Monge and Optimal Transport...

Mathematicians are ignorant of their history. They know the names of the greats but generally can repeat only one or two (erroneous) stories about them. For example, if asked for a mathematical story from the French Revolution, then many would plump for a story about Galois, shot at dawn. However, there were many other revolutionary mathematicians. One such is Monge. His name is familiar to differential geometers through Monge form and to analysts of PDEs through the Monge-Ampere equation. Information about him is scant in English. All the important books are in French and so my attempts to study his life in more depth have been thwarted by my lack of fluency. I think I first became interested in him when I was a student. I read a book (the name escapes me) which stated that he had studied the concept of the optimal transport of soil when constructing fortifications. The fact that caught my imagination was that the answer was constrained by the observation that the paths of two particles should not cross. At the time I came up with a counter-example but that was because I didn’t really understand the parameters of the problem. I was greatly interested then when I heard that one of this year’s Hardy Lectures would be about Monge and the optimal transport problem. I was greatly disappointed when I discovered that it clashed with previous commitments. When I was at the conference in Liverpool to celebrate the birthdays of Bill Bruce and Terry Wall Andrew Ranicki told me that the lecture had been recorded. And I’m glad it was, it’s a great talk. It’s given by Etienne Ghys. He takes in the cutting of stones, including how Monge designed a never used plan for the ceiling of...

The origin of x in maths...

I’m still wrestling with the fallout from marking exams. Despite this, I found time to watch a short TED video posted recently that features Terry Moore explaining why x is used as an unknown in mathematics. Watch the video – only 4 minutes – or jump to the spoilers below if you want to know more. The main idea is that the we use x because the Spanish used (chi) as the first sound of the Arabic word for “something” because they couldn’t say the correct “sh” sound. I recollect a similar argument made somewhere else with slightly different details, though, sadly, I cannot remember them precisely, nor their location. I was unable to track down references at the time to verify this argument and so dismissed it as a Just So Story. Since it has resurfaced it would be interesting to see evidence. Does anyone know of any? My reason for asking is that I’ve spent the last few years learning about Greek mathematics and I am interested in how it has been transmitted to us via the Arabic scholars and scientists. So far I’ve only read the popular accounts, Science and Islam: A History by Ehsan Masood and Pathfinders: The Golden Age of Arabic Science by Jim Al-Khalili. The latter is good but spends the first few chapters explaining the history of Islam and various empires. Furthermore, one of the first bits, maybe the first bit, of science to be explained in depth is Eratosthenes’ measurement of the Earth, i.e., a high point of Hellenistic science. Obviously providing a context is important in a book but I still feel as though I don’t know much about the science from the Arabic world between the end of the Greek era and the beginning...

William Noel at TED on the Archimedes Codex...

Regular readers will know that I’m interested in the history of mathematics and am a fan of Archimedes. Well, here’s another video on the Archimedes Codex, this time by William Noel at TED rather than one by his coauthor, Reviel...

Euclid and Eratosthenes — Greek or African?...

Last month the mathematics author John Derbyshire wrote an online article not about mathematics but on his personal views regarding race. Views which eventually got him sacked as a columnist for the publisher. The Guardian newspaper responded through an article by Jonathan Farley. My post today is not about race but rather about some points made in the comments to Farley’s article. In his article Farley said … Euclid, Eratosthenes and other African mathematicians outshone Europe’s brightest stars for millennia. In the comments section it was asked They are known as Greek mathematicians. Why are they quoted in an article about Black mathematicians? Now, one should avoid getting involved in fights in comments section and fortunately someone had replied in a comment later highlighted by Guardian staff Well, Euclid is ‘Euclid of Alexandria’ which is in Egypt of course and Eratosthenes was born in Cyrene (modern Libya). So I don’t think its erroneous to say they were black mathematicians. Various arguments were made later in the comments about whether North African counted as black. I’m not going to get into this argument either. Instead my post is about the precise origins of Euclid and Eratosthenes. In my geometry and history of mathematics courses I tell my students that when we talk of Greek mathematicians, we should not think of them as swanning around in Athens dressed in togas. Instead they came from all over the Mediterranean, from what we now call Italy, Egypt, Turkey, Libya and so on. Some even came from Greece. (And they didn’t wear togas. That was the Romans. The Greeks wore a chiton, a type of tunic.) A good example of this is the greatest Greek scientist, Archimedes, who was from Syracuse in Sicily. But what about Euclid and Eratosthenes?...

Thales’ Theorem and Lockhart’s Lament...

The Yorkshire Branch of the Mathematical Association recently hosted a talk by David Acheson entitled Proof, Pizza and Guitar. (By the way I’ll be giving a talk on card cheating for the YBMA on Wednesday 8th February at 7.30pm in School of Mathematics. All welcome but a small charge of a pound may be made.) During David’s talk he gave a proof of Thales’ Theorem. This is a theorem that states the following. For any point in a semi-circle, the angle formed by the lines from that point to the two edge points of the base is right-angled. This is a good theorem in that, to me at least, it does not seem intuitively obvious (what, it’s always a right angle? Really?) and yet it is easy to convince yourself it’s true by doing some examples. Thales (c624BC – c547BC) is often considered to be the first scientist because he was the first person (we know of) who looked for non-supernatural reasons for phenomena. Rather than believing lightning or earthquakes were caused by gods he considered more natural explanations. However, his solution to the latter involved the way that the land floated on the sea, i.e., he was totally wrong but here it is the concept of avoiding invoking the gods that counts. In the case of mathematics he is credited with a number of theorems and the main point is that, allegedly, he provided proof. He is also credited with measuring the pyramids in Egypt. His method is interesting because it does not involve a brute force use of measuring instruments, i.e., get out measuring rods and send people up the pyramids with them. His proof is more elegant than that. He measured the height of a slave and when the sun was...

Gilbreath Conjecture

Within the card magic community the Gilbreath Principle is a well-known but much misunderstood mathematical principle. Few magicians know much about its creator, Norman Gilbreath, and in particular they are unaware of his other mathematical work. Following a recent email conversation with him about the principle (always go to the source!) he kindly sent me an offprint of a recent paper on the Gilbreath Conjecture. The Gilbreath Conjecture is a conjecture about primes and is fairly easy to state. Consider the sequence of primes 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, … Now work out the (absolute) difference between neighbouring terms 1, 2, 2, 4, 2, 4, 2, 4, 6, 2, 6, … Work out the absolute difference between terms for this sequence and keep doing this: 1, 0, 2, 2, 2, 2, 2, 2, 4, 4, … 1, 2, 0, 0, 0, 0, 0, 2, 0, … 1, 2, 0, 0, 0, 0, 2, 2, … 1, 2, 0, 0, 0, 2, 0, … 1, 2, 0, 0, 2, 2, … 1, 2, 0, 2, 0, … 1, 2, 2, 2, … 1, 0, 0, … 1, 0, … 1, … The conjecture is that the first term on a line, after the first line, is always a 1. Gilbreath’s paper, Processing process: The Gilbreath conjecture, has recently been published in the Journal of Number Theory. (You can find it at http://dx.doi.org/10.1016/j.jnt.2011.06.008 but unless you have a subscription to the journal you will have to pay for it.) The introduction states the following There is one very important aspect of history that is often left out – the process. This is even true of the history of mathematics. I will give an example. A number of years ago...