The Oxford Concise Dictionary of Mathematics, edited by Christopher Clapham and James Nicholson, covers both pure and applied mathematics and statistics and includes linear algebra, optimization, nonlinear equations, and differential equations.  With over 3,000 authoritative entries it is a one-stop math resource.  Since I usually don’t deal with math I thought it would be fun to excerpt some entries.  Below are some samples from the “A”, “B”, “C”, “Q”, “R”, and “S” sections.  Enjoy!

Achilles paradox: The paradox which arises from considering how overtaking takes place.  Achilles gives a tortoise a head start in a race.  To overtake, he must reach the tortoise’s initial position, then where the tortoise had moved to, and so on *ad infinitum.  The conclusion that he cannot overtake because he has to cover an infinite sum of well-defined non-zero distances is false, hence the paradox.

Break-even point: The point at which revenue begins to exceed cost.  If one graph is drawn to show total revenue plotted against the number of items made and sold and another graph is drawn with the same axes to show total costs, the two graphs normal intersect at the break-even point.  To the left of the break-even point, costs exceed revenue and the company runs at a loss while, to the right, revenue exceeds costs and the company runs at a profit.

Cuboctahedron: One of the *Archimedean solids, with 6 square faces and 8 triangular faces.  It can be formed by cutting off the corners of a cube to obtain a polyhedron whose vertices lie at the midpoints of the edges of the original cube.  It can also be formed by cutting off the corners of an *octahedron to obtain a polyhedron whose vertices lie at the midpoints of the edges of the original octahedron.

QED: Abbreviation for quod erat demonstrandum.  Latin for ‘which was to be proved’.  Often written at the end of a proof.

Radius (radii): A radius of a circle is a line segment joining the centre of the circle to a point on the circle.  All such line segments have the same length, and this length is also called the radius of the circle.  The term also applies in both senses to a sphere.

Subtraction: The mathematical operation which is the inverse operation to *addition which calculates the difference between to numbers or quantities.  So 7-2=5, and (3x+5y)-(x+2y)=2x+3y.

What do road and railway systems, electrical circuits, mingling at parties, ancient mazes, and the Internet have in common? The answer is that they all involve interconnections – they are what mathematicians call networks. Peter M. Higgins is the author of Nets, Puzzles, and Postmen: An Exploration of Mathematical Connections, which examines the hidden mathematical structures that underpin many real world phenomena. Below is an excerpt from the first chapter of the book, which reveals just how important networks are.

Who discovered networks? The question is almost like asking who discovered drawing—the urge to start doodling pictures of networks is almost overwhelming as soon as we begin thinking about a situation in which there is a multitude of connections. The advantage of the picture is that it allows you to see all the connections at once and we can remind ourselves of any one of them simply by flitting our eyes around the diagram.

Perhaps networks have been underestimated because they are so common, yet at the same time they seem to lack any structure. The mathematical topics that have been studied extensively for thousands of years are numbers and geometry. Numbers are pervasive, they allow us to tally and compare, and have an undeniable natural order. Geometric objects are pretty and visual, providing all manner of symmetries that can strike you before a word is said, so the attraction of geometry is very powerful and immediate. Networks on the other hand are none of these things. Networks are not numbers of any kind, nor are they truly geometrical even though we can draw pictures of them. They represent quite a different realm of mathematics. And not only of mathematics, for everyone appreciates the importance of networking—the real measure of our comprehension of the world is our understanding of how all the various parts come together and affect one another.

Moreover, the use of the word ‘network’ in this context is more than just a metaphor. Some of the most difficult and technically demanding research in the social and political sciences centres on studying the nature of networks of international organizations of all kinds, whether they be legal, cultural, and diplomatic, or scientific, commercial, and sporting. Relatively small nations and organizations can have profound influence on world affairs. Sometimes this can be tracked to their strategic or cultural importance or to dominant individuals. However, substantial and sometimes less visible influence often stems from the way they are placed within relevant networks and how they draw from and feed into these webs.

It is fair to say that the first genuine problem in networks dates to the eighteenth century when the famous Swiss mathematician, Leonhard Euler, showed how to solve the now celebrated riddle of the Bridges of Königsberg by finding a simple general principle that dealt with any question of that kind. But more of that later. This does alert us however to the fact that networks have been studied from the mathematical viewpoint for centuries. None the less, it is striking how their relative importance keeps growing and growing. In part this is due to examples of networks springing up in modern life—we need look no further than the internet to find a massive and important instance of a network that has come into being almost spontaneously. This network pervades most aspects of the modern world and has taken on a life of its own. The internet acts as a vehicle for another network, the World Wide Web. These networks differ in two ways, one physical and the other mathematical. The Web is visible but intangible and floats on top of the internet, which is a physical array consisting of routers and their connections. The Web is also a directed network for there are links directed from one page to another, but not necessarily in the reverse direction. This gives it a very different character from networks in which all connections are mutual and two-way.

It all goes much deeper than that however. Professional mathematicians have tended to have a similar reaction to that of the general public to the underlying idea. The notion of a network of connections is so simple and natural that there looks to be not much to it. To be sure, even in the eighteenth century Euler showed that even a simple example can yield an interesting problem. All the same, it was felt that the depth and interest of the mathematics involved could hardly be on a level comparable with really serious science, such as that which explains how the Earth and the Heavens move. Since the time of Isaac Newton, calculus, the mathematics of change and movement, has been a well-spring of scientific inspiration and was seen as the heir to classical Greek geometry, representing the pinnacle of mathematical practice and sophistication. Indeed Leonhard Euler himself perhaps did more than anyone who has ever lived to develop the methods of Newton, the so called differential and integral calculus. By comparison, problems about networks were regarded as a poor relation, little more than recreational puzzles, fit only for those who could not contribute to the really tough stuff.

Networks, however, spring many surprises. And they truly are surprises because no one would expect objects with virtually no mathematical structure to yield anything of interest. After all, a network is any array of points on a page with lines drawn between some of them in any fashion at all. The idea would seem to be far too general to yield anything that went much beyond the obvious. However, there is a whole world to be explored by those prepared to search and the results have consequences for real networks of people and telephone lines. For instance, at any party that ever there was, or ever will be, or ever could be, there will be two people with the same number of friends at the gathering—this, and many results like this, are unavoidable consequences of the nature of networks, as we shall soon witness.

Part of the trouble has been that mathematics itself has been slow to wake up to what was happening. Problems about networks keep arising irresistibly, even when you are not looking for them. I myself spend a lot of time on my own speciality that is a certain area of algebra. What has happened in my own field has been mirrored elsewhere. Certain intractable problems have arisen and, in the end, progress is only made when they are represented in terms of networks whereupon it transpires that what is holding you up is a question about whether or not certain patterns can or cannot arise in a network. No use sneering—it turns out that nets were really what you have been studying all along.

Robert and Ellen Kaplan have taught courses ranging from Sanskrit to Gödel’s Theorem, to students ranging in age from four to seventy. In 1994, the Kaplans, along with Tomás Guillermo, began The Math Circle an esteemed, popular mathematics learning program. On their website they state that, “We are careful to choose topics which are unlikely to be in the school curriculum – we see our role as widening and deepening the river, rather than accelerating its flow between narrow banks.” In keeping with their mission, to supplement math education, they have written a book, Out of the Labyrinth: Setting Mathematics Free that reveals the secrets behind the Math Circle’s success. Below we have excerpted from the first chapter of Out of the Labyrinth. This piece illustrates just how much fun math can be for children. Be sure to check out DailyKos on Thursday when Peter Flom will explore math circles himself, in a new feature titled, Education Uprising, an effort to reform education from the ground up.

When I walked into the Harvard classroom, about eight students were already there, as I’d expected. But people passing by were surprised to see them – not because there was anything unusual in early arrivals at the beginning of a semester but because of what these students looked like: sitting in their tablet armchairs, their little legs sticking straight out in front of them. They were all about five years old.

This was the first class for these new members of The Math Circle. The parents, ranged at the back, may have been nervous but their children weren’t, because it was simply the next thing to happen in a still unpredictable life.

“Hello,” I said. “Are there numbers between numbers?”

I don’t know what you’re talking about,” said Dora, in the front row.

“Oh – well, you’re right – I’m not too sure myself.” I drew a long line on the blackboard and put a 0 at one end, a 1 near the other. “Is there anything in there?”

Sam jumped up and down. “No, there’s nothing there at all,” he exclaimed, “except of course for one half.” This wasn’t as surprising a remark as you might think, since Sam had just turned an important five and a half.

“Right,” I said, and made a mark very close to the 0 and carefully labeled it 1/2.

“It doesn’t go there!” said Sonya, sitting next to Dora.

“Really? Why not?”

“It goes in the middle.”

“Why?”

“Because that’s what one half means!”

I erased my mark and, with a show of reluctance, moved 1/2 to the middle, acting now as no more than Sonya’s secretary.

“Well,” I said, “is there anything else in there?”

Silence…five seconds of silence, which in a classroom can seem like five minutes. Ten seconds. Tom, at the back, got up and began to put on his jacket, because clearly the course was over: we’d found all there was to find between 0 and 1.

“I guess it’s just a desert,” I said at last, “with maybe a camel or a palm tree or two,” and began sketching in a palm tree on the number line.

“Now that’s ridiculous!” said Dora, “there can’t be any palm trees there!”

“Why not?”

“Because it isn’t that sort of thing!” This is an insight many a philosopher has struggled over.

Obediently I began erasing my palm tree, when I felt a hand grabbing the chalk from mine. It was Eric, who hadn’t said anything up to this point. He started making marks all over the number line – most of them between 0 and 1 but a fair number beyond each.

“There are kazillions of numbers in here!” he shouted.

“Is kazillion a number?” Sonya asked – and we were fully launched.

In the course of the semester’s ten one-hour classes, these children invented fractions (and their own notation for them), figured out how to compare them – as well as adding, subtracting, multiplying, and dividing them; and with a few leading questions from me, turned them into decimals. In the next to last class they discovered that the decimal form of a fraction always repeats – and in the last class of all, came up with a decimal that had no fractional equivalent, since they made it guaranteed that it wouldn’t repeat (0.12345678910111213…)

The conversations involved everyone, even a parent or two, who had to be restrained. Boasting and put-downs were quickly turned aside (this wasn’t that sort of thing), and an intense and delightful back-and-forth took their place…

…You probably think this was a class of young geniuses, hand picked for special training. In fact it was like any other Math Circle class: we take whoever comes – math lovers and math loathers; we don’t advertise but rely on word of mouth. True, this is the Boston area, with its high percentage of academic and professional families, but we’ve had the same sort of classes with inner-city kids, suburban teenagers, college students, and retirees – from coast to coast, in London, Edinburgh, and (in German!) in Zurich and Berlin. The human potential for devising math, with pleasure, is as great as it is for creative play with one’s native language: because mathematics is our other native language.