The Number Mysteries: A Mathematical Odyssey through Everyday Life

by Marcus du Sautoy

  WHY MIGHT CATCHING AN ICOSAHEDRON KILL YOU?

  In 1918, the Spanish flu pandemic killed at least fifty million people—far more than the casualties of the First World War. Such devastation concentrated scientists’ minds on determining the mechanism of this dangerous disease, and they soon realized that the cause was not bacteria but something much smaller that couldn’t be seen under the microscopes of the time. They called these new agents “viruses,” after the Latin word for poison.

  Uncovering the true nature of these viruses had to wait for the development of a new technology, called X-ray diffraction, which gave scientists a way of penetrating the underlying molecular structure of the organisms that were causing such havoc. A molecule can be visualized as a collection of Ping-Pong balls joined together with toothpicks. Although this sounds too simple to be real science, every chemistry lab is equipped with ball-and-stick kits to help students and researchers explore the structure of the molecular world. In X-ray diffraction, a beam of X-rays is passed through the material being investigated, and the X-rays are deflected through various angles by the molecules they encounter. The pictures that are produced are a bit like the shadow you get when you shine a light on one of these ball-and-stick structures.

  Math was a powerful ally in the battle to unravel the information contained in these shadows. The game is to identify what three-dimensional shapes could possibly give rise to the two-dimensional shadows that X-ray diffraction produced. Quite often, progress depends on finding the best angle at which to “shine the light” and reveal the molecule’s true character. A silhouette of someone’s head from the front gives little information beyond whether that person’s ears stick out, but a profile tells you much more about who you’re looking at. It’s the same with molecules.

  Having cracked the structure of DNA, Francis Crick and James Watson, along with Donald Caspar and Aaron Klug, turned their attention to what the two-dimensional pictures from X-ray diffraction could reveal about viruses. To their surprise, they found shapes full of symmetry. The first images showed dots arranged in triangles, which implied that the virus had a three-dimensional shape that could be spun by a third of a turn and look the same. When the biologists looked in the mathematicians’ cabinet of shapes, it was the Platonic solids that seemed the best candidates for the form of these viruses.

  The problem was that all five of Plato’s shapes had an axis about which you could spin the shape by a third of a turn so that all the faces realign. It was only when the biologists obtained another diffraction image that they got a view that enabled them to pin down the shapes of these viruses more precisely. Suddenly, dots arranged in pentagons appeared, and that allowed them to home in on one of the more interesting of Plato’s dice: the icosahedron—the shape made of 20 triangles with five triangles meeting at each point.

  Viruses like symmetrical shapes because symmetry provides a very simple means for them to multiply, and that is what makes viral diseases so infectious—in fact, that’s what virulent means. Traditionally, symmetry has been something people have found aesthetically appealing, whether it is seen in a diamond, a flower, or the face of a supermodel. But symmetry isn’t always so desirable. Some of the deadliest viruses in the biological books—from influenza to herpes, from polio to the AIDS virus—are constructed using the shape of an icosahedron.

  Imagining Shapes

  Imagine hanging a cube-shaped decoration on a Christmas tree, with the string attached to one of the corners, or vertices. If you cut the cube horizontally between the point at the top and the point at the bottom, you get two pieces, each with a new face. What is the shape of that new face? The answer is at the end of the chapter.

  IS THE BEIJING OLYMPIC SWIMMING CENTER UNSTABLE?

  The swimming center built for the Beijing Olympics is an extraordinarily beautiful sight, especially lit up at night, when it looks like a transparent box full of bubbles. Its designer, the firm Arup, was keen to capture the spirit of the aquatic sports played inside but wanted also to give the building a natural, organic look.

  The firm began by looking at shapes that can tile a wall—like squares or equilateral triangles or hexagons—but decided that these were all too regular and didn’t capture the organic quality the firm was after. It explored other ways in which nature packs things together, like crystals or cell structures in plant tissue. In all these structures, there are examples of the sort of shapes that Archimedes discovered made such good soccer balls, but Arup was particularly drawn to the shapes made by lots of bubbles packed together to make foam.

  Considering that it took until 1884 to prove that the sphere is the most efficient shape for a single bubble, it may not come as a surprise that sticking more than one bubble together to make foam leads to some tough questions that are still vexing mathematicians today. If you have two bubbles that contain the same volume of air, what shape do they make when they join together? The rule is always that bubbles are lazy and look for shapes with the least energy. Energy is proportional to surface area, so they try to make a shape that has the smallest surface area of soap film. Since two joined bubbles share a boundary, they can make a shape with smaller surface area than just two bubbles touching.

  If you blow bubbles and two bubbles of the same volume fuse together, then the combination looks like this:

  Figure 2.9

  The two partial spheres will meet at an angle of 120 degrees and be separated by a flat wall. This is certainly a stable state—if it wasn’t, nature wouldn’t let the bubbles stay as they are. But the question is whether there might be another shape that has even less surface area—and therefore less energy—that would make it even more efficient. It might require putting some energy into the bubbles to take them out of their current stable state, but perhaps there is an even lower energy state that two bubbles could assume. For example, perhaps the two fused bubbles could be bettered by some weird configuration with less energy in which one bubble takes the shape of a bagel and wraps itself around the other bubble, squeezing it into the shape of a peanut, like this:

  Figure 2.10

  The first proof that the fused bubbles couldn’t be bettered was announced in 1995. Although mathematicians don’t really like asking for help from a computer (because that doesn’t appeal to their sense of elegance and beauty), they needed one to check through the extensive numerical calculations that were involved in their proof.

  Five years later, a pencil-and-paper proof of the double-bubble conjecture was announced. It actually proved a more general conjecture: if the bubbles do not enclose the same volume, but rather one is smaller than the other, then the bubbles fuse together so that the wall between the bubbles is no longer flat but bent into the small bubble. The wall is part of a third sphere and meets the two spherical bubbles in such a way that the three soap films have angles of 120 degrees between them:

  Figure 2.11

  In fact, this 120-degree property turns out to be a general rule for the way soap bubbles fuse together. It was first discovered by Belgian scientist Joseph Plateau, who was born in 1801. While he was doing research into the effect of light on the eye, he stared at the sun for half a minute, and by the age of 40, he was blind. Then, with the help of relatives and colleagues, he switched to investigating the shape of bubbles.

  Plateau began by dipping wire frames into bubble mixture and examining the different shapes that appeared. For example, when you dip a wire frame in the shape of a cube into the mixture, you get 13 walls that meet at a square in the middle:

  Figure 2.12

  This “square,” however, isn’t quite a square—the edges bulge out. As Plateau explored the various shapes that appeared in different wire frames, he began to formulate a set of rules for how bubbles join together.

  The first rule was that soap films always meet in threes at an angle of 120 degrees. The edge formed by these three walls is called a Plateau border in his honor. The second rule was about the way these borders can meet. Plateau borders meet in fours at an a
ngle of about 109.47 degrees (cos–1 – ⅓, to be precise). If you take a tetrahedron and draw lines from the four vertices to the center, you get the configuration of the four Plateau borders in foam. So the edges in the bulging square at the center of the cube wire frame actually meet at 109.47 degrees:

  Figure 2.13

  Any bubble that didn’t satisfy Plateau’s rules was believed to be unstable and would therefore collapse to a stable configuration that did satisfy these rules. It was not until 1976 that Jean Taylor finally proved that the shape of bubbles in foam had to satisfy the rules laid down by Plateau. Her work tells us how the bubbles connect together, but what about the actual shapes of the bubbles in foam? Because bubbles are lazy, the way to the answer is to find the shapes that enclose a given amount of air in each bubble in the foam while minimizing the surface area of soap film.

  Honeybees have already worked out the answer to the problem in two dimensions. The reason they construct their hives using hexagons is that this uses the least amount of wax to enclose a fixed amount of honey in each cell. Yet again, it was only a very recent breakthrough that confirmed the honeycomb theorem: there is no other two-dimensional structure that can beat the hexagonal honeycomb for efficiency.

  Once we step up to three-dimensional structures, though, things become less clear. In 1887, the famous British physicist Lord Kelvin suggested that one of Archimedes’s soccer balls was the key to minimizing the surface area of the bubbles. He believed that while the hexagon was the building block of the efficient beehive, the truncated octahedron—a shape made by cutting the six corners off of a standard octahedron—was the key to constructing foam:

  Figure 2.14

  The rules that Plateau developed for how bubbles in foam must meet show that the edges and faces are not actually flat—they’re curved. For example, the edges of a square meet at 90 degrees, but by the second of Plateau’s rules, that isn’t permitted. Instead, the edges of the square bulge out as they do in the cube wire frame, so that the two soap films meet at the requisite 109.47 degrees.

  Figure 2.15 A foam of truncated octahedra.

  Many believed that Kelvin’s structure must be the answer to how to build bubbles with minimum surface area, but no one could prove it. But in 1993, Denis Weaire and Robert Phelan at the University of Dublin found two shapes that packed together to beat the Kelvin structure by 0.3 percent (a warning to anyone who thinks that proving things in mathematics is a waste of time).

  Figure 2.16 The shapes that Denis Weaire and Robert Phelan discovered.

  The shapes turn out not to be on Archimedes’s list. The first is made up of irregular pentagons built into a distorted dodecahedron. The second shape is called a tetrakaidecahedron and consists of two elongated hexagonal faces and 12 irregular pentagonal faces of two different sorts. Weaire and Phelan found that they could pack these shapes together to create a more efficient foam than the one proposed by Kelvin. Again, to satisfy the rules of Plateau, the edges and faces need to be curved rather than straight. Interestingly, it’s quite hard to get inside foam to see what’s really happening, and the shapes were discovered thanks to experiments the two scientists did by using computers to simulate foams.

  Is this the best that bubbles can do? We don’t know. We believe that this is the most efficient network of shapes. But then again, Kelvin thought that he had the answer.

  The designers at Arup had been looking at mist, icebergs, and waves in their search for interesting natural shapes that evoked the sports that would take place inside the Olympic swimming center. When they chanced upon the foams of Weaire and Phelan, they realized that here was the potential to create something that had never previously been attempted in the architectural world. In order to create shapes that didn’t look too regular, they decided to slice the foam at an angle. What you get instead is actually the shapes the bubbles would make if you inserted a sheet of glass through the foam at an angle.

  Although Arup’s structure looks quite random, it does repeat itself across the building; but it still has the organic feel that the designers were seeking. If you look closely, though, there is one bubble that doesn’t appear to satisfy Plateau’s rules, because there are angles of 90 degrees in the shape, in addition to the 120 degrees or 109.47 degrees that Plateau demands. (You can see the odd bubble here: http://www.flickr.com/photos/88315679@N00/2243512961/in/photostream/.) So is the Water Cube stable? If it were really made of bubbles, the answer would be no. This one right-angled bubble would have to change its shape to satisfy the mathematical rules all bubbles must obey. However, the authorities in China shouldn’t worry. The Water Cube is likely to stay standing, thanks to the mathematics that went into creating such a beautiful structure.

  It isn’t just Arup and the Chinese authorities who are interested in the shape of lots of bubbles squashed together. Understanding the structure of foam helps us to understand much else in nature—for example, the structure of plant cells, the structure of chocolate and whipped cream, and the structure of the head on a pint of beer. Foam is used to put out fires, protect water from radioactive spills, and process minerals. Whether you are interested in fighting fires or making sure that the head on your Guinness doesn’t vanish too quickly, the answer lies in understanding the mathematical structure of foam.

  WHY DOES A SNOWFLAKE HAVE SIX ARMS?

  One of the first people to try to give a mathematical answer to this question was the seventeenth-century astronomer and mathematician, Johannes Kepler. He got his idea for why snowflakes have six arms by looking inside a pomegranate. The seeds in a pomegranate start off as spherical balls. As anyone who sells fruit knows, the most efficient way to fill space with spherical balls is to arrange them into layers of hexagons. The layers fit neatly on top of one another, with each ball nestling over three balls in the layer below. Together, the four balls are arranged so that they are at the corners of a tetrahedron.

  Kepler conjectured that this was the most efficient way to pack space—in other words, the arrangement in which the spaces between the balls take up the least volume. But how can you be sure that there isn’t some other complicated arrangement of balls that would improve upon this hexagonal packing? The Kepler conjecture, as this innocent statement came to be known, would obsess generations of mathematicians. A proof did not appear until the end of the twentieth century, when mathematicians joined forces with the power of the computer.

  Now, back to the pomegranate. As the fruit grows, the seeds begin to squash together, morphing from spheres into shapes that fill space completely. Each seed at the heart of the pomegranate is in contact with 12 others, and as they squash together, the seeds change into shapes with 12 faces. You might think that the dodecahedron with its 12 pentagonal faces would be the shape the seeds adopted, but you can’t put dodecahedrons together so that they stack perfectly, filling all the available space. The only Platonic shape that stacks perfectly to fill space is the cube. Instead, the 12 faces on the seed form a kite shape. Called the rhombic dodecahedron, it is a shape often found in nature:

  Figure 2.17

  Crystals of garnet have 12 faces looking like kites. The word garnet actually comes from the Latin for pomegranate because the seeds of the fruit also form tiny red 12-sided solids with kite-shaped faces.

  Analyzing the kite-shaped faces of the pomegranate seed inspired Kepler to start investigating all the possible symmetrical shapes that could be built out of this slightly less symmetrical kite-shaped face. Plato had considered shapes made from one perfectly symmetrical face; Archimedes took things a step further by looking at shapes made from two or more symmetrical faces. Kepler’s investigations sparked off a whole industry dedicated to different shapes that extend the ideas of Plato and Archimedes. We now have the Catalan solids, the Poinsot solids, the Johnson solids, shaky polyhedra, and zonohedra—and many more exotic objects.

  Kepler believed that the hexagons at the heart of the way balls pack together were responsible for the snowflake having six arms. His analysis fo
rmed the subject of a book he dedicated to an imperial diplomat named Matthaeus Wackher as a New Year’s present—an astute move by a scientist always on the lookout for funding. As spherical raindrops froze in clouds, Kepler thought, they were somehow packing themselves together like the pomegranate seeds. A nice idea, but it turned out to be wrong. The real reason for the snowflake’s six arms relates to the molecular structure of water—something that would be revealed only with the invention of X-ray crystallography in 1912.

  A molecule of water is made up of one oxygen atom and two hydrogen atoms. When water molecules bind together to form crystals, each oxygen atom shares its hydrogen atoms with neighboring oxygen atoms and, in turn, borrows two other hydrogen atoms from other water molecules. So an ice crystal assembles itself with each oxygen atom connected to four hydrogen atoms. In a ball-and-stick model, four balls representing hydrogen atoms are arranged around each oxygen atom in a shape that ensures that each hydrogen atom is as far away as possible from the three other hydrogen atoms. The mathematical solution to such a requirement is to position each hydrogen atom at the corner of a tetrahedron—the Platonic shape made up of four equilateral triangles—with the oxygen atom at its center: