Who wouldn’t want to be Santa? You work one day a year, people give you mince pies and you spend the rest of your time deciding who’s naughty and who’s nice. However, Santa’s job is more difficult than you might think and he is certainly a mathematical genius who could win $1,000,000!
Efficient packing is an Elf's forte.
Let’s focus on just two of his important tasks. Firstly, he has to pack his sleigh full of toys. Annoyingly, toys are not all the same shape so how does he efficiently pack his sleigh? Either he is excellent at Tetris, or he has managed to solve an extremely difficult maths problem known as “bin packing”. The problem is, simply stated, ‘given a fixed amount of space and packages of different sizes, is there a complete list of instructions that will allow you to make the most of the space?’
Santa's problems are bigger than his belly.
The second problem he faces is finding the quickest route around Earth that visits every house. This is known as the “travelling salesperson problem”. Is it possible to find a recipe that will always give you the fastest route?
In both cases no one knows the answer. What is worse is that we don’t even know if there is an answer! Mathematicians describe these problems as “NP” (Not Polynomial). This means that the difficulty of finding the best solution increases dramatically whenever you add another package or house.
Although these problems may be placed in fantasy, real life logistic companies face these obstacles every day and solving them would save a fortune. In fact, the solutions are so important that the Clay Mathematics Institute offers one million dollars to anyone who can either produce a method that works flawlessly, or show that one doesn’t exist.
So until Santa decides to retire and give up his secrets, that million is waiting for the right mind. Who knows? It could even be yours.
Merry Christmas from the Laughing Mathematician and see in 2013 for a new set of posts on the gömböc.
A very seasonal Turing pattern.
Addendum
Although we may not able to solve the problem yet we can still take a peak at Santa's solution.The United States and Canada aerospace defence organisation, known as NORAD, take it upon themselves to track Santa every year as he travels around the globe. You can follow Santa's journey through their website, http://www.noradsanta.org/.
I've recently been involved with a new mathematical project called Everything is Mathematical. A group of Spanish Mathematicians have written a comprehensive set of popular maths books, which have proved so successful that they're being translated into other languages, including English. The team behind the series have joined up with The Times newspaper and Marcus du Sautoy to present the series to the British audience.
But where do I fit it? Well, to generate some interest in the series they're producing a number of maths challenges that can be found on their website (mine is below).
Some of the challenges are
fairly standard and well known, however, there are a couple of gems
in there that I had never seen before! Keep a look out on the
website for the weekly challenges. If you do solve one of the problems there is a
competition you can enter. The winners receive a free subscription to the
entire set of books.
Now since there are 40 in the set and each one is £10, you may need come convincing before you part with your cash. To get you interested I thought I would review one of the books. Of course since I'm working with these guys you'll have to weight my review with the fact that I'm biased :) (although I'll try to remain neutral).
Who are they for?
Based on what I've read I think these books are more suitable for an
older audience. It could certainly appeal to those who were
educated in numerical subjects and then left due to career decisions, but never lost their interest, as
well as those who never really got on with mathematics but realise its
importance. Having said that I could also also see parents buying a set to enhance their children's education as school as well as benefit their own interests.
Appearance
The first thing I noticed about the series is the sheer comprehensiveness of the books. To be honest, I'd be hard pressed to even name 40 different mathematical fields! The books cover all the standard aspects such as geometry, symmetry, probability, etc. but then goes further with books on the mathematics of our senses and perception, artificial intelligence and, of course (my favourite) mathematical biology.
You may think (as I did) that, by spanning so many subjects, each one would be a pamphlet in size. Here, again, I am pleased to say I was surprised. Each book I've seen is a really nice hard back spanning over 100 pages in length. They contain: nice clear diagrams; formulas; anecdotes about the history; fact boxes and even problems to challenge you.
Book number 1.
What about the content?
Well here is my first admission: I haven't actually read the first book. The first one is all about the golden ratio and its geometry and to be honest, I've never been a fan of the golden ratio because even though it has some extremely nice mathematical properties I find that discussions quickly descend into numerology [Rant 1]. So, instead of starting with The Golden Ratio, I jumped straight in with "When Straight Lines Meet" a book all about non-Euclidean geometry.
Now I would've expected the book to have started with easy stuff, y'know setting up the cartesian coordinates and linear geometry leading on to hyperbolic and elliptic geometry later. How wrong could I have been? The first chapter drops you straight in by explaining the taxi cab metric, using the city of el Ensanche as an illustration.
el Ensanche's very regular city set up.
This is certainly one of the strengths of the books. They use of definite, tangible examples allowing them to discuss complex ideas and very quickly outstrip even my knowledge! For example I had never heard of Girolamo Saccheri's contribution to non-Euclidean geometry.
Each chapter is completely different and could potentially be read independently. For getting a deep overview about a specific aspect (history of geometry, curved space in relativity, uses in computing, etc.) then this format is very appealing. However, it lacks an overarching sense of story that will keep you reading all the way through like the best popular science books can.
What is good to see is they're not afraid of displaying mathematical functions and formulas and then challenging you to calculations that actually mean something, e.g. why calculate the length of a line of a sphere when you can calculate the distance between New York and Sydney on the Earth?
Summary
£400 is a steep price for a comprehensive set of popular maths books, although calling them "popular" perhaps does not do them credit. Think of them more as a set of encyclopaedias of the mathematical world, but with each chapter being eminently more readable than a fact driven summary of events.
There is no better set that will give you the range and depth of subjects available. Due to their diverse nature you will discover topics that interest, entertain and even surprise you. Thus, if you have want to see what mathematics can really do outside of the classroom then these books are you.
[Rant 1] Of course you're going to find things in the golden ratio if you look for it. Any ratio can be found in nature if you look hard enough. I freely admit that in certain cases you can attribute inspiration to the golden ratio, e.g. Le Corbusier's architecture or Dali's and Da Vinci's paintings. However, these are explicit uses. Stating that certain natural phenomena occur in the golden ratio is much dodgier.
To really believe that the ratio is there I would want to understand WHY it is there. This is the essence of mathematical biology. Biologists observe phenomena, mathematicians try to understand its cause. For instance, there are reasons why the golden ratio/ Fibonacci's sequence should appear in sunflower seed packing as it is the optimal packing ratio. Compare this to the claim that your fingers are in the golden ratio and the suggestion that beautiful people are more in the golden ratio than not. Not only are you left asking why but you're suggesting that people who don't fit your rigid proportions are ugly!
The golden ratio supposedly appearing in the human body.
This week concludes the serialisation of my interview with Andrew
Hodges. The final questions I asked looked at the much bigger picture of Alan's legacy.
Once again, if you are interested in reading more about Andrew Hodges you can find his website here and his book can be bought from here.
Well I disagreed with the wording of the petition because it
was too specific to Turing; it didn’t take into account the hundreds of
thousands of other people who were in similar positions. But on the other hand
I warmed to it a lot as Gordon
Brown’s apology was very, very, good and brought out all of these wider
points and linked them in a similar way to that which I had done in my book.
Gordon Brown's apology to Alan Turing. Signed and donated to Bletchley Park by Gordon Brown.
My point of view, coming from the gay rights perspective, is
that Turing’s case was simply an extreme example of what society was like, what
the effect of legislation was like here and what it will be like in other
countries now. The evil is not what happened to the individual it is what it
happens generally and Alan illustrates this very clearly with his story. That
is how it struck me in ’73 and I haven’t changed from that point of view. The
story is very vivid but the message is much more general. So I think this
apology is too much mixed up in people thinking that we have to rescue this
great scientist, rather than seeing a human rights question in general.
It is hard to know what Alan would have thought but I
believe he would have thought the whole system was wrong rather than simply
wanting an excuse for himself.
Ah well let me put you right there. They didn’t get his
papers back. They simply got Max Newman’s copies of Turing’s published papers. More
positively, its great that Turing’s actual papers are preserved at King’s
College, Cambridge.
Back to your original question, I revived my interest in the
mid 90s as the internet was beginning to grow and I’ve written another shorter
book on Turing as a philosopher as well as a number of smaller academic papers.
On the whole I am very happy to leave everything to other people who have
become great experts in some particular area of his life, whereas my strength
was to see all of these links as a whole.
When/ how/ why did you start thinking about writing a biography about Turing?
Well, when I learned I had these three connections with the man and he had such a story I knew I had to tell it. However, it would’ve been very difficult simply to write an article. All of the connections as I’ve just described were hidden at this time. No one even knew the story of the computer because of Bletchley Park being so secret!
I also sensed that there was a big fascinating story to tell about how all of these links came together in just one person. I had a feeling that it would be possible to present something modern in its social and political point of view but about something unexpected, namely the history of technology, science and the Second World War. Although at the beginning I didn’t realise just how big the story would become and how difficult it would be to get all of the information.
Because it was so difficult did you ever get frustrated?
It was difficult to do, but the discipline of writing a biography means that you have the unifying feature of a single person’s vision and you don’t have to cover everything that was going on in the world at that time, which is what historians have to do. I liked the idea of working through someone else’s life and remembering that they never knew what was going to happen next. That is one of the things that kept me going during the difficult times. It was hard work interviewing people about subjects that, at the time, where either secret or difficult to talk about. I also acquired a lot of scientific material, which in the beginning I knew nothing about.
What kind of character was Turing?
He was certainly odd. But in mathematics we have a slightly different picture of what you expect people to be like. Many of the things that strike people as odd came across to me as traits that anyone involved in deep concentration, not just mathematicians, would have.
One way of putting it is that he lived a lot like people did 20 years later.
That is quite an interesting point as, having read your book, Alan seems to come across as a relatively normal person.
Yes, but normal for a number of decades later. I would say that rather as his ideas (computation and morphogenesis) were ahead of his time, so was his lifestyle. We just wouldn’t have made such a fuss about his oddness during the sixties and seventies.
Earlier this year I was lucky enough to have the chance of interviewing Andrew Hodges about his experience of writing a biography of Alan Turing, "Alan Turing: The Enigma". Since we spoke for quite a while I've decided to serialise the interview over the next few weeks. This week I introduce Andrew and ask what set him on his path of writing the book.
If you are interested in reading more about Andrew Hodges you can find his website here and his book can be bought from here.
Firstly, for those who do not know of your career, could you just give an outline of how you came to where you are?
I am in an unusual situation, because the Turing material is something that took me off from my normal trajectory. It was not a direct part of my normal progress as I was already a post doctoral researcher in quite a different field, mathematical physics; working with Roger Penrose. Although I took time off to do it I managed to mix it in with my research over the six years, from 1977 to 1983
In 1985 I received an advanced Science and Engineering Research Council fellowship, which really established me in Oxford, where I have been ever since.
You did your Phd in 1975 in twistor theory and continue to work in this area to this day. Can you explain a little about what this is? It is linked to string theory isn’t it?
It is really very different from string theory, although there is more overlap now.
String theory is about adding in extra dimensions to space and time and structures within those. Twister theory is simply a different way of describing the four dimensions that we know about. So in that way it is less radical!
What do you get out this description that the normal view of space and time doesn’t give you?
Instead of thinking about the four dimensions as physical description of three space dimensions and one dimension of time we instead think about of two dimensions plus two dimensions. It is based on a set of light rays which are the fundamental objects.
This description fits very nicely with the way the whole of fundamental physics has gone which puts emphasis on objects that have zero mass. The ideas behind the Higgs field and gluons also fit beautifully into this point of view.
Since you work in such a different field to Turing how did you first hear about him
In 1969 I was a Cambridge undergraduate and at that time his name wasn’t well known to most undergraduates. I read a lot of maths that was not on the syllabus and I discovered his work on Turing machines. So his name at least meant something to me at that time and I was very surprised during 1972-73 when his name came up in quite a different way. I met many people in the Gay Liberation movement who had actually known him.
So Alan was more famous in the gay community than the mathematical one?
Oh no. He was by no means a household name. It was simply coincidence of meeting the older generations and having seen his name before. Very few people knew of this connection to Turing. Of course, it is well known now, but at the time it was something no one wanted to talk about.
The third connection was that in the mid 70s, books started leaking the details of the work done in Bletchley Park. Also the BBC had a very good programme on how the Enigma was broken but it didn’t really highlight Turing’s role. It mentioned that he was there but didn’t discuss his work, whereas through my connections I gathered that he wasn’t just “there” but actually he was the most important figure in the whole project.
And is that true? Turing is often praised as being the genius behind the breaking of Enigma but there were plenty of others there working with him who should also not be forgotten.
I don’t think any of the other big people (Donald Michie, Max Newman, Jack Good and Shaun Wylie to name but a few) would have questioned his importance… on the scientific side of course. There were many important people running the infrastructure, engineering of the machines and linguists, but on the scientific side there is no question of his influence. He was both the first serious scientific person into the field of decryption and the most innovative.
He had a very unobvious idea of how the Enigma machine worked and also developed all of the statistical theory which they used.
My friend Andrew Holding presents a show on CamFM called the Science of Fiction. He has spoken about many different subjects including comic book science and the use of computers in movies. On 29th May 2011, he did a whole show on the science of zombie invasion. I was lucky enough to join Andy and Frank Swain (the Science Punk) to discuss with them the whys and hows of the zombie holocaust. If you are interested I grabbed the show out of the airways and you can listen to it below.
