What’s the million-dollar prime problem?
The million-dollar question is about the nature of these dice: are the dice fair or not? Are the dice distributing the primes fairly through the universe of numbers, or are there regions where they are biased, sometimes giving too many primes, sometimes too few? The name of this problem is the Riemann Hypothesis.
Bernhard Riemann was a student of Gauss’s in the German city of Göttingen. He developed some very sophisticated mathematics which allows us to understand how these prime number dice are distributing the primes. Using something called a zeta function, special numbers called imaginary numbers and a fearsome amount of analysis, Riemann worked out the mathematics that controls the fall of these dice. He believed from his analysis that the dice would be fair, but he couldn’t prove it. To prove the Riemann Hypothesis, that is what you have to do.
Another way to interpret the Riemann Hypothesis is to compare the prime numbers to molecules of gas in a room. You may not know at any one instance where each molecule is, but the physics says that the molecules will be fairly evenly distributed around the room. There won’t be one corner with a concentration of molecules, and at another corner a complete vacuum. The Riemann Hypothesis would have the same implication for the primes. It doesn’t really help us to say where each particular prime can be found, but it does guarantee that they are distributed in a fair but random way through the universe of numbers. That kind of guarantee is often enough for mathematicians to be able to navigate the universe of numbers with a sufficient degree of confidence. However, until the million dollars is won, we’ll never be certain quite what the primes are doing as we count our way further into the never-ending reaches of the mathematical cosmos.
The Story of the Elusive Shape
The great seventeenth-century scientist Galileo Galilei once wrote:
The universe cannot be read until we have learnt the language and become familiar with the characters in which it is written. It is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word. Without these, one is wandering about in a dark labyrinth.
This chapter presents the A–Z of nature’s weird and wonderful shapes: from the six-pointed snowflake to the spiral of DNA, from the radial symmetry of a diamond to the complex shape of a leaf. Why are bubbles perfectly spherical? How does the body make such hugely complex shapes like the human lung? What shape is our universe? Mathematics is at the heart of understanding how and why nature makes such a variety of shapes, and it also gives us the power to create new shapes as well as the ability to say when there are no more shapes to be discovered.
It isn’t only mathematicians who are interested in shapes: architects, engineers, scientists and artists all want to understand how nature’s shapes work. They all rely on the mathematics of geometry. The Ancient Greek philosopher Plato put above his door a sign declaring: ‘Let no one ignorant of geometry enter here.’ In this chapter I want to give you a passport to Plato’s home, to the mathematical world of shapes. And at the end I’ll reveal another mathematical puzzle, one whose solution is worth another million dollars.
Why are bubbles spherical?
Take a piece of wire and bend it into a square. Dip it in bubble mixture and blow. Why isn’t it a cube-shaped bubble that comes out the other side? Or if the wire is triangular, why can’t you blow a pyramid-shaped bubble? Why is it that, regardless of the shape of the frame, the bubble comes out as a perfect spherical ball? The answer is that nature is lazy, and the sphere is nature’s easiest shape. The bubble tries to find the shape that uses the least amount of energy, and that energy is proportional to the surface area. The bubble contains a fixed volume of air, and that volume does not change if the shape changes. The sphere is the shape that has the smallest surface area which can contain that fixed amount of air. That makes it the shape that uses the least amount of energy.
Manufacturers have long been keen to copy nature’s ability to make perfect spheres. If you’re making ball bearings or shot for guns, getting perfect spheres could be a matter of life and death, since a slight imperfection in the spherical shape could lead to a gun backfiring or a machine breaking down. The breakthrough came in 1783 when a Bristol-born plumber, William Watts, realized that he could exploit nature’s predilection for spheres.
When molten iron is dropped from the top of a tall tower, like the bubble the liquid droplets form into perfect spheres during their descent. Watts wondered whether, if you stuck a vat of water at the bottom of the tower, you could freeze the spherical shapes as the droplets of iron hit the water. He decided to try his idea out in his own house in Bristol. The trouble was that he needed the drop to be further than three floors to give the falling molten lead time to form into spherical droplets.
So Watts added another three storeys on top of his house and cut holes in all the floors to allow the lead to fall through the building. The neighbours were a bit shocked by the sudden appearance of this tower on the top of his home, despite his attempts to give it a Gothic twist with the addition of some castlelike trim around the top. But so successful were Watts’s experiments that similar towers soon shot up across England and America. His own shot tower stayed in operation until 1968.
Although nature uses the sphere so often, how can we be sure that there isn’t some other strange shape that might be even more efficient than the sphere? It was the great Greek mathematician Archimedes who first proposed that the sphere was indeed the shape with the smallest surface area containing a fixed volume. To try to prove this, Archimedes began by producing formulas for calculating the surface area of a sphere and volume enclosed by it.
FIGURE 2.01 William Watts’s clever use of nature to make spherical ball bearings.
Calculating the volume of a curved shape was a significant challenge, but he applied a cunning trick: slice the sphere with parallel cuts into many thin layers, and then approximate the layers by discs. Now, he knew the formula for the volume of a disc: it was just the area of the circle times the thickness of the disc. By adding together the volumes of all these different sized discs, Archimedes could get an approximation for the volume of the sphere.
FIGURE 2.02 A sphere can be approximated by stacking different sized discs on top of one another.
Then came the clever bit. If he made the discs thinner and thinner until they were infinitesimally thin, the formula would give an exact calculation of the volume. It was one of the first times that the idea of infinity was used in mathematics, and a similar technique would eventually become the basis for the mathematics of the calculus developed by Isaac Newton and Gottfried Leibniz nearly two thousand years later.
Archimedes went on to use this method to calculate the volumes of many different shapes. He was especially proud of the discovery that if you put a spherical ball inside a cylindrical tube of the same height, then the volume of the air in the tube is precisely half the volume of the ball. He was so excited by this that he insisted a cylinder