-dimensional measurement).

We might look at the section of a fractal by a straight line, that is the set of points of the fractal that lie on the line. Dimensions of sections are generally well-behaved. As an illustration, draw a straight line through the Sierpiński triangle, Figure 11, in an arbitrary direction. The section will be a dust-like fractal consisting of those points scattered along the line that are in the Sierpiński triangle. The dimension of this fractal will almost certainly be 0.585 = 1.585 – 1, that is the dimension of the Sierpiński triangle minus 1. Of course, as can be seen from the picture, for some lines the overlap with the Sierpiński triangle will be an entire line segment and therefore of dimension 1, for example in certain positions when the line is horizontal. However, such positions are exceptional, and if the line is drawn ‘at random’ a section of dimension 0.585 is likely. In general, ‘typical’ or ‘random’ sections of a fractal by a straight line have a dimension that is 1 less than that of the fractal, unless this number is negative, in which case the fractal is so sparse that it will have no point draw a straight line through theNixs in common with ‘most’ lines. Looking at part of the fractal within a 1-dimensional line rather than at the whole within 2-dimensional space results in this dimension decrease of 1.

Dimensions also behave well under projections ‘or shadows’. Here it is simplest to think of a fractal in 3-dimensional space, a plant perhaps. Looking at its shadow or taking a photograph gives a flat picture of the plant (a ‘projection onto a plane’) whose dimension might be found by box-counting. The dimensions of a typical photograph and the original plant are related: if the dimension of the plant is greater than 2 then its picture will have dimension 2 and have positive area; if the dimension of the plant is less than 2 then the dimension of the picture will be the same as that of the plant. (This is a rather vague paraphrase of a precise and important mathematical theorem about projections.) This principle enables the dimension of some spatial objects to be estimated in a practicable way by box-counting on a photograph.

Limitations of dimension

Whilst dimension provides fundamental information about the character of a fractal and its appearance under ever-increasing magnification, a single number can still only provide limited information about objects that come in a enormous variety of shapes and forms.

Fractals of a given dimension may have widely differing structures. For example, a fractal of dimension 1.26 could be a continuous curve such as the von Koch curve, it might be full of holes rather like the Sierpiński triangle, it might be a ‘dendrite’ or ‘twig-like’, it could be totally disconnected or ‘dust-like’, or it may be ‘striated’, consisting of irregularly spaced parallel line segments. There is a major branch of mathematics known as topology concerned with distinctions of this nature.

Dimension tells us little abations with an

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, –ј, 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 Ч i = –1. This enables us to express the square roots of all negative numbers in terms of i. For example, , since (3i)² = 3² Ч i² = 9 Ч (–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, Ѕ + 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 complex numbers work in a natural way so that all the usual rules of arithmetic and algebra hold, except that we replace i² by –1 whenever it is encountered, consistent with i being the square root of –1. Here are some examples:

(1 + 3i) + (2 + i) = (1 + 2) + (3i + i) = 3 + 4i

(1 + 3i) + (2 + i) = (1 – 2) + (3i – i) = –1 + 2i.

To multiply two complex numbers we multiply each part of the first number by each part of the second number, in a similar way to multiplying out an algebraic expression. For instance, here’s how we square the complex number 2 + i:

where we have used that i² = –1. Thus (2 + i)² = 3 + 4i.

Complex numbers have a natural pictorial or geometrical representation. We simply think of the complex number z = x + yi as located at the point with coordinates (x, y) in the Euclidean plane, which in this context is referred to as the complex plane. Thus we may regard points in the plane in two ways: either given by coordinates or as complex numbers. The point with coordinates (4, 2) can equally well be thought of as the complex number 4 + 2i. The locations of various complex numbers are shown in Figure 24. Note that the origin is the complex number 0 = 0 + 0i.

24. Complex numbers in the complex plane, with the magnitude and angle of 4 + 2i indicated

There is another convenient way of specifying the position of a complex number z in the plane. We simply draw a line from the origin to z. The length of this line, in other words the distance of z from the origin 0, is called the magnitude (or modulus) of z, and the angle between this line and the horizontal axis (measured in a counter-clockwise direction) is called the angle (or argument) of z, see Figure 24. If we know both the angle and magnitude of a complex number z, we know in which direction and how far away it is from the origin, and this is enough to determine its location.

Arithmetic operations on complex numbers have geometric interpretations in the complex plane. One way to view addition of two complex numbers, for example (1 + 3i) + (2 + i), is to start at 1 + 3i and add 2 to the real part and 1 to the imaginary part, which corresponds to a shift from 1 + 3i parallel to and of after the French mathematician. This ix length equal to the line joining 0 to 2 + i, see Figure 25(a). Equivalently, the sum 3 + 4i is located at the fourth vertex of the parallelogram whose other three vertices are 0, 2 + i and 1 + 3i. In general, addition of any two complex numbers may be thought of geometrically by shifting one of them in a direction and distance given by the line repesenting the other.