42. Processes with different Hurst indices

Another, maybe more obvious, way of distinguishing the different characters of the graphs of Figure 42 is by the fractal dimensions of the plots. It can be shown mathematically that, typically, a graph with Hurst index H has box-counting dimension 2 – H. Thus a Hurst index close to 0 corresponds to a wildly erratic g case with the

Chapter 6 Fractals in the real world

Most of the fractals encountered so far in this book lie in the idealized world of the mathematician, where it is theoretically possible to repeat a construction step forever, or view an object at arbitrarily fine scales. Of course, the real world is not like that—in reality, we only encounter approximate fractals. If we zoom in too closely on a real object any self-similarity will be lost, and eventually we encounter molecular or atomic structure. Nevertheless, it can be very useful to regard natural objects as fractals if they exhibit irregularities or self-similarity when viewed over a significant range of scales. But all scientific descriptions or ‘models’ of reality are approximate, and this is a further instance. Planets are not perfect spheres but, for many purposes such as calculating their orbits, little accuracy is lost by assuming that they are. Similarly, Newtonian mechanics is completely adequate for most conventional calculations involving the structure or motion of everyday objects, though it breaks down at very small or very large scales, when quantum theory or relativity come into play.

Real objects tend to be described as ‘fractal’ if they contain parts that at two or more smaller scales appear in some way similar to the whole, indeed this is often enough for them to look very like mathematically precise fractals, such as for the fern computed in Figure 17. Alternatively, if estimates of dimension suggest a power law over a range of box-counts where the ratio of the largest to smallest sensible box sizes is more than about ten, this power law exponent may be stated as the dimension of the object and, provided that this is interpreted within the appropriate scale range, it can reflect useful information about the object and its physical behaviour.

Here is a selection of phenomena where the fractal viewpoint has provided valuable insight.

Coastlines and landscapes

Geographical shapes, such as coastlines, landscapes, and river networks display many fractal features. We saw in Chapter 3 that the box-counting dimension of the coast of Britain, estimated over a range of scales, is about 1.2 as opposed to the value of 1 that would be the case for a smooth boundary. It was Lewis Richardson (1881–1953) who first presented in a quantitative form the observation that the length of a coastline depends on the scale at which it is measured. Richardson was interested in the causes of warfare, and he collected a wide range of data that he thought might be relevant, including the lengths of certain coastlines and land frontiers that might affect tensions between neighbouring countries. He measured coastlines on maps using various step lengths to get estimates of the total length. Shorter step lengths detected more of the bays and capes that had been missed by coarser measurements, leading to a longer overall length. He realized that the calculated lengths increased considerably as the scale at which the measurements were made was reduced, and he displayed this on a graph relating the logarithm of the calculated length to that of the scale. The log-log graph gave a straight line but, whereas for a circle or smooth curve the line was flat, that is with slope or gradient 0, for the west coast of Britain the slope was about 0.25, providing a clear demonstration of the ‘fractality’ of the coastline. Richardson’s statistics and graphs were not published until 1961, eight years after his death, although there is evidence that Multifractal spectrum of the distribution of the proportion of 0s in numbersin6Richardson had talked of these effects many years before. This data was publicized by Mandelbrot in 1967 in a paper entitled How Long is the Coast of Britain? where he pointed out that Richardson’s calculations essentially said that the coastline had dimension 1 + 0.25 = 1.25, valid over a wide range of scales. This article, which clearly demonstrated that fractional dimension was appropriate for describing a natural feature, played a key role in convincing scientists that such notions could be used to study real phenomena of an irregular nature.

Not surprisingly, fractals occur in other geographical features. The land surface of countries is an obvious example, which can display fractality from scales of hundreds of miles down to a few feet. Mountain ranges contain many peaks, each with subsidiary summits, with hillsides made up of large and small undulations resulting from local geology or erosion by streams, and smaller hillocks and tufts formed by soil irregularities. Of course, the dimension of the land surface varies from region to region, perhaps from about 2.25 for the Rockies to 2 (the dimension of a flat or smooth surface) for the Utah Salt Flats.

The form of a skyline is determined by that of the landscape, with higher hills and land obscuring lower features from view. A fractal landscape will present a fractal skyline; indeed the dimension of a skyline curve is typically exactly 1 less than that of the local land surface. Thus the dimension of a mountain skyline might be around 1.2 whereas for gentler countryside it would be closer to 1.

Mathematical fractal constructions have been used very effectively to simulate realistic looking landscapes. Methods used to generate random curves such as Brownian motion may be extended to produce random surfaces. One might start with a (low) pyramid and modify it by building on each face a smaller pyramid extending upwards or downwards a random amount. Repeating this procedure by replacing faces with ever smaller pyramids may give a fractal surface. Realistic landscapes may be produced by choosing the construction parameters appropriately, a surface of ‘rural’ appearance having dimension about 2.1 and a mountainous one having dimension up to 2.2. Fractal landscapes have been utilized widely in art and movies, with fractal planet and landscape simulations pioneered in the early 1980s films Star Trek II: The Wrath of Khan and Star Wars: Return of the Jedi and used in many later films. Usually it is the overall appearance of the landscape that is important rather than its precise form, and random fractal constructions can produce and vary such scenery very efficiently and at a much lower cost than creating elaborate film sets.

Turbulent fluids

A turbulent liquid or gas is one that behaves in a non-smooth, swirling fashion. Often when a tap is first turned on water emerges in a smooth stream, but then it breaks up into a gushing, irregular, turbulent, flow. Turbulent fluids can be difficult to control or predict and often have a violent association—a stream in spate, gusting winds, a bumpy aeroplane ride. Yet, despite having been studied intensively by scientists for hundreds of years, turbulence is far from understood. A first step in analysing turbulence would be to understand better the behaviour of solutions to the Navier-Stokes equations, the fundamental equations of fluid flow that were derived early in the 19th century. The importance and difficulty of this is indicated by the million dollar ‘Millennium’ prize offered by the Clay Mathematics Institute in the year 2000 for establishing some seemingly basic properties of the solutions of these equations.

One way turbulence may be realized is as a hierarchy or ‘cascade is completely defined by the s shapes’ of eddies or whirls of decreasing sizes superimposed on each other. This idea, which was proposed by Lewis Richardson in 1922, encompasses a notion of self-similarity which is neatly summed up in his oft quoted parody:

Big whirls have little whirls that feed on their velocity, And little whirls have lesser whirls and so on to viscosity.

(The original lines from Jonathan’s Swift’s On Poetry: A Rhapsody had previously been reworded, already with a fractal flavour, by the mathematician Augustus De Morgan (1806–71) as:

Great fleas have little fleas upon their backs to bite ’em, And little fleas have lesser fleas, and so ad infinitum.)

In 1941 the Russian mathematician Andrey Kolmogorov (1903–87) formulated a quantitative theory based on this idea, applicable to relatively thin fluids. This depends on energy being passed down from the largest eddy via the intermediate ones, to reach the smallest eddies where the energy is dissipated as heat. Under the assumptions that the turbulence is isotropic, that is, there is no preferred direction, and homogeneous, that is, it fills the entire fluid, Kolmogorov developed a theory which fitted in reasonably well with physical observations. Nevertheless, there were observable differences, notably that when examined at a very fine scale the energy dissipation was not homogeneous, but varying considerably across the fluid in a rather clumpy manner, a phenomenon known as ‘intermittency’. In the early 1970s, Benoit Mandelbrot proposed that this might be explained if the turbulent activity was not homogeneous, but was concentrated on a fractal region. Assuming such a model, experiments suggest that the turbulent region within the fluid should be a fractal of dimension around 2.4.

Fractals in our bodies

Some of the most extensive examples of branching fractal networks are to be found within the bodies of humans or other mammals, in particular in the breathing, blood circulation, and nervous systems. Whilst these remarkable structures lead to efficient physiological operation of the body, it is still not clear what evolutionary or biophysical mechanisms underlie this branching growth.

In the breathing or respiratory system, the windpipe, or trachea, splits into two bronchial tubes leading into the two lungs. These tubes split into narrower tubes, which continue to split repeatedly until, after about 11 levels of branching, they reach numerous very fine tubes called bronchioles which end in microscopic thin-walled sacs called alveoli. A lung contains around 400 million very closely spaced alveoli. Air breathed through the mouth or nose passes down the trachea and into the lung to reach the alveoli from whence oxygen is passed into the bloodstream and carbon dioxide is absorbed from the blood to be exhaled. Adult human lungs are about 12 inches long and 5 inches wide, but because of their branching fractal structure have an enormous surface area of about 100 square yards. It is the fractal structure that achieves this large area within a confined space and thus enables oxygen to be supplied to the blood efficiently and in adequate quantity for the whole body.

Blood vessels which transport blood through the body form another branching network. Blood is carried away from the heart by arteries leading to all parts of the body, and these divide into narrower arterioles and end in fine capillaries of about 0.01mm in diameter. Oxygen and nutrients pass out through the thin capillary walls to body tissue and waste products are absorbed into the capillaries. For this to function, ever copies of itself at scaleNixy cell in the body must be within around 0.1mm of a blood vessel. Moreover, for the circulatory system to function efficiently, the distance between the heart and the capillaries should be no longer than necessary. This requires an intricate branching fractal network, comprising a total length of blood vessels of around 60,000 miles. In one component, the blood vessels in the retina of the eye have a striking branching structure, not unlike the fractal growth structure of Figure 41, and indeed the dimension of the network within the retina has been estimated at 1.7 over a fair range of scales.

Medical diagnosis

A graph of a person’s heart rate measured every few seconds and plotted over a period of, say, half an hour, provides a great deal of information about the state of the heart. The heart does not beat at a constant rate: there is some variation in the rate in both the short term and longer term and the graph for a healthy heart has a fractal form not unlike the graph of share prices. This reflects the complex feedback system within the body, with information from all facets of its operation continually processed to determine the response from the heart rate. We have seen that even simple mathematical systems can lead to fractal outcomes, and the body is far from simple. Too erratic variation can indicate conditions such as atrial fibrillation when the chambers of the heart contract rapidly and randomly; on the other hand, too little variation can be a sign of congestive heart failure when the heart is unable to respond to the needs of the body. A fractal analysis of the heart beat graph can identify such problems. Very roughly, a healthy heart has a graph of dimension around 1.5 with lower or higher values indicating problems. Fractal analysis is used alongside other techniques, such as frequency analysis of heart signals, in diagnosing problems with the heart and blood circulation.

Considerable research has been done in the past few years to develop fractal methods for the early detection and classification of cancer tumours. Fractal analysis on scans of parts of the body and medical samples can indicate the development of tumour growth. For example, in vascular cancer, the arrangement of blood vessels, that is the vascular network, changes with the presence of tumoral cells which stimulate the production of the proteins needed to form new blood vessels. This can lead to the network becoming fragmented into a large number of very short branching links with a much more chaotic form than a healthy network. Fractal dimension can provide a good method of quantifying such changes, which are reflected in an increase in the box-counting dimension estimated over a suitable range of scales on images of the network. Fractal analysis has been incorporated into sophisticated automated methods of detecting a variety of cancers including breast and skin cancers and leukaemia.

Clouds

Clouds consist of microscopic droplets of water or crystals of ice suspended in the atmosphere. As warmer air rises from the earth it cools and eventually gets to a temperature at which it can no longer hold all its moisture. The surplus water vapour condenses as tiny drops on particles of dust or other material in the atmosphere to form visible clouds. The cooler air returns to earth, setting up convection or circulating air currents. Clouds appear in many characteristic shapes, which depend on the behaviour of the currents and the prevailing atmospheric pressure and temperature. Factors such as the proximity of mountains and nature of the ground covering further affect the highly complex climatic system.

Of the many types of cloud, several display fractal features. Cirrus clouds,">A on the circle and perform random walks until they reach a square next to one already shaded,

Cumulus clouds, sometimes called ‘fair weather clouds’, appear as large, white, fluffy heaps, with irregular curvy sides and top, but with a fairly flat base usually below 6,000ft. Cumulus clouds have fractal boundaries, and box-counting on a photograph of a cumulus cloud gives the dimension of a cloud’s outline as about 1.16. There are theoretical reasons to expect that the surface of a three-dimensional object will typically be one more than the dimension of its outline on a photograph. For example, a photograph of a sphere is bounded by a circle and a circle has dimension 1 corresponding to the spherical surface having dimension 2. In the same way we would expect the cloud surface to have dimension 1.16 + 1 = 2.16. By reflecting both the sun’s rays and the heat coming up from the earth as well as absorbing heat and re-radiating it, clouds have a major effect on the temperature of the atmosphere and the earth’s surface. As fractals, these clouds have a very large and convoluted surface, with far more exposed boundary available for heat absorption, radiation, and reflection than there would be if the cloud was, say, spherical. The fractality of clouds is a major factor in the immensely complex behaviour of the global climate.

Galaxies

The notion that the universe itself might have some form of hierarchical structure goes back at least 200 years, with astronomers such as John Herschel (1792–1871) and philosophers such as Immanuel Kant (1724–1804) suggesting that the portion of space that we can see around us might be replicated at much vaster scales. In 1907, Edmund Fournier d’Albe (1868–1933) published a remarkable book entitled Two New Worlds which proposed a hierarchical arrangement of stars and included a diagram of an idealized formation which was an exactly self-similar fractal dust. In the 1970s, with improved technology, it became possible to pinpoint many distant galaxies. Hubble’s law states that the speed at which galaxies are receding from Earth is proportional to their distance away, and this speed can be measured fairly accurately from the redshift, that is the amount that light from the galaxies shifts towards the red end of the spectrum. This technique provides a picture of what the universe looks like millions of light years away (a light year is the distance that light travels in a year, some 6,000 million million miles).

The distribution of galaxies is fundamental to theories of cosmology, that is the evolution and structure of the universe. By the early 1980s galaxies had been mapped up to distances of about 50 million light years away. Based on this data, Jim Peebles, Benoit Mandelbrot, and others proposed that galaxies might have a hierarchical, fractal distribution, rather than a uniform ‘homogeneous’ distribution across space. There is now general agreement amongst cosmologists that, over a range of ‘relatively small’ scales, maybe up to about 300 million light years, galaxies are distributed in a fractal-like fashion with a dimension in space of around 2. Thus billions of stars group together to make up galaxies, the galaxies form clusters, and the clusters are arranged in superclusters. However, views diverge about what the universe looks like at even larger scales, copies of itself at scaleNix with some believing that fractality continues indefinitely but others claiming that there is a cut-off scale above which the universe becomes rather homogeneous with dimension close to 3. Very recent observations at scales of up to 3 billion light years seem to favour homogeneity, but fractal clustering still cannot be ruled out even at large scales.

Fractal antennae

Antennae, or aerials, collect and radiate radio waves, and are an essential part of radio and television transmitters and receivers and, more recently, mobile phones and GPS systems. Antennae design is a complex problem: antennae normally operate at a specific ‘resonant’ wavelength or frequency, and whilst they may be of some use at other wavelengths, their efficiency decreases markedly. Ideally, antennae have sizes that are comparable with the wavelength of operation, typically or of the wavelength, but there may be constraints requiring a rather smaller size. For example, mobile phones typically operate at frequencies of 900–1,800 MHz, corresponding to wavelengths of about 17–33cm, but their antennae need to be much smaller than this.

To overcome these difficulties fractal antennae have been introduced. One approach is to use a fractal shape to fit a large length of material that can receive or transmit radio waves within a small space. We have seen that a good approximation to a von Koch or other fractal curve can be very long, so a conductor in the form of a fractal can achieve this easily. A second possibility is to make a conductor in the form of a self-similar fractal and get resonances at several wavelengths corresponding to the similarity ratios of the fractal. For example, the Sierpiński triangle comprises similar copies of itself at scales 1, , , , , … and would have high efficiency at these multiples of some fundamental wavelength. This idea has led to the ‘Sierpiński dipole’, consisting of two Sierpiński triangles placed next to each other with the input at adjacent corners. Varying the construction parameters of the Sierpiński triangle can provide antennae suitable for other sequences of wavelengths. On the other hand, a randomly fingering fractal tree avoids any such exact scaling repetition, so may be appropriate as the basis for an antenna over a continuous range of wavelengths.

Multifractals

This account would be incomplete without at least some mention of multifractals—the idea that a widely varying distribution can lead to a whole hierarchy of fractals. For example, if a drop of coloured dye (food colouring will do) is put into water, then it spreads irregularly through the water—in some places the dye remains highly concentrated, elsewhere it is less intense and in some places almost negligible. The locations at which the dye has a given concentration density are far from regular, and maybe a fractal. Different density values will give different fractals whose dimensions might be found, and plotting a graph of these dimensions for various densities gives a ‘spectrum’, known as a multifractal spectrum, which codes information about the distribution of the dye though the water.

The following example, in which we analyse the decimal expansions of all the numbers between 0 and 1, illustrates the multifractal idea, with the distribution of decimal digits in numbers leading to a whole range of fractals. If we write down a ‘typical’ number, perhaps by choosing each digit at random, such as

0.40986721504290346318650372909536258741121853608794 …

each of the digits 0,1,2,3,4,5,6,7,8,9 appears about 10 per cent of the time. For example, 9.994 per cent of the first 10 million digits of p are 0, and taking even more digits the proportion of 0s in a ‘typical’ number will become closer and closer to 10 per cent. Nevertheless there are plenty of numbers where this limiting proportion is very different. A number with 0s in alternate places like

0.60104050703080104020303090706040…

has 0 occurring 50 per cent of the time, though such numbers are much less common than those with 10 per cent of the digits 0. So how ‘large’ are the sets of numbers between 0 and 1 with 10 per cent, 20 per cent, or 30 per cent, etc., of their digits 0s? It turns out that the set of numbers with any given proportion of 0s is a fractal, so the natural way to measure the size is by a fractal dimension. Each percentage defines a different fractal set of numbers and their dimensions may be calculated and are given in the following table (for technical reasons we need to use Hausdorff’s definition of dimension here rather than box dimension).

Chapter 7 A little history

Geometry, with its highly visual and practical nature, is one of the oldest branches of mathematics. Its development through the ages has paralleled its increasingly sophisticated applications. Construction, crafts, and astronomy practised by ancient civilizations led to the need to record and analyse the shapes, sizes, and positions of objects. Notions of angles, areas, and volumes developed with the need for surveying and building. Two shapes were especially important: the straight line and the circle, which occurred naturally in many settings but also underlay the design of many artefacts. As well as fulfilling practical needs, philosophers were motivated by aesthetic aspects of geometry and sought simplicity in geometric structures and their applications. This reached its peak with the Greek School, notably with Plato (c 428–348 BC) and Euclid (c 325–265 BC), for whom constructions using a straight edge and compass, corresponding to line and circle, were the essence of geometric perfection.

As time progressed, ways were found to express and solve geometrical problems using algebra. A major advance was the introduction by Renй Descartes (1596–1650) of the Cartesian coordinate system which enabled shapes to be expressed concisely in terms of equations. This was a necessary precursor to the calculus, developed independently by Isaac Newton (1642–1727) and Gottfried Leibniz (1646–1714) in the late 17th century. The calculus provided a mathematical procedure for finding tangent lines that touched smooth curves as well as a method for computing areas and volumes of an enormous variety of geometrical objects. Alongside this, more sophisticated geometric figures were being observed in nature and explained mathematically. For example, using Tycho Brahe’s observations, Johannes Kepler proposed that planets moved around ellipses, and this was substantiated as a mathematical consequence of Newton’s laws of motion and gravitation.

The tools and methods were now available for tremendous advances in mathematics and the sciences. All manner of geometrical shapes could be analysed. Using the laws of motion together with the calculus, one could calculate the trajectories of projectiles, the motion of celestial bodies, and, using differential equations which developed from the calculus, more complex motions such as fluid flows. Although the calculus underlay Graph of a Brownian process8I to think of all these applications, its foundations remained intuitive rather than rigorous until the 19th century when a number of leading mathematicians including Augustin Cauchy (1789–1857), Bernhard Riemann (1826–66), and Karl Weierstrass (1815–97) formalized the notions of continuity and limits. In particular, they developed a precise definition for a curve to be ‘differentiable’, that is for there to be a tangent line touching the curve at a point. Many mathematicians worked on the assumption that all curves worthy of attention were nice and smooth so had tangents at all their points, enabling application of the calculus and its many consequences. It was a surprise when, in 1872, Karl Weierstrass constructed a ‘curve’ that was so irregular that at no point at all was it possible to draw a tangent line. The Weierstrass graph might be regarded as the first formally defined fractal, and indeed it has been shown to have fractal dimension greater than 1.

In 1883, the German Georg Cantor (1845–1918) wrote a paper introducing the middle-third Cantor set, obtained by repeatedly removing the middle thirds of intervals (see Figure 44). The Cantor set is perhaps the most basic self-similar fractal, made up of 2 scale copies of itself, although of more immediate interest to Cantor were its topological and set theoretic properties, such as it being totally disconnected, rather than its geometry. (Several other mathematicians studied sets of a similar form around the same time, including the Oxford mathematician Henry Smith (1826–83) in an article in 1874.) In 1904, Helge von Koch introduced his curve, as a simpler construction than Weierstrass’s example of a curve without any tangents. Then, in 1915, the Polish mathematician Wacław Sierpiński (1882–1969) introduced his triangle and, in 1916, the Sierpiński carpet. His main interest in the carpet was that it was a ‘universal’ set, in that it contains continuously deformed copies of all sets of ‘topological dimension’ 1. Although such objects have in recent years become the best-known fractals, at the time properties such as self-similarity were almost irrelevant, their main use being to provide specific examples or counter-examples in topology and calculus.

44. The middle-third Cantor set F. Each stage of the construction, E₁, E₂, … is obtained by removing the middle third of each interval in the preceding stage

It was in 1918 that Felix Hausdorff proposed a natural way of ‘measuring’ the middle-third Cantor set and related sets, utilizing a general approach due to Constantin Carathйodory (1873–1950). Hausdorff showed that the middle-third Cantor set had dimension of log2/log3 = 0.631, and also found the dimensions of other self-similar sets. This was the first occurrence of an explicit notion of fractional dimension. Now termed ‘Hausdorff dimension’, his definition of dimension is the one most commonly used by mathematicians today. (Hausdorff, who did foundational work in several other areas of mathematics and philosophy, was a German Jew who tragically committed suicide in 1942 to avoid being sent to a concentration camp.) Box-dimension, which in many ways is rather simpler than Hausdorff ">628941dimension, appeared in a 1928 paper by Georges Bouligand (1889–1979), though the idea underlying an equivalent definition had been mentioned rather earlier by Hermann Minkowski (1864–1909), a Polish mathematician known especially for his work on relativity.

For many years, few mathematicians were very interested in fractional dimensions, with highly irregular sets continuing to be regarded as pathological curiosities. One notable exception was Abram Besicovitch (1891–1970), a Russian mathematician who held a professorship in Cambridge for many years. He, along with a few pupils, investigated the dimension of a range of fractals as well as investigating some of their geometric properties.

Appendix

This appendix sets out mathematical details of some properties that were taken on trust in the main part of the text and which may be of interest to readers a little more at home with algebra.

Powers and logarithms

Some of the dimension arguments in Chapter 3 involved raising numbers to powers other than whole numbers. To appreciate what this means, we must first consider whole number powers, obtained by repeatedly multiplying the number by itself. Raising a number to the power 2 is just squaring, so, for example, 5² = 5 Ч 5 = 25, where the superscript 2 indicates that 5 is raised to the power 2. To find the 3rd power, or cube, requires a further multiplication, so 5³ = 5 Ч 5 Ч 5 = 125. We can find any whole number power of a number by repeated multiplication, so in general a^b = a Ч a Ч a Ч … Ч a where a occurs b times in the multiplication, thus 5⁶ = 5 Ч 5 Ч 5 Ч 5 Ч 5 Ч 5 = 15,625.

Grouping together terms in this product gives

5⁶ = 5Ч5Ч5Ч5Ч5Ч5 = (5Ч5)Ч(5Ч5Ч5Ч5) = 5²Ч5⁴

and similarly

5⁶ =5Ч5Ч5Ч5Ч5Ч5=(5Ч5Ч5)Ч(5Ч5Ч5)= 5³Ч5³

This illustrates the very useful sum rule for powers, that multiplying together two powers of the same number equals the number raised to the sum of the powers, in symbols

a^bЧa^c=a^b+c.

Now suppose we raise a number to one power and then raise the result to another power, for example, if we square 5 and then cube the answer,

(5²)³ =5²Ч5²Ч5² =(5Ч5)Ч(5Ч5)Ч(5Ч5)=5Ч5Ч5Ч5Ч5Ч5 =5⁶.

The new power 6 is the product of the powers 2 and 3. In the same way, for whole number powers in general, to raise a number to a power and then raise the result to another power we multiply the powers, that is

(a^b)^c=a^bЧc,

termed the product rule for powers.

The key idea in defining powers of other numbers is to do so in such a manner that the template completely defines wh to think ofsum and product rules remain valid, even when b and c are not whole numbers. So how must we define 5 raised to the power Ѕ? For the product rule to hold we require (5^1/2)² = 5^1/2Ч2 = 5¹ = 5. Thus 5^1/2 is the number such that when squared gives 5, in other words the (positive) square root of 5, which is about 2.236. In the same way 5^1/3 = 1.710 and 5^1/4 = 1.495 are, respectively, the numbers which when cubed and raised to the fourth powers give 5, that is the cube and fourth roots of 5. In general, the powers 1/2, 1/3, 1/4, … just represent square roots, cube roots, and fourth roots, etc., so that, when b is a positive whole number and a is positive, a^1/b is the bth root of a, the number that when raised to the power b gives a.

Now let’s think about fractional powers where the numerator is not 1—for example what does 5^3/2 or 5^1.5 mean? Noting that 3/2 = (1/2) Ч 3, if the product rule for powers is to hold we must have that

5^3/2 = 5^1/2Ч3 =(5^1/2)³.

But 5^1/2 is just the square root of 5, and we can find its cube, so that

5^1/2 = (5^1/2)³= (2.236)³ = 11.180.

In a similar way, for any fraction p/q and any positive number a we must define a^p/q to be (a^1/q)^p, thus the p/qth power of a number is its q th root raised to the power p.

Finally, not every positive number can be expressed as a fraction, for example pi, the ratio of the circumference to the diameter of a circle, is an ‘irrational’ number, that is, a never ending decimal π = 3.14159265…. Nevertheless, we can find fractions as close as we like to π and, as above, raise a number to such fractional powers, to get arbitrarily close approximations to the πth power. For example, 5^3.1416 = (5^1/10000)³¹⁴¹⁶ = 156.9944 which is close to the actual value of 5^π = 156.9925 … which may be found by taking π to more decimal places. Defining powers of numbers in this way ensures that the sum and product rules of powers remain true, even if the powers are not whole numbers.

Logarithms, which were used in Chapter 3 and elsewhere, are very closely related to powers. Indeed, the product and power rules for logarithms are little more than the two rules for powers written in a different way. Recall that the logarithm of a number is the power to which 10 must be raised to give that number, thus a = 10^c and c = log a are just different ways of expressing more than this is true. The Nix the same relationship.

To see why the product rule for logarithms holds, if c = log a and d = log b, then by definition, a = 10^c and b = 10^d. Multiplying, a Ч b = 10^c Ч 10^d = 10^c+^d by the sum rule for powers. Hence c + d is the power to which 10 must be raised to give a Ч b; in other words

log(aЧb) = c+d = loga + logb.

For the power rule, if c = loga^b then, from the definition of logarithms, a^b = 10^c. Using the product rule for powers twice,

a = a^bЧ1/b=(a^b)^1/b = (10^c)^1/b = 10^cЧ1/b = 10^c/b.

From the definition of logarithms log a = c/b, so multiplying by b gives

b loga = c = log a^b

which is the power rule for logarithms.

Squaring complex numbers

The claim in SierpiChapter 4 that squaring a complex number squares its magnitude and doubles its angle needs some justification. To square a general complex number z = x + yi we multiply out the brackets, so that

using that i² = −1 so that y²i² = −y². Thus, in coordinate notation, where the complex number x + yi is thought of as the point in the plane with coordinates (x, y), the squaring function z → z² is given by

(x, y) → (x² − y², 2xy).

To find the magnitude of (x + yi)² we apply Pythagoras’ Theorem to the coordinates, so the square of its magnitude is given by

which 0" height="408" alt="image"/>8941is just the square of the magnitude of x + yi. Thus squaring a complex number squares its magnitude.

To show that squaring a complex number doubles the angle is a little trickier. Whilst this can be deduced from standard formulae from trigonometry, here we give a direct algebraic approach. We first address the special case where z = x + yi has magnitude 1, so that x² + y² = 1. Then the magnitude of z² is also 1. Consider the isosceles triangle with vertices 0, 1, and z, which have coordinates by (0, 0), (1, 0), and (x) must take a

Further reading

Several other books in this ‘Very Short Introduction’ series include material that complements or elaborates on topics mentioned in this book. Timothy Gowers’ wide-ranging overview Mathematics includes a broad discussion of geometry and dimension. Peter Higgins’ Numbers includes discussions on the real numbers that form the middle-third Cantor set and the complex numbers that underlie Julia sets. Chaos by Leonard Smith shows how fractals often arise from ‘chaotic’ systems.

Those wishing to study fractals further should certainly take a look at Benoit Mandelbrot’s The Fractal Geometry of Nature (W. H. Freeman, 1982), which, apart from its historical significance, provides a broad scientific and philosophical overview of fractals, with little technical mathematics but with many illustrations that are remarkably fine for the early 1980s. There are several books covering fractals and chaos with mathematics kept to a basic level, including Does God Play Dice? (Penguin, 2nd edn, 1997) by Ian Stewart, author of many popular mathematics books, and Fractals: Images of Chaos by Hans Lauwerier (Penguin, 1991). Introducing Fractals: A Graphic Guide by Nigel Lesmoir-Gordon and Will Rood, illustrated by Ralph Edney (Icon Books, 2009), provides a short, entertaining overview of fractals with insightful cartoons on every page. The Colours of Infinity, edited by Nig height="48" a

Index

Page numbers in bold type refer to definitions

A

angle 64, 65–6, 124–5

antennae 111–12

arbitrage 88–9

area 35–6, 54–5

argument 64

attractor

of a function 14

of an iterated function system 27, stage of the constructionRing a simple step over and over again27–8

of a template 20, 23–5

strange 15

axes 10

B

Bachelier, Louis 88

bacterial growth 96

Barnsley, Michael 28, 34

bell-shaped distribution 86, 90

Besicovitch, Abram 119

bifurcation 79, 81

Black, Fisher 89

Black-Sholes model 89–90

blood vessels 107

Bouligand, Georges 119

box-count 42

box-counting 38–59

number 42

Brahe, Tycho 117

Brooks, Robert 83

Brown, Robert 93

Brownian process 87–93, 100

dimension 87, 93

planar 91– Computer simulation of fractal fingering: successive particles are released from random points 3, 94

spatial 91–3

statistically self-similar 87–8

C

calculus 6

call option 89

cancer 108–9, 115

Cantor, Georg 117

Cantor set, middle-third 117–18

Carathйodory, Constantin 118

Cauchy, Augustin 117

chaos game 28

clouds 109–10

coastline 51, 103–4

complex arithmetic Sierpi62–6, 124–5

complex numbers Sierpi61–6, Sierpi62, 124

complex plane 63, 64–6

computer pictures 28, 32–4, 119

congruent 17

contracting 28

coordinates Sierpi10, 62

Cartesian 10

cubed 37

D

Decartes, Renй 10, 116

decimal expansion 112–14

De Morgan, Augustus 106

diffusion limited aggregation 94, 94–7

dimension 35–50, 44

box-counting 35, 37–54, 47

fractal 35–60, 47

Hausdorff 118–19

of self-similar fractal 54

topological 118

distortion 58

Douady, Adrien 74, 84

Douady rabbit 74, 79

drunkard’s walk 85–7

dust-like set 57

E

electrolysis 51, 94–7

Einstein, Albert 93

Euclid 17, 116

exponent 44

F

Fatou, Pierre 77, 83

Fatou set 83

fern 29, 51

Fields Medal 84

figure 9

fine structure 4, 7

fixed point 70

Fournier d’Albe, Edmund 110

fractal 2, 8, 120

dimension 35–60

fingering 94, 94–7

image compression 32–4

networks 106

real 2–3, 102

self-affine 29

self-similar after the French mathematician a6 fu 5, 5–7, 17–28, 53–4

statistically self-similar 30–2, 87–8

time records 98–101

function 11, 25–8, 65–6

G

galaxies 110–11

generator 8

gradiant 50

grass 8

grid 42

H

Hausdorff, Felix 56, 118–19

heart rate 108

Hele-Shaw cell 97

Hele-Shaw, Henry 97

Hйnon, Michael 13

Hйnon function 13

Hйnon attractor 13–15, 114

Herschel, John 110

Hubbard, John a conductor in the form of a 894184

Hurst, Harold 98

Hurst index/exponent 98, 98–101

I

image analysis 115

image compression 32–4

imaginary part Sierpi62

intermittency 106

iterate 12, 67–72, 81–2

iterated function system 28

fundamental theorem of 28

iteration 12, 15, 67–72, 81–2

itinerary 12, 67–72, 81–2

J

Joseph effect 90–1

Julia, Gaston 77, 82–3

Julia set 67–84, 69, 71

dimension 82

filled-in 71, 71–2

K

Kant, Immanuel 110

Kolmogorov, Andrey 106

L

lacunarity 59–60

landscape 103–5

Laplace, Pierre-Simon 97

Laplace’s equation 97

Leibnitz, Gottfried 1, 116

length 6, 35–7, 54–5

infinite 37

Lichtenberg figures 96–7

Lichtenberg, Georg 96

lightning 96–7

limit 47

line 50

line segment 38–44

logarithm 44, 44–6

product rule 45, 123

power rule 45, 123–4

long-term dependence 100

lung 107

M

magnitude 64, 65–6, 124–5

Mandelbrot, Benoit 2, 8, 83, 104, 106, 111, 119–20

Mandelbrot set Sierpi61, 74–84, 75

boundary 79–80, 82

buds 77, 79

cardioid 77, 79

connected 80

dimension 80

fundamental theorem of 77–8

locally connected 80

universality 81

manganese deposits 96

map, mapping 25

Matelski, Peter 83

Maxwell, James Clark 1

McMu after the French mathematician a6 fullen, Curt 84

measurement, d-dimensional 54–6

Merton, Robert 89

Minkowski, Hermann 119

Misiurewicz, Michal 79

Misiurewicz point 79

modulus 64

motif 8

multifractals 112–15

multiplier 44

N

Navier-Stokes equations 105

Newton, Isaac 1, 116

Noah effect 90–1

Nobel Prize 89, 93

normal distribution 86

O

object 9

orientation 21, 21–5,

origin 10

P

Peebles, Jim 111

Perrin, Jean 93

plane 9

Plato 116

porosity 59

power 37, 40–2, 121–3

law 43, 43–4, 50, 102–3

product rule 122

sum rule 122

product 45

projection 59

Pythagoras’ theorem 65, 124

R

random walk 85, 85–7

range of fractality 49

real number Sierpi61

real part Sierpi62

recursive construction 7

rescaled range analysis 98–101

Richardson, Lewis 103–6

Riemann, Bernhard 117

river levels 98–101

S

Sander, L. M. 94

scale 7, 17

scaling ratio 17, stage of the construction. pe55–6

scaling transformation 20

Scholes, Myron 89

section 58

self-affine set 29

self-similar, self-similarity 5, 7, 17–34, 53–4

self-similar set 18

set 9

shape 9

share prices 88–91, 100

Shishikura, Mitsuhiro 80

Sierpiński triangle 20–1, 26–7, 28–9, 40, 42–3, 57, 59–60, 112, 118

Sierpiński carpet 118

Sierpiński, Wacław 118

similar 17

similarity transformation 20 stage of the construction. pe

slope 50

Smith, Henry 118

snowflake 7, 20, 22, 48

spiral 20, 22, 48

square 21, 38–44

square root Sierpi61

of-1 Sierpi62

squared 13, 36

squig curve 8

Star Trek 105

Star Wars 105

strike price 89

Swift, Jonathan 106

symmetry 21–3

of the square 21

T

tangent 6

template 18, 18–26, 20, 33

three squares 23–4

topology 59

totally-disconnected 57, 77

transformation 25,

class="nounder" href="kindle:pos:fid:000G:off:000000048E" aid="F89OG">25–8

affine 29

contracid="F89OM">27, 27–8

tree 29

turbulence 105–6

V

viscous fingering 97

volume 54–5

von Koch curve 3–7, 4, 18–20, 28, 36–7, 43,

"nounder" href="kindle:pos:fid:000N:off:00000003E3" aid="F89PG">47–8, 59, 112, 118

random 30

von Koch, Helge 4, 118

W

Weierstrass, Karl 117

Weierstrass curve 117

Wiener, Norbert 93

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