Wednesday, March 4, 2009
Wednesday Math, Vol. 62: How things get named in math
Of all the things I have done creatively in my life, all the songs I've written, all the video games I've designed, the mathematical concepts and proofs I've come up with on my own, the one thing that gives me the greatest pride of ownership is the website Pascal's Triangle From Top To Bottom, which is dedicated to putting as much information as possible in one place about Pascal's Triangle and the numbers that comprise it, known as the binomial coefficients.
On the bottom of the title page, the credit for design, programming and research goes to Matthew Hubbard and Tom Roby. The fact is, Tom Roby paid for this thing and did a little editing of my prose, two facts for which I am eternally grateful. Everything else was done by me, except drawing the logo above, which was done by my friend Jodi Soares based on an idea I had rattling around in my head. Thanks, Jodi!
You probably saw Pascal's Triangle at some time in your academic career. The first few rows look like this.
1 2 1
1 3 3 1
1 4 6 4 1
You might recall that you get a row of Pascal's Triangle by adding consecutive entries from the previous row. The rows always start with a 1, then then entries for the next row would be
1+4 = 5
4+6 = 10
6+4 = 10
4+1 = 5
1+... = 1
While it is nearly always written as an infinite triangular array, hence the name, it could be an infinite rectangular array instead, where all the entries to the right of the triangle are zeros, so 1+... is really 1+0 = 1.
Pascal's Triangle is a very, very useful thing in math. It's a fair question to ask how it got its name. The answer is that it has three names, depending on what language you speak. In most of the world, it's Blaise Pascal who gets the credit, but in Italy it is called Tartaglia's Triangle and in China it is Yanghui's Triangle.
The triangle has been studied all over the world and the earliest know work predates Jesus Christ. The earliest work was published in India. Of the three guys whose names get stuck on the thing, the Chinese mathematician Yanghui published first. His work was done in 1261 A.D., long before Tartaglia or Pascal is born, but Yanghui does not take credit, and he readily admits the earlier work of Chia Hsien, well over a century before him. It is Chu Shi-Chieh who calls it Yanghui's Triangle, in the book The Precious Mirror of the Four Elements, giving credit to his teacher. This happens many time in the history of math around the world and through the ages. It's considered bad form to name a thing after yourself, but completely kosher to honor the person who taught you. The name of the book comes from the fact that the triangle is symmetrical (hence the precious mirror) and produces the answers to a lot of questions, so Chu considered it the object that unlocks the secrets of the universe, which is of course made up of the Four Elements.
In China, and many other places, the triangle is used to count the number of ways to combine stuff, but in Italy, the mathematicians there are interested in algebra, most especially in how to factor polynomials. An Italian algebraist would have called himself a cosista, which translates to "thing-ist", because finding the roots of a polynomial was called "finding the thing" in Italian. You likely learned (and quickly forgot) the quadratic formula when you were in high school. That finds the roots of a quadratic equation of the form ax^2 + bx + c = 0, where a, b and c are constants and x is the unknown you want to solve for. This formula has been around since the 1200's at least. In the 1500's, Tartaglia finds the next important formula, the cubic formula, which finds the roots for ax^3 + bx^2 + cx + d = 0. The triangle is very useful in this search, and Italians give their man Tartaglia (whose name means "stammerer") the credit when they discuss the triangle. I do not know the history of who first called the thing Tartaglia's Triangle.
Then we come to Pascal. His work is called The Treatise on the Arithmetical Triangle. Here, the adjective is pronounced "a-rith-MET-i-cal", and it refers to of the property of adding stuff together to get the next row. He was working on a problem of how to best play a gambling game, proposed to him by a nobleman named the Chevalier de Mere. Pascal made no claim of ownership, but he wrote down everything he knew about the numbers and the relationships between them in one place, so Pascal's Treatise became the text to turn to when looking at the numbers for about a century after it was published in 1665. Monmort gave Pascal credit in a 1708 text, deMoivre quotes him in 1730, and little by little everybody reads this stuff and starts calling the arithmetical triangle Pascal's Triangle.
Pascal himself never got to bask in this glory. The Treatise was published a year after his death.
More stuff from the triangle next week.