Although I did not expect to become a mathematician when I was growing up—my first desire had been to be a train driver, and later it was (secretly) to be a brain surgeon—mathematics had intrigued and excited me from a young age. My father was not a professional mathematician, but he used mathematics in original ways in his statistical work in human genetics. He clearly enjoyed mathematics for its own sake, and he would often engage me and my two brothers with mathematical puzzles and with aspects of the physical and biological world that had a mathematical character....

In the fourth annual volume ofThe Best Writing on Mathematics series,we present once again a collection of recent articles on various aspects related to mathematics. With few exceptions, these pieces were published in 2012. The relevant literature I surveyed to compile the selection is vast and spread over many publishing venues; strict limitation to the time frame offered by the calendar is not only unrealistic but also undesirable.

I thought up this series for the first time about nine years ago. Quite by chance, in a fancy, and convinced that such a series existed, I asked in a...

First let me explain my use of the phrase “multimedia civilization.”* I mean it in two senses. In my first usage, it is simply a synonym for our contemporary digital world, our click-click world, our “press 1,2, or 3 world”, a world with a diminishing number of flesh-and-blood servers to talk to. This is our world, now and for the indefinite future. It is a world that in some tiny measure most of us have helped make and foster.

In my second usage, I refer to the widespread and increasing use of computers, fax, e-mail, the Internet, CD-ROMs, iPods, search...

In this opening verse of William Blake’s “The Tyger” from hisSongs of Experienceof 1794, the poet is using “symmetry” as an artistic metaphor, referring to the great cat’s awe-inspiring beauty and terrible form. But the tiger’s form and markings are also governed by symmetry in a more mathematical sense. In 1997, when delivering one of the Royal Institution’s televised Christmas Lectures, I took advantage of this connection to bring a live tiger into the lecture theatre. I have never managed to create quite the same focus from the audience in any lecture since then.

Taking the term literally,...

Modern mathematics is a powerful tool to model any number of real-world situations, whether they be natural—the motion of celestial bodies, for example, or the physical and chemical properties of a material—or man-made: for example, the stock market or the voting preferences of an electorate.¹ In principle, mathematical models can be used to study even extremely complicated systems with many interacting components. However, in practice, only simple systems (ones that involve only two or three interacting agents) can be solved precisely. For instance, the mathematical derivation of the spectral lines of hydrogen, with its single electron orbiting the...

The idea of six degrees of separation—that is, that every person in the world is no more than six people away from every other person on earth—has fascinated social scientists and laymen alike ever since Hungarian writer Frigyes Karinthy introduced the concept in 1929. (The story has since been reprinted in 2006 in Princeton University Press’sThe Structure and Dynamics of Networksby Mark Newman, Albert-László Barabási, and Duncan J. Watts, translated by Adam Makkai and edited by Enikö Jankó.)

For the greater public, the cultural touchstone of the theory was the 1990 play entitledSix Degrees of...

Our very brains revolt at the idea of randomness. We have evolved as a species to become exquisite pattern-finders; long before the advent of science, we figured out that a salmon-colored sky heralds a dangerous storm or that a baby’s flushed face likely means a difficult night ahead. Our minds automatically try to place data in a framework that allows us to make sense of our observations and use them to understand and predict events.

Randomness is so difficult to grasp because it works against our pattern-finding instincts. It tells us that sometimes there is no pattern to be found....

Patterns that are perfectly pure and mathematically exact have a strong aesthetic appeal, as advocated by Pythagoras and Plato and their innumerable intellectual descendants. Yet a bit of irregularity and unpredictability can make a pattern even more beautiful. I was reminded of this fact as I passed by two decorative walls while walking yesterday from my home to my office: One wall, newly built, tries to emulate the regular rectangular pattern of a grid, but it looks sterile and unattractive to my eyes; the other wall consists of natural stones that fit together only approximately yet form a harmonious unity...

One of the most powerful tools a designer can use when developing games is probability, using random chance to determine the outcome of player actions or to build the environment in which play occurs. The use of luck, however, is not without its pitfalls, and designers should be aware of the tradeoffs involved—what chance can add to the experience and when it can be counterproductive.

One challenge with using randomness is that humans are notoriously poor at evaluating probability accurately. A common example is thegambler’s fallacy,which is the belief that odds even out over time. If the...

On a snowy day in Princeton, New Jersey, in March 1956, a short, owlish-looking man named Kurt Gödel wrote his last letter to a dying friend. Gödel addressed John von Neumann formally even though the two had known each other for decades as colleagues at the Institute for Advanced Study in Princeton. Both men were mathematical geniuses, instrumental in establishing the U.S. scientific and military supremacy in the years after World War II. Now, however, von Neumann had cancer, and there was little that even a genius like Gödel could do except express a few overoptimistic pleasantries and then change...

The Jerusalem Chords Bridge, in Israel, was built to make way for the city’s light rail train system. However, its design took into consideration more than just utility—it is a work of art, designed as a monument. Its beauty rests not only in the visual appearance of its criss-cross cables, but also in the mathematics that lies behind it. Let us take a deeper look into these chords.

The Jerusalem Chords Bridge is a suspension bridge, which means that its entire weight is held from above. In this case, the deck is connected to a single tower by powerful...

Albrecht Dürer (1471–1528), master painter and printmaker of the German Renaissance, never thought of himself as a mathematician. Yet he used geometry to uncover nature’s hidden formulas for beauty. His efforts influenced renowned mathematicians, including Gerolamo Cardano and Niccolo Tartaglia, as well as famous scientists such as Galileo Galilei and Johannes Kepler.

We praise Leonardo da Vinci and other Renaissance figures for embracing art and science as a unity. But for artists such as Leonardo and Dürer, there was little science to embrace. Efforts to draw or paint directly from nature required an understanding of physiology and optics that...

Try the following experiment. Get a tangerine and attempt to take the peel off in one piece. Lay the peel flat and see what you notice about the shape. Repeat several times. This can be done with many types of citrus fruit. Clementines work especially well.

Cornell mathematics professor William P. Thurston used this experiment to help students understand the geometry of surfaces. Thurston, who won the Fields Medal in 1982, was well known for his geometric insight. In the early 1980s, he made a conjecture, called thegeometrization conjecture,about the possible geometries for three-dimensional manifolds. Informally, an n-dimensional...

The classical Jordan curve theorem (JCT) says,

Every Jordan curve (a non-self-intersecting continuous loop in the plane) separates the plane into exactly two components.

It is often mentioned just in passing in courses ranging from liberal arts mathematics courses, where it is an illuminating example of an “obvious” statement that is difficult to prove, to undergraduate and graduate topology and complex analysis, where it tends to break the flow. In complex analysis, it is especially given short shrift. There are several reasons for this short shrift. For one, a professor has bigger fish to fry. There are the theorems of...

“Why do I have to learn mathematics? What do I need it for?” When I was a school student, it never occurred to me to ask these questions, nor do I remember hearing it from any of my classmates. “Why do I need history?”—yes. “Why Latin?” (yes, as a high school student I was supposed to study this ancient language)—certainly. But not, “Why mathematics?” The need to deal with numbers, geometric figures, and functions was beyond doubt, and mathematics was unassailable.

Things changed. Today, every other student seems to ask why we need mathematics. Over the years, the...

For people with math anxiety, opening a math textbook or even entering a math classroom can trigger a negative emotional response, but it does not stop there. Activities such as reading a cash register receipt can send those with math anxiety into a panic. Math anxiety is an adverse emotional reaction to math or the prospect of doing math [1]. Despite normal performance in most thinking and reasoning tasks, people with math anxiety perform poorly when numerical information is involved.

Why is math anxiety tied to poor math performance? One idea is that math anxiety is simply a proxy for...

Recently a belief has spread that the set of five Platonic solids has been known since prehistoric times, in the form of carved stone balls from Scotland, dating from the Neolithic period. A photograph of a group of these objects has even been claimed to show mathematical understanding of the regular solids a millennium or so before Plato. I argue that this is not so. The archaeological and statistical evidence do not support this idea, and it has been shown that there are problems with the photograph. The high symmetry of many of these objects can readily be explained without...

For the purpose of reviewing the history of mathematical instruments and the place the subject might command in the history of science, if we take “early modern” to cover the period from the 16th to the 18th century, a first impression may well be one of a change from vigorous development in the 16th century to relatively mundane stability in the 18th. More careful scrutiny suggests that this perception is relative and depends more on our priorities as historians than on the interests of the instrument manufacturers or users. In fact, the role of mathematical instruments and the agenda of...

The physical sciences all went through ‘‘revolutions”: wrenching transitions in which methods changed radically and became much more powerful. It is not widely realized, but there was a similar transition in mathematics between about 1890 and 1930. The first section briefly describes the changes that took place and why they qualify as a “revolution,” and the second section describes turmoil and resistance to the changes at the time.

The mathematical event was different from those in science, however. In science, most of the older material was wrong and discarded, whereas old mathematics needed precision upgrades but was mostly correct. The...

This article outlines some of the mistakes made in the calculus of probability, especially when the discipline was being developed. Such is the character of the doctrine of chances that simple-looking problems can deceive even the sharpest minds. In his celebratedEssai Philosophique sur les Probabilités(Laplace 1814, p. 273), the eminent French mathematician Pierre-Simon Laplace (1749–1827) said,

. . . the theory of probabilities is at bottom only common sense reduced to calculus.

There is no doubt that Laplace was right, but the fact remains that blunders and fallacies persist even today in the field of probability, often...

Probability theory and the metaphysical category of possibility are based on the notion of “states of the world” (or possible worlds). In the market, the only states of the world are prices. Contingency is a very general category that is independent of thelaterdivision of the world into identifiable states or the recognition of the different possible worlds that the world might be. Metaphysical thought later works contingency into the notion of separable possible states. However, pure and absolute (and initial) contingency only minimally says that the world or that the things in it could have been different.

A...

On August 30, 2012, a Japanese mathematician named Shinichi Mochizuki posted four papers to his faculty website at Kyoto University. Rumors had been spreading all summer that Mochizuki was onto something big, and in the abstract to the fourth paper Mochizuki explained that, indeed, his project was as grand as people had suspected. Over 512 pages of dense mathematical reasoning, he claimed to have discovered a proof of one of the most legendary unsolved problems in math.

The problem is called the abc conjecture, a 27-year-old proposition considered so impossible that few mathematicians even dared to take it on. Most...