Andrew Wiles and Fermat’s Last Theorem

In 1995, Andrew Wiles proved Fermat’s Last Theorem, a problem that had stumped mathematicians for over 350 years.

Andrew Wiles (Left) and Pierre de Fermat (Right)

Never heard of Fermat’s Last Theorem? Well, to explain, let’s start with the Pythagorean Theorem. As you may know, the Pythagorean Theorem states that for right angle triangles, a2+b2=c2, where c  is the length of the hypotenuse, and a and b are the lengths of the other 2 sides. We know that there are a number of positive whole number solutions to the Pythagorean Theorem: 3, 4, and 5; 6, 8, and 10; etc.

In 1637, Pierre de Fermat, a French lawyer and amateur mathematician, conjectured that there are no positive whole number solutions for an+bn=cn for values of n>2. So, in other words, there are no positive whole number solutions for a3+b3=c3, or a4+b4=c4, etc. He claimed to have a proof, but did not have room in the margin of the book (Arithmetica) where he wrote this conjecture. For over 350 years, mathematicians tried to discover Fermat’s proof but were unsuccessful. Wiles finally proved the theorem, but he used some modern techniques that would have been unknown to Fermat, which makes ones wonder if Fermat had some other simpler proof that has yet to be discovered.

If you are interested in learning more about Fermat’s Last Theorem, there is a great book and a documentary (both by Simon Singh) that you should check out.

The book, Fermat’s Enigma, provides a detailed and interesting account of Andrew Wiles quest to solve Fermat’s Last Theorem, including explanations of the math appropriate for non-mathematicians. Singh gives us the entire story of Fermat’s Last Theorem starting from Fermat himself and covering the mathematicians along the way that attempted to solve the proof or provided important breakthroughs used by Wiles in his successful proof. Simon Singh wrote the book after producing a documentary about Wiles’ journey for the BBC. The documentary, Fermat’s Last Theorem does not provide as much detail, or course, but I love how, through the interviews with Wiles, you really get a sense of how much of himself he pored into this problem. This problem, which he first learned about from a library book at the age of 10, truly was his life’s work. You can hear the emotion in his voice when he describes how important it was to him and how he felt when he finally completed the proof. The documentary was also produced in the U.S. as The Proof by NOVA, but it isn’t available online. You can see the BBC version in full on YouTube (embedded below).

NOVA Exposes the Hidden Dimension of Fractals

Mandelbrot Set created by Wolfgang Beyer with the program Ultra Fractal 3.

Whether or not you’ve heard of fractals before, the NOVA documentary, Hunting the Hidden Dimension, will amaze you with how cool they really are. A fractal is a geometric pattern that is repeated at smaller and smaller scales, producing shapes that can’t be represented by classical geometry. When mathematicians first started toying with the idea of fractals, they seemed so strange and foreign they were known as “monsters”. Now we see that they aren’t so foreign. In fact, they are everywhere – the branching of trees and blood vessels in our bodies, coastlines, clouds. Isn’t it amazing how even the strangest mathematical concepts seem to lead back to the natural world?

Hunting the Hidden Dimension is a fascinating look at fractals, covering the history of their study, from the 19th century, when they were known as “monsters”, to current applications, such as CGI and cell phone antennae. We also learn about the life and work of Benoit Mandelbrot, the man who developed fractal geometry as a field of mathematics and coined the term “fractal”, his advantage being that he came along at a time when computers were becoming available to tackle such problems. The “hidden dimension” in the title refers to the “fractal dimension”. You’ve heard of things being 2-dimensional or 3-dimensional, but fractal geometry can describe shapes with non-integer dimensions like 1.3 or 2.6.

I watched this documentary with my kids and I have shown clips of it to my after-school math group. The kids especially like seeing the beautiful images of the Mandelbrot set and seeing how fractals were used in the making of the latest Star Wars movies.

Hunting the Hidden Dimension is available for free on Hulu and is currently available on YouTube (embedded below).

Probability, Poker, and God

WNYC’s Radiolab recently aired a segment called “Dealing with Doubt”. Jad and Robert spoke to Professional poker players, Annie Duke and brother Howard Lederer, and learned that reading other players’ “tells” is a very small part of the game. The way to win at poker is to use math – to calculate the probability that you will get a winning hand, your “hand odds”, and compare it to the “pot odds”, the ratio of the current size of the pot to the cost of a call.

They also discuss how 17th century French mathematician, Pascal, applied probabilities to a very big question.

You can listen to the segment here: Dealing with Doubt. (Sorry I can’t embed – WordPress won’t allow it. )

The First Computer is Older Than You Thought

Antikythera Mechanism Reproduction
Mogi Vicentini

PBS recently aired a fascinating episode of NOVA, titled “Ancient Computer,” which details the discovery and study of a geared machine, dubbed the Antikythera Mechanism. Fragments of this machine, determined to be from around 100 BC, were discovered on the floor of the Mediterranean Sea off the coast of the Greek island of Antikythera in 1900. But, it wasn’t until a century later, that a team of scientists were able to use modern technology to unravel what the machine did and how it worked.

After much study, the scientists determined that this ingenious system of 30 (or more) gears was able to predict solar and lunar eclipses and the movement of the planets known at that time. It could even take into account the elliptical orbits of the moon and planets. This machine is believed to be the first analog computer.

This episode is currently available to view at PBS.org and on YouTube (embedded below).

And just for fun…