The Math Book

by DK

  When he died in 1665, French mathematician Pierre de Fermat left behind a well-thumbed copy of Arithmetica by the 3rd-century CE Greek mathematician Diophantus, its margins marked with Fermat’s ideas. All the questions posed in Fermat’s marginal scribbles were later solved, except for one. He left a tantalizing note in the margin: “I have discovered a truly marvelous proof, which this margin is too small to contain here.”

  Fermat’s note related to Diophantus’s discussion of Pythagoras’s theorem—that in a right-angled triangle the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares on the other two sides, or x2 + y2 = z2. Fermat knew that this equation had an infinity of integer solutions for x, y, and z, such as 3, 4, and 5 (9 + 16 = 25) and 5, 12, and 13 (25 + 144 = 169), known as “Pythagorean triples.” He then wondered if other triples could be found to the power of 3, 4, or any integer beyond 2. The conclusion Fermat reached was that no integer greater than 2 could stand for n. Fermat wrote: “It is impossible for a cube to be the sum of two cubes, a fourth power [number to the power of 4] to be the sum of two fourth powers, or in general for any number that is a power greater than the second to be the sum of two like powers.” Fermat never revealed the proof he claimed to have for his theory and so it remained unsolved, becoming known as Fermat’s last theorem.

  Many mathematicians attempted to reconstruct Fermat’s claimed proof after his death, or to find their own. But despite the seeming simplicity of the problem, no one was successful, although a century later Leonhard Euler did prove the theory where n = 3.

  Finding a solution

  Fermat’s last theorem remained one of the great unsolved problems in mathematics for more than 300 years, until it was proved by British mathematician Andrew Wiles in 1994. Wiles had first read about Fermat’s challenge when he was ten. He had been amazed that he, just a boy, could make sense of it, and yet the best mathematical minds in the world had failed to prove it. It made him want to study mathematics at the University of Oxford, and then to get his PhD at Cambridge. There, he chose elliptic curves as the area of study for his doctoral thesis—a subject that seemed to have little to do with his interest in Fermat. Yet it was this branch of mathematics that would enable Wiles to prove Fermat’s last theorem.

  In the mid-1950s, Japanese mathematicians Yutaka Taniyama and Goro Shimura had made the bold step of linking two apparently unrelated branches of mathematics. They claimed that every elliptic curve (an algebraic structure) could be associated with a unique modular form, one of a class of highly symmetrical structures belonging to number theory.

  The potential importance of their conjecture was gradually understood over the next three decades and it became part of an ongoing program to link different mathematical disciplines. However, no one had any idea how to prove it.

  In 1985, German mathematician Gerhard Frey made a link between the conjecture and Fermat’s last theorem. Working from a hypothetical solution to the Fermat equation, he constructed a curious elliptic curve that appeared not to be modular. He argued that such a curve could only exist if the Taniyama–Shimura conjecture were false, in which case Fermat’s last theorem would also be false. On the other hand, if the Taniyama–Shimura conjecture were true, Fermat’s last theorem would follow. In 1986, Ken Ribet, a professor at Princeton University, in New Jersey, managed to prove Frey’s conjectured link.

  Wiles’s investigation of Fermat’s last theorem began with his study of elliptic curves, which are described by the equation y2 = x3 + Ax + B, where A and B are constants (fixed).

  Proving the unprovable

  Ribet’s proof electrified Wiles. Here was the chance he had been waiting for—if he could prove the seemingly impossible Taniyama–Shimura conjecture, then he would also prove Fermat’s last theorem. Unlike most mathematicians, who like to work collaboratively, Wiles decided to pursue this goal on his own, telling no one except his wife. He felt that to talk openly about working on Fermat would stir up excitement in the mathematics community, and perhaps lead to unwanted competition. However, as the proof reached its final stages, in the seventh year of working on it, Wiles realized he needed help.

  At the time, Wiles was employed at the Institute for Advanced Study (IAS) in Princeton, home to some of the world’s finest mathematicians. These colleagues were completely astounded when Wiles revealed that he had been working on Fermat while still carrying out his daily tasks of lecturing, writing, and teaching.

  Wiles recruited the help of these colleagues for the final step in compiling his proof. He turned to American mathematician Nick Katz to check his reasonings. Katz could find no errors, so Wiles decided to go public. In June 1993, at a conference at the University of Cambridge, Wiles delivered his results. Tension rose as he piled his results one on top of the other, with only one end in view. He delivered his final line, “Which proves Fermat’s last theorem,” smiled, and added, “I think I’ll leave it there.”

  Some mathematics problems look simple. There’s no reason why these problems shouldn’t be easy, and yet they turn out to be extremely intricate.

  Andrew Wiles

  Fixing an error

  The next day, the world’s press was full of the story, transforming Wiles into the world’s most famous mathematician. Everyone wanted to know how this problem had finally been solved. Wiles was delighted, but then came a twist; there was a problem with his proof.

  The results had to be verified before they could be published—and Wiles’s proof covered scores of pages. Among the reviewers was Wiles’s friend Nick Katz. For a whole summer Katz went through the proof line by line, querying and questioning until the meaning was clear. One day, he thought he had spotted a hole in the argument. He emailed Wiles, who replied, but not to Katz’s satisfaction. More emails followed, before the truth emerged—Katz had found a flaw at the heart of Wiles’s work. A vital point in the proof contained an error that undermined Wiles’s method.

  Suddenly Wiles’s approach was brought into question. Had he worked with others rather than alone, the error might have been identified earlier. The world believed that Wiles had resolved Fermat’s last theorem, and it was waiting for the finished, published proof. Wiles was under immense pressure. His mathematical achievements so far had been impressive, but his reputation was at stake. Day after day, Wiles tried different approaches to the problem, which proved futile—as his fellow IAS mathematician Peter Sarnak said, “It was like pinning down a carpet in one corner of a room, only for the carpet to pop up in another.” Eventually, Wiles turned to a friend, British algebra specialist Richard Taylor, and they worked together on the proof for the next nine months.

  Wiles was close to having to admit that he had claimed a proof prematurely. Then, in September 1994, he had a revelation. If he took his present problem-solving method and added its strengths to an earlier approach of his, then one might fix the other, allowing him to solve the problem. It seemed a small insight, but it made all the difference. Within weeks, Wiles and Taylor had plugged the gap in the proof. Nick Katz and the wider mathematical community were now convinced there were no mistakes, and Wiles emerged for a second time as the conqueror of Fermat’s last theorem—this time on solid ground.

  I had this rare privilege of being able to pursue in my adult life what had been my childhood dream.

  Andrew Wiles

  After the theorem

  Fermat was amazingly far-sighted in his original conjecture, but it is unlikely that the “marvelous proof” he claimed to have discovered existed. The idea that every mathematician since the 1600s could have missed a proof that a mathematician from Fermat’s time could have discovered is inconceivable. In addition, Wiles solved the theorem using advanced mathematical tools and ideas invented long after Fermat.

  In many ways, it is not the proving of Fermat’s last theorem that has significance, but rather the proofs used by Wiles. A seemingly impossible problem about integers had been solved by marrying number theory to algebraic geometry, using new
and existing techniques. This in turn opened up new ways of looking at how to prove many other mathematical conjectures.

  ANDREW WILES

  The son of an Anglican priest who later became a professor of divinity, Wiles was born in Cambridge in 1953, and was a passionate problem-solver in mathematics from an early age. Awarded his first degree in mathematics at Merton College, Oxford, and his doctorate at Clare College, Cambridge, he took up a post at the Institute for Advanced Study in Princeton in 1981, and was appointed professor there the following year.

  While in the US, Wiles made contributions to some of the most elusive problems in his field, including the Taniyama–Shimura conjecture. He also began his long solo attempt to prove Fermat’s last theorem. His eventual success led to him receiving the Abel Prize—the highest honor in mathematics—in 2016.

  Wiles has also taught in Bonn and Paris, and at the University of Oxford, where he was appointed Regius Professor of Mathematics in 2018. A new mathematics building at Oxford—as well as an asteroid—9999 Wiles—have been named after him.

  See also: Pythagoras • Diophantine equations • Probability • Elliptic functions • Catalan’s conjecture • 23 problems for the 20th century • Finite simple groups

  IN CONTEXT

  KEY FIGURE

  Grigori Perelman (1966–)

  FIELDS

  Geometry, topology

  BEFORE

  1904 Henri Poincaré states his conjecture on the equivalence of shapes in 4-D space.

  1934 British mathematician Henry Whitehead stirs interest in Poincaré’s conjecture by publishing an erroneous proof.

  1960 American mathematician Stephen Smale proves the conjecture is true in the fifth and higher dimensions.

  1982 Poincaré’s conjecture is proved in four dimensions by American mathematician Michael Freedman.

  AFTER

  2010 When Perelman rejects the Clay Millennium Prize, the £1 million award is used to set up the Poincaré Chair for gifted young mathematicians.

  In 2000, the Clay Mathematics Institute in the US celebrated the millennium with seven prize problems. Among them was the Poincaré conjecture, which had challenged mathematicians for nearly a century. Within a few years, it was solved—by a little-known Russian mathematician, Grigori Perelman.

  Poincaré’s conjecture, conceived by the French mathematician in 1904, is stated as: “Every simply connected, closed 3-manifold is homeomorphic to the 3-sphere.” In topology, a field that studies the geometrical properties, structure, and spatial relations of shapes, a sphere (a 3-D object in geometry) is said to be a 2-manifold with a 2-D surface existing within a 3-D space—a solid ball, for example. A 3-manifold, such as the 3-sphere, is a purely theoretical concept: it has a 3-D surface and exists in a 4-D space. The description “simply connected” means that the figure has no holes, unlike a bagel or hoop shape (torus), and “closed” means the shape is limited by boundaries, unlike the open endlessness of an infinite plane. In topology, two figures are homeomorphic if they can be distorted or stretched into the same shape. While the question of whether every closed 3-manifold could be deformed to take the shape of a 3-sphere is hypothetical, Perelman has claimed that it holds the key to understanding the shape of the Universe.

  Finding a solid proof

  Initially, it proved easier to substantiate the conjecture for manifolds of the fourth, fifth, and higher dimensions than it was for 3-manifolds. In 1982, American mathematician Richard Hamilton attempted to prove the conjecture using Ricci flow, a mathematical process that potentially allows any 4-D shape to be distorted to an increasingly smooth version, and ultimately to a 3-sphere. However, the flow failed to handle spikelike “singularities”—deformities including “cigars” and infinitely dense “necks.”

  Perelman, who learned much from Hamilton during a two-year fellowship at Berkeley in the early 1990s, continued to study Ricci flow and its application to the Poincaré conjecture when he returned to Russia. He masterfully overcame the limitations that Hamilton encountered by using a technique called surgery, in effect cutting out the singularities, and was able to prove the conjecture.

  A 3-sphere is the 3-D equivalent of a spherical surface, that is a two-dimensional surface, or 2-sphere, such as the ball shown here. To appreciate the shape of the ball, it has to be viewed in 3-D space. To see a 3-sphere requires 4-D space.

  Surprising the math world

  Perelman had achieved success quietly. Unconventionally, he posted his first 39-page paper on the subject online in 2002, emailing a summary to 12 mathematicians in the US. He published two more installments a year later. Others reconstructed his results and explained them in the Asian Journal of Mathematics. Finally, his proof was fully accepted by the mathematical community in 2006.

  Since then, Perelman’s work has been closely studied, fuelling new developments in topology, including a more powerful version of his and Hamilton’s technique for using Ricci flow to smooth singularities.

  Perelman’s proof… solved a problem that for more than a century was an indigestible seed at the core of topology.

  Dana Mackenzie

  American science writer

  GRIGORI PERELMAN

  Born in 1966 in St. Petersburg, Grigori Perelman developed a passion for mathematics from his mother, who taught the subject. Aged 16, he won a gold medal at the International Mathematical Olympiad in Budapest, achieving a perfect score. A successful academic career followed, including a spell at several research institutes in the US, where he solved a major geometry problem called the Soul conjecture. While there, he met Richard Hamilton, whose work influenced his proof of the Poincaré conjecture.

  The reclusive Perelman did not enjoy the fame his proof brought him. He turned down the two greatest accolades for a mathematician: the Fields Medal in 2006 and the Clay Mathematics Institute prize (and its $1 million award) in 2010, saying it belonged as much to Hamilton.

  Key works

  2002 “The entropy formula for the Ricci flow and its geometric applications”

  2003 “Finite extinction time for the solutions to the Ricci flow on certain 3-manifolds”

  See also: The Platonic solids • Graph theory • Topology • Minkowski space • Fractals

  DIRECTORY

  In addition to the mathematicians covered in the preceding chapters of this book, many other men and women have made an impact on the development of mathematics. From the ancient Egyptians, Babylonians, and Greeks to the medieval scholars of Persia, India, and China and the city-state rulers of Renaissance Europe, those looking to build, trade, fight wars, and manipulate money realized that measuring and calculating were crucial. By the 19th and 20th centuries, mathematics had become a global discipline, with its practitioners involved in all the sciences. Math remains crucial in the 21st century as space exploration, medical innovations, artificial intelligence, and the digital revolution press ahead, and more secrets about the Universe are revealed.

  THALES OF MILETUS

  C. 624–C. 545 BCE

  Thales lived in Miletus, an ancient Greek city in what is now Turkey. A student of mathematics and astronomy, he broke with the tradition of using mythology as a way of explaining the world. Thales used geometry to calculate the height of pyramids and the distance of ships from the shore. The theorem named after him states that where the longest side of a triangle contained within a circle is the diameter of the circle, that triangle has to be a right-angled triangle. The astronomical discoveries attributed to Thales include his forecast of the 585 BCE solar eclipse.

  See also: Pythagoras • Euclid’s Elements • Trigonometry

  HIPPOCRATES OF CHIOS

  C. 470–C. 410 BCE

  Originally a merchant on the Greek island of Chios, Hippocrates later moved to Athens, where he first studied, then practiced mathematics. References by later scholars suggest that he was responsible for the first systematic compilation of geometrical knowledge. He was able to calculate the area of crescent-shaped figures conta
ined within intersecting circles (lunes). The Lune of Hippocrates, as it was later called, is bounded by the arcs of two circles, the smaller of which has as its diameter a chord spanning a right angle on the larger circle.

  See also: Pythagoras • Euclid’s Elements • Trigonometry

  EUDOXUS OF CNIDUS

  C. 390–C. 337 BCE

  Eudoxus lived in the Greek city of Cnidus (now in Turkey). He developed the “method of exhaustion” to prove statements about areas and volumes by successive approximations. For example, he was able to show that the areas of circles relate to each other according to the squares of their radii; that the volumes of spheres relate to each other according to the cubes of their radii; and that the volume of a cone is one-third that of a cylinder of the same height.

  See also: The Rhind papyrus • Euclid’s Elements • Calculating pi

  HERO OF ALEXANDRIA

  C. 10–C. 75 CE

  A native of Alexandria in the Roman province of Egypt, Hero (or Heron) was an engineer, inventor, and mathematician. He published descriptions of a steam-powered device called an aeolipile, a wind wheel that could operate an organ, and a vending machine that dispensed “holy” water. His mathematical accomplishments included describing a method for computing the square roots and cubic roots of numbers. He also devised a formula for finding the area of a triangle from the lengths of its sides.