The Yanghui Triangle

What is that? Well, it's used to find the coefficients of a binomial expansion. Just like Pascal's triangle. Actually, it is the same thing. Yanghui was a Chinese mathematician from the thirteenth century who did work with binomial expansions and created a triangular array to contain them. One interesting thing about the document we have of the ancient Chinese version of the triangle is that it appears to contain an error. Here is a high resolution image where you can try to find it:


An ancient scribe must have been having an off day. Notice also that for some reason, the symbol for "ten" looks a lot like the Arabic numerals version, just turned on its side. There is more history of the triangle, and the answer to where the mistake is, here.

Mathematical Personalities: Gerolamo Cardano

Sorry about the lack of updates, but the life has been busy.

Down to business.

The Italian Renaissance was one of those interesting times in world history in which the great increase in knowledge and trade led to an explosion of wealth, which led to more knowledge and trade, and so on...but we have to remember that these are not modern people. This was a time that was altogether more violent and chaotic than anything we see today (in the Western world, anyway) because for some reason the people of that time and place really knew how to work the chaos into something rather creative. I mentioned Cardano before, but in the context of the solution to the cubic equation, which is a great story. Now I want to mention some other things about the man. The events of his life illustrate the differences between that era and this.

Remember that the modern world comes largely from the Enlightenment with heavy doses of Modernism and Postmodernism thrown into the mix. This all happened after the Renaissance, and the 16th century was more like the medieval world than like ours. His mother fled the plague. He had trouble starting a career in medicine because his parents were not married. He was a gambler and wrote a book on the subject that was the first to make probability into a science. While he contributed to the sciences and mathematics, he was an astrologer as well and was arrested for heresy after casting the horoscope of Jesus.

People who are brilliant enough to make lasting contributions to mathematics have a tendency to be eccentric. Add to this the fact that many discoveries were made before what we would call the modern era, and we find that the people of the history of mathematics are incredible to study.

mathematics, astronomy, religion

I found something on the Library of Congress website: a collection of images from the Vatican Library of mathematical manuscripts from between the ninth and fifteenth centuries.

Saturday Comic:


The background is an ultra deep field photo taken by the Hubble Space Telescope and credit is due to ESA and NASA.

Pascal's quote is from the Pensées, numbers 205-206.

Reputation

Rooster image from here.
All kinds of quotations about mathematics and by mathematicians can be found here.


According to St. Augustine:

The good Christian should beware of mathematicians, and all those who make empty prophecies. The danger already exists that the mathematicians have made a covenant with the devil to darken the spirit and to confine man in the bonds of Hell.

This makes much more sense when you realize that in this context, a "mathematician" is what we would call an "astrologer".

An interesting essay and a comic.

I found something interesting on Reddit today, which is somewhat connected to the content of yesterday's posting. Here is an essay by a teacher and mathematician arguing that math education is too focused on notation and formulas instead of teaching kids about the art of discovering elegant numerical patterns and relationships. He suggests that mathematics should be art and play.

Also, here is the math subreddit.

Solving the Cubic Equation

Stories like this are why I like math history. Students, often rightly, think that the subject they study is too abstract and removed from the real human experience. Most high school math is not especially related to the everyday life of teenagers. Furthermore, it does rather seem like whenever a teacher presents examples of how the abstractions can be applied to practical problems regarding quantities such as mass or distance, the students quail at the prospect of the dreaded Word Problems. To be fair these do tend to be more complicated than the abstract ones, because life is complicated and the abstractions of math tend to simplify matters for the purpose of examining certain underlying numerical relationships. The question of whether education should inspire students to look past their daily life and expand their scope of vision is for later.

At any rate, the story of how Gerolamo Cardano came to publish the solution to certain cubic equations is full of rich human drama: secrecy, competition, betrayal, and an obscure poem, (which apparently sacrificed a certain amount of accuracy to maintain the proper Italian rhyme scheme.) The people who developed the mathematics we use today were people much like us, and the facts and techniques we try to teach to our teenagers were slowly developed over thousands of years. Mathematics is relevant to our lives not because we will need to solve equations in the course of daily life, but because using and developing the concept of number is a fundamental human activity. Much like poetry.

Theorem Loafing

Time for the Wednesday comic.