But what does it mean to raise a number to an imaginary power? How do we multiply 23, or indeed any other number, by itself √–1 times? Surely this is mathematics run amok.
Again, the trick is not to think of numbers but functions. Euler knew a way to write the function exp(x) as an infinite sum. With this equation, it was a simple matter for Euler to substitute ix in place of x, keeping in mind that i2 = –1, i3 = –i, i4 = 1, and so forth. The result is:
Separating the terms without i from the terms with i, Euler instantly recognized the second and third most important functions of calculus, the sine and cosine functions:
This is the formula that Euler himself considered important! It is featured in his calculus textbook of 1748. Nowhere in that textbook, or anywhere else, did he write the equation that has become associated with his name (eiπ = –1). Euler understood that calculus was about functions, not about numbers. However, we can get the “number version” of his formula easily enough by the final step of substituting x = π. Then eiπ = exp(iπ) = cos(π) + i sin(π) = –1 + 0i = –1.
Beauty is, of course, in the eye of the beholder. The readers of Mathematical Intelligencer preferred the numerical formula because it relates the five fundamental constants of mathematics. One could argue that the version with exp, cos, and sin is much more beautiful, because it relates the three most fundamental functions of calculus, functions that have likewise been studied for centuries. Furthermore, it explains the meaning of the otherwise opaque equation, eiπ = –1. Surely a formula that helps us understand mathematics better is much more beautiful than a formula that only mystifies us.
Above A cube has 8 vertices, 12 edges, and 6 faces.
V – E + F = 2. Second in the Intelligencer poll was this elegant formula, which relates the number of vertices (V), edges (E), and faces (F) of any polyhedron. For example, a cube has 8 vertices, 12 edges, and 6 faces; and, sure enough, 8 – 12 + 6 = 2. As it turns out, this equation has exceptions that Euler was not aware of. For a doughnut-shaped polyhedron, for instance, V – E + F = 0, not 2. With hindsight, it is clear that this equation marked the beginning of a new branch of mathematics called topology, which flourished in the twentieth century. The number V – E + F is now called the Euler characteristic. It is a “topological invariant” that distinguishes one two-dimensional surface from another. Sphere-shaped surfaces always have Euler characteristic 2; doughnut-shaped surfaces always have Euler characteristic 0; pretzel-shaped surfaces have Euler characteristic –4, and so on.
The infinitude of prime numbers. This was an ancient discovery, known to Euclid, but Euler discovered a radically different proof, which not surprisingly uses the concepts of functions and infinite series that were so dear to him. The proof involves the zeta function, ζ(x) = 1 + 1/2x + 1/3x + 1/4x + … Euler showed that this infinite sum is also equal to Euler’s product:
The product in the denominator runs over all prime numbers (2, 3, 5, 7, …). For number theorists, Euler’s product is probably the most important formula ever discovered. Most of what we know about the distribution of prime numbers comes from the careful study of the zeta-function: a theme that will be returned to later.
The Basel Problem. Finally, the fourth of Euler’s equations, which was fifth place in the list, was the formula, already mentioned, that cemented his reputation:
The discerning reader will notice that the left-hand side is actually ζ(2), and might even wonder if Euler’s product for the zeta function has something to do with this formula. To the best of my knowledge, the answer is no. Euler actually derived it from an infinite product representation for the sine function, rather than the zeta function.
THE INTELLIGENCER LIST was, possibly, somewhat biased toward pure mathematics. There aren’t very many formulas on this list that are used in non-mathematical applications. And that’s a pity, because Euler could do it all. He developed the first theory of hydrodynamics; he studied the buckling of elastic rods; he even worked on the optimal placement of masts in ships (a very important practical problem of the day). It is doubtful that he saw much of a distinction between mathematics that was done for its own beauty and mathematics that was done to solve a practical problem.
Finally, as math historian Jeremy Gray points out, one of Euler’s most important contributions to math was not an equation at all. Part Two has described throughout how controversies arose and progress slowed because mathematicians, for one reason or another, were reluctant to share their secrets. It is the one thing that del Ferro, Tartaglia, Galileo, Fermat, and Newton all had in common. Euler was the one shining exception. He published abundantly; he was willing to step aside and give others credit; his articles routinely delivered more than they promised. He led by example, and helped transform mathematics into what it is today—a profession where information is not proprietary but is (with some unusual and unfortunate exceptions) openly shared.
* * *
* For well over a century, the most prestigious honors in mathematics were international competitions, organized by the national academies of science, in which papers were solicited on a particular topic. Euler won his first Grand Prix de Paris in 1738 for a paper on the nature of fire, and subsequently won eleven more (a record, of course).
PART THREE
equations in a promethean age
If you ever go to Dublin, Ireland, take a bus to Broombridge Road and get off at the Royal Canal. You may not realize it, but you have just arrived at the site of the most famous mathematical graffiti in history.
From street level, the stone bridge that the road takes its name from is small and nondescript, but if you descend to canal level and walk to the west side of the bridge, you will find (along with lots of modern, spray-painted graffiti) a plaque with the following inscription:
“Here, as he walked by on the 16th of October 1843, Sir William Rowan Hamilton in a flash of genius discovered the fundamental formula for quaternion multiplication i2 = j2 = k2 = ijk = –1 & cut it on the stone of this bridge.”
To be honest, no one knows if Hamilton really did carve his formula into Brougham (pronounced “broom”) Bridge. The source of the story is a letter that he wrote to his son, Archibald, many years later, and like many family stories it may have been embellished. However, there is no doubt of Hamilton’s excitement over the discovery of quaternions, which he considered the greatest of his life.
Hamilton became so besotted with his creation that he spent the rest of his life studying the equations. Viewed from more than a century later, the discovery was in fact a turning point in mathematical history, but in a subtler way than Hamilton could have anticipated. They were the first example of a new algebra, created entirely out of one person’s imagination. This step, along with the nearly simultaneous discovery by other mathematicians of new geometries and new functions, liberated mathematicians from traditional structures (and strictures). For the first time, they could venture beyond the real world—they were free to invent entire new worlds.
Before the nineteenth century, there was only one algebra and one geometry. The idea did not even occur to mathematicians to invent anything different. It is true that the concept of “number” had gradually expanded over the centuries, first to include irrationals, then zero and negatives, and finally imaginary numbers. But these new kinds of number were annexed only with great difficulty, and only after bitter debate. They were accepted only because they were indispensible. Similarly, calculus was a revolutionary technique but it did not involve the creation of a new geometry. Newton’s concept of space was exactly the same as Euclid’s.
All of this changed in the nineteenth century. It was a revolutionary era, in mathematics as in the outside world. Beginning with the French Revolution, European societies were scrapping their old political structures and creating new ones. Likewise, mathematicians began trying out new structures that directly contradicted axioms they had been using for centuries. It was an era when Mary Shelley could write her novel, Frankenstein; or the Modern Prometheus,
warning of the dangers of scientists playing God. Mathematicians became modern-day Prometheans, like Dr. Frankenstein, although their creations were not made of flesh and blood.
Hamilton would have deplored this development. Socially he was conservative, a supporter of the English Crown in an Ireland that was starving and chafing under English domination. Mathematically, he invented quaternions in order to understand Euclidean space, not in order to create a new algebra. Nevertheless, revolutions are often begun by people who have no inkling of what they are starting.
13
the new algebra hamilton and quaternions
Born in 1806, William Rowan Hamilton was a child prodigy who knew all the European languages, as well as Hebrew, Latin, Greek, and others, by the time he was ten years old. Hamilton was an enthusiastic amateur poet throughout his life, and was a close friend of the English poet William Wordsworth. It was Wordsworth who delicately, but wisely, advised Hamilton that he had more to offer the world as a scientist than as a poet.
In 1827, Hamilton was appointed Royal Astronomer of Ireland—even though he had not yet graduated from university! The appointment had more to do with his research on optics than his interest in astronomy. He had already published papers on optics as an undergraduate, and five years later he made a sensation with his discovery of conical refraction. Certain kinds of crystals, which are called birefringent, can split a light beam up into two separate beams. Hamilton proved mathematically that if the angle of incidence was just right, the beam would split up not just into two beams, but into a hollow cone of light. Later that year Humphrey Lloyd demonstrated conical refraction in his laboratory. It was one of the first times that a new physical phenomenon had been deduced by pure mathematics first, and confirmed by experiment second. After this breakthrough, Hamilton was no longer just a prodigy; he was one of Great Britain’s scientific heroes.
i, j, k represent imaginary units. Multiples of these units can be added to real numbers to form quaternions, a + bi + cj + dk. The above multiplication rules will then uniquely define the product (and, with a little work, the quotient) of any two quaternions.
Hamilton’s other great discovery, quaternions, had a much longer and stranger history. Sometime around 1830 Hamilton began looking for a way of multiplying number triplets together. By this time the multiplication and division of number pairs, or complex numbers, had proven itself to be not only possible but an essential part of mathematics. Any two such pairs, say (a + bi) and (c + di), can be multiplied or divided, using the rules of algebra plus the miraculous identity i2 = –1:
But there was another motivation for multiplying number triples. Hamilton knew that complex multiplication has a geometric meaning, quite apart from its origins in algebra. For example, the algebraic operation “multiply by i” is the same as the geometric operation “rotate by 90 degrees counterclockwise.” More generally, the instruction “multiply by (a + bi)” can be broken down into two steps—a rotation and a dilation. This interpretation takes a lot of the mystery out of complex numbers. Some people may have trouble imagining a number whose square is –1, but everybody knows that two 90-degree rotations give a 180-degree rotation. Not only that, this description makes the invertibility of complex multiplication completely obvious. To undo the operation “rotate counterclockwise by 72 degrees,” you simply rotate clockwise the same amount. To undo the operation “enlarge to 150 percent,” you reduce to 67 percent.
Alas, complex numbers are limited to representing operations in a plane. They are great for manipulating two-dimensional photographs, but not three-dimensional reality. Hamilton was convinced that there must be an algebra of three dimensions as powerful as the two-dimensional algebra of complex numbers. But until 1843, he was stymied. The rock on which all his attempts foundered was division. No matter how he defined the multiplication of number triples, he was not able to divide them.
Above The plaque on Brougham Bridge, Dublin, commemorating Hamilton’s quaternion equation.
And then came Hamilton’s Monday stroll across Brougham Bridge. What he suddenly realized—although his calculations must have subconsciously been leading him to this point—was that introducing a fourth number makes both multiplication and division possible.
Thus Hamilton proclaimed: Let there be three imaginary units, i, j, and k. Let them go forth and multiply using the following rules:
Then one can add, subtract, multiply, and divide any two quaternions, (a + bi + cj + dk) and (w + xi + yj + zk), just by following the normal rules of algebra. Division is the trickiest part, of course. It turns out that 1/(a + bi + cj + dk) equals (a – bi – cj – dk)/(a2 + b2 + c2 + d2), as you can check by multiplying both sides by (a + bi + cj + dk). Don’t even think about what the imaginary numbers i, j, and k mean—just do it and you’ll see that it works.
SUCH A WANTONLY Promethean act had almost never been seen before in mathematics. Hamilton’s colleagues were aghast, though they could not find anything wrong with it. “There is still something in the system which gravels me,” wrote his friend John Graves. “I have not yet any clear view as to the extent to which we are at liberty arbitrarily to create imaginaries, and to endow them with supernatural properties.”
Hamilton spent the remaining 22 years of his life proselytizing the importance of quaternions. He wrote a 700-page book about them but then, convinced it was too difficult, started a shorter “manual” for students—which grew to more than 800 pages and lay unfinished when he died.
However, quaternions fell from popularity for a variety of reasons. First, Hamilton had intended to find an algebra of three-dimensional space. What, then, did the fourth dimension of a quaternion mean? Hamilton argued that it could represent time*—and in so doing, he became the first scientist to merge time and space into a single “spacetime.” However, physics had not yet matured to the point where it needed this concept; it would have to wait for the twentieth century and Albert Einstein.
A second blow to quaternions was the development of vector analysis in the 1870s, by Oliver Heaviside (an Englishman) and Josiah Willard Gibbs (an American). Heaviside and Gibbs dispensed with the imaginary quantities entirely, and simply represented points in space by a triple of numbers, (a, b, c), called a vector. Instead of one multiplication, they defined two different vector multiplications, the dot product and the cross product. Neither one of them is invertible, and in fact the dot product of two vectors isn’t even a vector—it’s a real number. However, what they lack in elegance, vectors make up for in practicality. They are well adapted to the problems of physics and engineering. After a series of polemics in the 1890s between the quaternionists and the vector analysts, the analysts won. A typical example is this scathing comment from William Thomson, Lord Kelvin, in 1892: “Quaternions came from Hamilton after his really good work had been done; and though beautifully ingenious, have been an unmixed evil to those who have touched them in any way, including Clerk Maxwell.” Nowadays, you will never see a quaternion in a freshman physics book.
One reason that vectors won is that neither Hamilton nor his followers really understood what quaternions were, and therefore they were trying to use them the wrong way. Just as complex numbers represent geometric operations (rotations and dilations in a plane), quaternions represent rotations and dilations of space. Thus, they are not vectors. A vector is something that is acted upon by rotations. Quaternions are the rotation itself.
HAMILTON NEVER FIGURED OUT the difference. That was left to a twentieth-century mathematician, Elie Cartan, who named this kind of quantity a spinor (discussed in Part Four) to distinguish it from vectors. His findings vindicate Hamilton’s belief in quaternions, even though he did not grasp their significance. The key point to bear in mind is that quaternions are the very best way to represent anything that spins in three dimensions. That includes protons, neutrons, and electrons—the building blocks of our physical world. Of course, the existence of these subatomic particles was not even suspected in Hamilton’s time. He had discovered
the right mathematics almost a century before it would be needed.
But one immediate effect of quaternions, as mentioned above, was to liberate mathematicians to think about other kinds of algebra. Quaternion multiplication violates one previously unquestioned rule of algebra. It is not commutative; that is, the product of two quaternions is sensitive to the order in which they are multiplied. For example ij is equal to k, but ji is equal to –k. This was the first known example of a non-commutative algebra.
Hamilton’s friend Graves soon got over his uneasiness and discovered an algebra of 8-tuples, or octonions. These are even more finicky than quaternions, because products of three octonions are sensitive not only to the order of the octonions but also to their grouping. If a, b, and c are real numbers, or complex numbers, or even quaternions, then (ab) c = a(bc), but for octonions, (ab)c is usually not equal to a(bc). The independence of grouping was a property that mathematicians had always assumed without even realizing it; Hamilton had to make up a new word for it—the associative law. It might seem as if the next step would be an algebra of 16-tuples. However, each doubling of the number of dimensions comes with a sacrifice. Going from 2 dimensions to 4, you lose commutativity. Going from 4 dimensions to 8, you lose associativity. And going from 8 dimensions to 16, you lose division. At this point, Hamilton’s program of defining hypercomplex numbers breaks down, because the one thing he always insisted on was division.
The Universe in Zero Words Page 9