The Number Mysteries: A Mathematical Odyssey through Everyday Life
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Figure 2.18
The crystal structure that emerges has something in common with a stack of oranges at the grocery store, in which three oranges in one layer have a fourth orange placed on top to make a tetrahedron. But if you look instead at each layer of oranges, you’ll see hexagons everywhere. These hexagons, which appear in the ice crystals, are the key to the snowflake’s shape. So Kepler’s intuition was right—stacking oranges and the six-armed snowflake were related, but it wasn’t until we were able to look at the atomic structure of snow that we could see where the hexagons were hidden. As the snowflake grows, the water molecules attach themselves to the six points of the hexagon, building up the six arms of the snowflake.
It is in this passage from the molecular to the large snowflake where the individuality of each snowflake asserts itself. And while symmetry is at the heart of the creation of the water crystal, it is another important mathematical shape that controls the evolution of each flake: the fractal.
HOW LONG IS THE COASTLINE OF BRITAIN?
Is the coastline of Britain 18,000 km long? Or 36,000 km? Or is it even longer? Surprisingly, the answer to this question is far from obvious and is related to a mathematical shape that wasn’t discovered until the middle of the twentieth century.
Of course, with the tides coming in and out twice a day, the length of Britain’s coastline is constantly varying. But even if we fix the coastline, it’s still not clear how long it is. The subtlety arises from the question of how finely you measure the length of the coast. You could start by laying meter rulers end to end and counting how many rulers you need to circumnavigate the country, but using rigid meter rulers is going to miss a lot of smaller-scale detail.
Figure 2.19 Measuring the coastline of Britain.
If you used a long piece of rope instead of rigid rulers, you would be able to follow more of the intricate shape of the coastline. When you pulled the rope out straight to make your measurement, the length of the coastline would be considerably longer than the estimate obtained with meter rulers. But there’s a limit to the flexibility of a rope, which can’t capture the intricate detail of the contours of the coastline on the centimeter scale. If we used a thin thread, we would be able to capture more of this detail, and then our estimate of the length of the coastline would be greater still.
The Ordnance Survey gives the length of the coastline of Britain as 17,819.88 km. But measure the coastline in finer detail, and you will get double that length. As an illustration of just how difficult it is to pin down geographical lengths, in 1961, Portugal claimed that its border with Spain was 1,220 km, while Spain said it was only 990 km. The same level of discrepancy was found between the borders of Holland and Belgium. In general, it’s always the smaller country that calculates the longer border . . .
So, is there any limit to this process? Perhaps the more we zoom in on the detail, the longer the coastline becomes. To show how this is possible, let’s build a piece of mathematical coastline. To make the coastline you’ll need a ball of string. Start by pulling out 1 m of string from the ball and laying it down on the floor:
Figure 2.20
This is too straight to be a real coastline, so let’s make a large inlet in this straight bit of coast. Pull out some more string from the ball so that the middle third of the string is replaced by two sides of the same length that go in and out:
Figure 2.21
How much extra string did you have to pull out of the ball to make this inlet? The first line was made up of three pieces of length m, while this new coast consists of four pieces of length m. So the new length is times the first length—that is, m.
This new coast is still not very intricate. So again, divide each of the smaller lines into three and replace the middle third of each line by two sides of the same length. Now we have this coastline:
Figure 2.22
How long is this coastline? Well, each of the four lines has again increased in length by a factor of . So the length of the coast is now × m = ()2 m.
You’ve probably guessed what we’re going to do next. Keep repeating this procedure by dividing the straight lines into three and replacing the middle section with two lines of the same length. Each time we do this, the shape grows in length by a factor of . If we do this one hundred times, the length of our coastline will have increased by a factor of ()100, which makes it just over three billion kilometers. Laid out straight, a piece of string that long would stretch from the earth to Saturn.
If we carried out this procedure an infinite number of times, we would get a length of coast that was infinitely long. Of course, physics prevents us from dividing things beyond a certain limit, determined by what is called the Planck constant. This is because, according to physicists, it is actually impossible to measure a distance smaller than 10–34 m without creating a black hole that would swallow up the measuring device. When we do our trick of repeatedly adding smaller and smaller inlets to our coastline, by the time we get to the seventy-second step, the length of the lines will already be smaller than 10–34 m. But mathematicians are not physicists—we live in a world in which you can divide a line up an infinite number of times and not vanish into a black hole.
Another way to see why the coastline has infinite length is to consider a piece of coastline between two points, A and B, on a coastline that runs from A to E. Let’s suppose this piece between points A and B has length L. Assuming that the length from A to B equals the length from B to C, from C to D, and from D to E, if we magnify this piece of the coast from A to B three times, the result is an exact copy of the whole coastline from A to E. So the complete coastline has length 3L. On the other hand, if we take four copies of this smaller piece, we can put them end to end to cover the complete coastline: A to B, B to C, C to D, and D to E. From this viewpoint, the length of the coast is 4L, because we need four copies of the small piece to build the coastline. But they have to have the same length, whichever way we measure it. So how can 4L = 3L? The only resolution to this equation is if L is either of length 0 or has infinite length.
Figure 2.23
This infinite coastline is actually one side of a shape called the Koch snowflake, named after its inventor, the Swedish mathematician Helge von Koch, who constructed it at the beginning of the twentieth century.
This mathematical shape has too much symmetry to look like a real coastline, and doesn’t look particularly natural or organic, but if you randomize whether the lines you add each time cut into the coast or jut out into the sea, then things begin to look far more convincing. Here are pictures made by the same procedure as was just described, except that a coin was tossed to decide each time whether the lines were added below or above the line that was
Figure 2.24
If you join several of these coastlines together, you get something that looks remarkably like a medieval map of Britain:
Figure 2.25
So if you are ever asked what the length of the coastline of Britain is, frankly, you can choose whatever answer you like. Isn’t that the kind of math question everyone dreams of in school?
WHAT DO LIGHTNING, CAULIFLOWER, AND THE STOCK MARKET HAVE IN COMMON?
In 1960, the French mathematician Benoit Mandelbrot was asked to give a talk to the economics department at Harvard University about his recent work on the distribution of large and small incomes. When he entered his host’s office, he was rather perturbed to see the graphs he had prepared for his talk drawn on the blackboard. “How come you’ve got my data in advance?” he asked. The curious thing is that the graphs weren’t anything to do with incomes—they were variations in cotton prices that his host had been analyzing in a previous lecture.
The similarity piqued Mandelbrot’s curiosity and led to his discovery that if you took the graphs of various unrelated sets of economic data, they appeared to have a very similar shape. Not only that, but whatever timescale you looked at, the shapes seemed to be the same. For example, variations in cotton prices over eight years looked like variat
ions over eight weeks, and they looked much the same as variations over eight hours.
The same phenomenon occurs with the coastline of Britain. Take, for example, the images in Figure 2.26. They are all sections of the coastline of Scotland. One is from a map with a scale of 1:1,000,000. The others are much more detailed maps—one with a scale of 1:50,000, the other with a scale of 1:25,000. But can you match the pictures to the scale? However much you zoom in or out, these shapes seem to retain the same level of complexity. This isn’t true of all shapes. If you draw a squiggly line and zoom in and magnify a portion of it, then at some point it will begin to look quite simple. What characterizes the shape of a coastline or Mandelbrot’s graphs is that however far you zoom in, the complexity of the shape is retained.
Figure 2.26 The coastline of Scotland at different magnifications. From left to right, original map scales of 1:1,000,000; 1:50,000; and 1:25,000.
As Mandelbrot began to look further afield, he found these strange shapes, which remain infinitely complex at whatever level of magnification you look at them, all over the natural world. If you break off a floret from a cauliflower and magnify it, it looks remarkably like the cauliflower you started with. If you zoom in on a jagged bolt of lightning, then instead of looking quite straight, the magnified section looks like a copy of the original bolt. Mandelbrot christened these shapes fractals, and referred to them as “the geometry of nature” since they represent a genuinely new sort of shape only really recognized for the first time in the twentieth century.
There is a practical reason for the natural evolution of these fractal shapes. The fractal character of the human lung means that even though it sits inside the finite volume of the rib cage, its surface area is huge, and it can therefore absorb a lot of oxygen. The same goes for other organic objects. Ferns, for example, are looking to maximize their exposure to sunlight while not taking up too much space. It harks back to nature’s great ability to find shapes with the greatest efficiency. Just as the bubble found that the sphere is the shape that suits its needs best, life-forms have instead gone to the other end of the spectrum, choosing fractal shapes of infinite complexity.
The remarkable thing about fractals is that although they have this infinite complexity, they are actually generated by very simple mathematical rules. At first glance, it is difficult to believe that the complexity of the natural world could be based on simple mathematics, but the theory of fractals has revealed that even the most complex features of the natural world can be created by simple mathematical formulas.
HOW CAN A SHAPE BE 1.26-DIMENSIONAL?
The shapes that mathematicians encountered before fractals appeared on the scene were one-, two-, or three-dimensional—a one-dimensional line, a two-dimensional hexagon, a three-dimensional cube. Yet one of the most amazing discoveries in the theory of fractals is that these new shapes turn out to have dimensions greater than 1 but smaller than 2. If you are feeling brave, here’s an explanation of how a shape can have a dimension between 1 and 2.
The trick is to come up with a clever way to capture why a line is one-dimensional while a solid square is two-dimensional. Imagine taking a sheet of transparent graph paper, laying it over a shape, and counting how many squares contain part of the shape. Next, take a sheet of graph paper whose squares are half the size of those on the first piece.
If the shape is a line, the number of squares on the graph paper goes up simply by a factor of 2. If the shape is a solid square, the number of squares goes up by a factor of 4, or 22. Each time we halve the size of the grid on the graph paper, the number of squares meeting a one-dimensional shape increases by 2 = 21, while for a two-dimensional shape, the number increases by 22. The dimension corresponds to the power of 2.
The curious thing is that if you apply this procedure to the fractal coastline we constructed earlier in the chapter, when we halve the grid size of the graph paper, the number of squares that contain part of the coastline goes up by a factor of approximately 21.26. So from this perspective, the dimension of the mathematical coastline we constructed deserves to be called 1.26. We have thus created a new definition of dimension.
Figure 2.27 How to calculate the dimension of a fractal using graph paper. The dimension measures the increase in the number of pixels as you decrease the size of the pixels.
Instead of graph paper, you can capture these shapes with a pixelated computer screen. Make a pixel black if it contains some of the shape, and leave it white if not. As we increase the screen resolution, the dimension keeps track of the increase in the number of black pixels appearing. For example, if you move from 16 × 16 pixels to 32 × 32, then for a line, the number of black pixels doubles. For a solid square, the number of black pixels increases by a factor of 4, or 22. The number of black pixels in a computer image of the Koch snowflake increases by a factor of 21.26.
In a sense, the dimension tells us how much this infinite fractal line is trying to fill the space it occupies. If we construct variants of our fractal coastline in which we make the angle between the two lines that we add to the coast smaller and smaller, then the resulting coastline fills more and more of the space. And when we calculate the dimension of each of these sequences of coastline variants, we find that it is creeping closer and closer to 2:
Figure 2.28 As you change the angle of the triangle, the resulting fractal fills more space, and its fractal dimension increases.
If you analyze the fractal dimension of naturally occurring shapes, some interesting things emerge. The coastline of Britain is estimated to have a fractal dimension of 1.25—quite close to that of the mathematical coastline we constructed. We can think of the fractal dimension as telling us how fast the length of the coastline is increasing as we use smaller and smaller rulers to measure the coast. The fractal dimension of Australia’s coastline is estimated to be 1.13, indicating that it is less intricate in some sense than the coastline of Britain. Rather strikingly, the fractal dimension of the coastline of South Africa is only 1.04, which is a sign that it is very smooth. Perhaps the most fractal of all coastlines is Norway’s, with all its fjords—it comes out at 1.52.
Figure 2.29 What is the dimension of the coastline of Britain?
For objects in three dimensions, we can imagine playing a similar trick, but replacing the graph paper with a mesh of cubes and looking at how the shape intersects these cubes as the mesh gets finer and finer. Cauliflower comes out as a shape with dimension 2.33; a piece of paper crumpled into a ball hits 2.5; broccoli is quite intricate at 2.66; and amazingly, the surface of the human lung has fractal dimension 2.97.
CAN YOU FAKE A JACKSON POLLOCK?
In autumn 2006, a painting by the twentieth-century artist Jackson Pollock became the most expensive ever sold. It was reported that a Mexican financier, David Martinez, paid $140 million for the painting, called simply No. 5, 1948.
The painting was created by Pollock’s trademark technique of splashing paint across the canvas, which led to his nickname, Jack the Dripper. Critics were shocked at the price that was paid for such a piece, declaring, “Well, I could have made one of those!” and at first sight, it certainly looks as though anyone could splash paint around and hope to become a millionaire. But mathematics has revealed that Pollock was actually doing something more subtle than you might expect.
In 1999, a group of mathematicians led by Richard Taylor of the University of Oregon analyzed Pollock’s paintings and discovered that the jerky technique he used actually creates one of the fractal shapes so loved by nature. Magnified sections of a Pollock still look very similar to the full-size version and appear to have the characteristic infinite complexity of a fractal. (Of course, progressively increasing the magnification will eventually reveal the individual spots of paint, but this happens only when you magnify the canvas one thousand times.) The idea of fractal dimension can even be applied to analyze how Pollock’s technique developed.
Figure 2.30 The fractal dimension of a painting increases as you splash on more paint.
Pollock started to create fractal pictures in 1943. His early paintings have fractal dimensions in the region of 1.45, similar to the value for the fjords in Norway, but as he developed his technique, so the fractal dimensions crept up, reflecting the fact that the paintings were becoming more complex. One of Pollock’s last drip paintings, known as Blue Poles, took six months to complete and has a fractal dimension of 1.72.
Psychologists have explored the shapes that people find aesthetically pleasing. We are consistently drawn to images whose fractal dimension is between 1.3 and 1.5, similar to the dimension of many of the shapes found in nature. Indeed, there may be good evolutionary reasons why our brains are drawn to these sorts of fractals, because they are shapes that the brain has become hardwired to recognize as we negotiate the jungle around us. Or maybe, just as the best music sits somewhere between the extremes of boring elevator music and random white noise, these shapes appeal to us because they have a complexity that lies between the too regular and the too random.
If Pollock was creating fractals, how easy is it to replicate his technique? In 2001, a Texas art collector was getting worried that his “Pollock” didn’t have a signature or date anywhere on the canvas, so he took it to the mathematicians who’d revealed the fractal dimension of Pollock’s style. Their analysis showed that this painting lacked the special fractal character of Pollock’s jerky style, so they suggested that it was probably a fake. Five years later, the Pollock-Krasner Authentication Board, set up by the artist’s estate to rule on disputed works, asked Richard Taylor and his team to apply their fractal analysis to a collection of 32 paintings that had recently been found in a storage locker and were believed to be by Jackson Pollock. The fractal analysis implied that they, too, were all fakes.