## 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...

## Never mind the bosons, here’s the Newton papers...

Today the people at CERN announced that they had found evidence that the Higgs boson exists. Mind you, they do also say that more data is needed before they can reach any firm conclusions. So not exactly overwhelming evidence. In other news, Cambridge University has been digitizing their collection of Isaac Newton papers and is putting it online. See http://cudl.lib.cam.ac.uk/. Having had a quick, almost cursory, look through, I think that most of us are going to have to wait for an expert to produce a handy guide and some translation. Any volunteers? It’s amazing to think that in those papers and notebooks Newton wrote down ideas that revolutionized science and had a huge impact in the world. Compare it to today’s news from CERN. There are hundreds of scientists working there and even if their combined effort does find the Higgs boson (and it’s not looking 100% certain), it won’t have the same impact as Newton’s discoveries. That’s a bit of a sobering...

## Alan Turing programme (and Maths Jam)...

Today will see the first MathsJam in Leeds, have a look at the MathsJam website if you want to know more about the concept. The event tonight will be at Dock Street Market on Dock Street in Leeds at 7pm. Anyhow, last night there was a documentary on Channel 4 on Alan Turing. You can watch it on their website by clicking here. My hopes were not high when the programme was introduced as a drama-documentary as I often find that the two don’t mix well. However, it worked very well in this case with the drama part mostly being conversations between Turing and a psychiatrist, Dr Franz Greenbaum. One aspect that particularly pleased me was that Turing’s paper on morphogenesis was discussed in more than passing detail. Quite often this seminal paper on how reaction-diffusion processes could explain stripes and other patterns in animals is overlooked in favour of the computing papers. This is understandable, the effect of computers in our lives has been far greater and besides the paper was probably too far ahead of its time. Only recently have mathematicians and biologists really begun to understand the ideas and even then the process is only a possible explanation of biological patterns. It has never been proved that this is what is really happening. The original paper, The Chemical Basis of Morphogenesis, A. M. Turing, Philosophical Transactions of the Royal Society of London. Series B, Biological Sciences, Vol. 237, No. 641. (Aug. 14, 1952), pp. 37-72, can be found at the JSTOR...

## Bletchley Park needs your money...

Mathematicians are very proud of the fact that a group of mathematicians, including Alan Turing, were responsible for breaking Nazi secret codes during World War 2. It has been claimed that their work shortened the war by at least two years. The code breakers worked in a country house, Bletchley Park, and the huts that they used have been decaying for years. When I visited the house in the late 90s the huts were looking rather shabby; what must they look like now? Well, the Heritage Lottery Fund has allocated 4.6 million pounds for the regeneration of Bletchley Park to restore the huts and create a visitor centre and exhibition. Good news to cheer! However… There is a catch. Bletchley Park has to raise 1.7 million to secure this money. To read more and to donate go to...

## Archimedes Codex – The Lost Palimpsest...

In recent years I have become interested in the history of mathematics. What is interesting is that a lot of what I learned about the history of maths, usually through asides from my lecturers, is actually wrong. And often what I hear on the radio and TV makes me cringe as it is usually innaccurate. One area I am developing expertise in is ancient Greek mathematics. (The name is misleading – most of the important maths happened outside of what we think of Greece although it was certainly influenced by Hellenistic culture.) Following my interest my favourite mathematical theorem is Archimedes Quadrature of the Parabola. Also, there are many great stories – see for example the Antikythera Mechanism. Another interesting story to emerge in recent years is that of a copy of work by Archimedes discovered written in a medieval prayer book. In the 12th century books were rare, precious and difficult to make so people would disassemble old books, scrape off the ink and write something new on the cleaned paper. This fate befell a text book on the work of Archimedes which was turned into a prayer book. Fortunately, some of the mathematical writing was still visible and the scholar Heiberg found it in a religious establishment, publishing a translation of some of it in around 1906. The book then mysteriously disappeared, resurfacing in the 90s in an auction where it was sold for \$2 million. The project to find more secrets in the book was detailed in Archimedes Codex by Reviel Netz and William Noel. (Be warned the book is told in the first person by two people so can be a bit confusing!). The following video of the first author tells the story and discusses the work of...