Wolfram Computation Meets Knowledge

Date Archive: 2010 June

Education & Academic

Mathematica: A Game Changer for Mathematics

Bruce Torrence, PhD, chair of the Department of Mathematics at Randolph-Macon College, says he's engaging his students in mathematics more than ever before thanks to a single Mathematica command. That command is Manipulate. Professor Torrence calls the ability to create instant dynamic interfaces a "real game changer" for helping students understand mathematics. He says, "Once you play with a Manipulate and interact with the sliders and buttons, you really develop your intuition as to how the underlying mechanisms are interacting and working." In this video, Professor Torrence shares an example of how he used Mathematica to turn a previously tedious lesson into a highly compelling, interactive classroom activity.
Computation & Analysis

Simulating the World Cup Knockout Stage

The knockout stage of the 2010 FIFA World Cup is about to begin in South Africa. At the time of writing, every team has one group stage match remaining, and most teams still have a chance to finish in the top two places in their group and progress to the knockout stage (see the tournament schedule and group stage standings). There are different approaches to ranking world football teams. The most well known is FIFA's official world rankings, which are derived from points gained and lost in each match according to a heuristic set of rules that generally reward winning against higher-ranked opponents in more-important tournaments. A simple alternative with a more statistical basis is an Elo rating system (described in more detail below). A handy property of Elo rating systems is that they directly provide an estimate of the probability that a given team will perform better than another. We can use Mathematica with that to set up simulations of the knockout stage of the World Cup. This lets us estimate things like the chance of each team winning the tournament. We'll also generate some nice visualizations of the results, such as the following simulated knockout stage (based on the current top two teams in each group):
Announcements & Events

Happy Birthday, Alan Turing

Today (June 23, 2010) would have been Alan Turing's 98th birthday---if he had not died in 1954, at the age of 41. I never met Alan Turing; he died five years before I was born. But somehow I feel I know him well---not least because many of my own intellectual interests have had an almost eerie parallel with his. And by a strange coincidence, Mathematica's "birthday" (June 23, 1988) is aligned with Turing's---so that today is also the celebration of Mathematica's 22nd birthday. I think I first heard about Alan Turing when I was about eleven years old, right around the time I saw my first computer. Through a friend of my parents, I had gotten to know a rather eccentric old classics professor, who, knowing my interest in science, mentioned to me this "bright young chap named Turing" whom he had known during the Second World War. One of the classics professor's eccentricities was that whenever the word "ultra" came up in a Latin text, he would repeat it over and over again, and make comments about remembering it. At the time, I didn't think much of it---though I did remember it. Only years later did I realize that "Ultra" was the codename for the British cryptanalysis effort at Bletchley Park during the war. In a very British way, the classics professor wanted to tell me something about it, without breaking any secrets. And presumably it was at Bletchley Park that he had met Alan Turing. A few years later, I heard scattered mentions of Alan Turing in various British academic circles. I heard that he had done mysterious but important work in breaking German codes during the war. And I heard it claimed that after the war, he had been killed by British Intelligence. At the time, at least some of the British wartime cryptography effort was still secret, including Turing's role in it. I wondered why. So I asked around, and started hearing that perhaps Turing had invented codes that were still being used. I'm not sure where I next encountered Alan Turing. Probably it was when I decided to learn all I could about computer science---and saw all sorts of mentions of "Turing machines”. But I have a distinct memory from around 1979 of going to the library, and finding a little book about Alan Turing written by his mother, Sara Turing. And gradually I built up quite a picture of Alan Turing and his work. And over the 30 years that have followed, I have kept on running into Alan Turing, often in unexpected places.
Education & Academic

The Circles of Descartes

Somewhere, you've likely been forced to learn how fractions work, and how to calculate 2/7 + 2/5. To some extent, fractions have been falling out of favor in the world, losing out to decimals. The New York Stock Exchange gave up fractions on April 9, 2001. Much of the time, a decimal is okay. Sometimes, though, especially in mathematics, exact values are desired. Instead of a value being 3.00000000...00727..., it is exactly 3. Or exactly 10/35 + 14/35 = 24/35. For fractions themselves, the Farey sequence is quite interesting—the reduced fractions between 0 and 1 where the denominator is less than or equal to a particular value, like 7. For example, the F7 Farey sequence is the the first row in the following block. The next row has the denominator. The third row is twice the reciprocal of the denominator squared. The fourth row is the denominator from the third row.
Announcements & Events

Announcing the Wolfram Data Summit

The creation of large data repositories has been a key historical indicator of social and intellectual development—and indeed perhaps one of the defining characteristics of the whole progress of civilization. And through our work on Wolfram|Alpha—with its insatiable appetite for systematic data—we have gained a uniquely broad view of the many great data repositories that […]

Announcements & Events

Get Ready for the Wolfram Technology Conference 2010

Let Wolfram technologies—present and future—be part of your long-term software solution. Get opportunities to learn from Mathematica experts. Gain hands-on experience that will enhance and expand how you use Wolfram|Alpha. You can do all of this and more at the Wolfram Technology Conference 2010, being held the second week of October in Champaign, Illinois, USA.