Tuesday, 22 May 2007


If you have ever wondered what the fallout would be, so to speak, of a nuclear weapon on a urban area then you must watch this film. Despite being over twenty years old it is chilling and certainly puts Britain's recent decision to renew Trident into some sort of context.

Friday, 4 May 2007

Hilbert's Hotel

I've written about a few eccentrics on this blog and David Hilbert is another one. Hilbert was a German mathematician and geometrician and heavily involved in number theory. One of his students committed suicide after struggling with a problem set by Hilbert and he was invited to speak at the student's funeral. At the side of the grave he addressed the crowd and explained that the problem he set was actually quite simple, the student simply looked at it the wrong way. Hilbert's Hotel or Hotel Infinity was an illustration created by the mathematician to highlight the problems created by dealing with infinity as a number and I shall try and describe it to the best of my abilities.

Imagine, though don't worry if you can't, an infinite hotel with an infinite number of rooms numbered 1,2,3,4... and so on ad infinitum, and when you get there to check-in you find out that it's full despite the fact that the neon 'rooms available' sign is flashing outside. So you call for the manager and he reassures you that despite being full, he can still accommodate you all that is required is that the guest in room 1 moves to room 2, the guest from room 2 moves to room 3 and so on leaving everyone with a room and room 1 vacant for you. An infinite hotel is quite impressive so using the wireless internet in your room you log onto myspace or facebook and spread the word. The next day a thousand people arrive at reception eager to see this amazing hotel and again the manager has no problem in fitting everyone in. Everyone is shifted a thousand rooms, so you in room 1 are now moved into room 1001.

The next day an unexpected party arrives, an infinite number of holiday makers stop off at the hotel on their way to Legoland Windsor, it is frightfully popular. To fit this rather large number of new guests, the manager moves the person from room 1 to room 2, room 2 to room 4 and room 3 to room six and so on leaving an infinite amount of odd numbered rooms available for the infinite number of new arrivals. Now can you imagine staying at an infinite hotel, the queues for the lifts are, well infinite, room service is lamentable and you are sharing the bandwidth of the wireless broadband with an infinite number of other computers so you can appreciate that many guests are rather disgruntled and the next day all the guests in the even numbered rooms pack their bags and leave though despite going down to 50% capacity, the hotel still has infinite guests.

The last twist of the plot is that the chain who own the hotel close down an infinite number of infinite hotels and send all their guests to the Hotel infinity for accommodation. The manager becomes desperate for a solution, how do you accommodate an infinite number of infinite numbers of guests? So he asks his guests for some help, since you've observed everything that has gone on from the beginning you propose a solution. Remembering that you've only got people staying in the odd numbered rooms you move them into the even rooms, then the first person from the first hotel goes into the first empty room, room 1, the second person from the first hotel and the first person from the second hotel get the next two empty rooms, rooms 3 and 5 and the third person from the first hotel, the second person from the second hotel and the first person from the third hotel get the next three empty rooms and so on until everyone has a room.

Despite having 100% capacity in an infinite hotel with infinite turnover, costs are infinite and despite the accountant's good work at securing a low rate of tax, the liability is still infinite. The manager brings in his accountant to explain the situation and comes away reassured because even paying infinite costs and an infinite tax bills, he'll still be left over with infinite profits. David Hilbert's illustration leaves you with the question of whether there is an infinity which is bigger than another infinity. The answer to this question came from another German mathematician, Georg Cantor whose diagonal theorem apparently proves the existence of larger infinities but you can research that one for yourself. The Open University did a film of Hilbert's Hotel with Susannah Doyle (the scary black haired one from 'Drop the Dead Donkey') as the lead and it is well worth watching if you ever get the chance.

Wednesday, 2 May 2007

The Golden Ratio

When I studied Philosophy at A-Level we were given a talk by a composer and he brought the golden ratio to my attention and it is one of those things, like pi for instance, that just blow your mind away. The ratio, 1:1.61803, is not just a mathematical constant but is a ratio that can be found in nature and art with such frequency that the Romans titled it 'sectio divina' or 'the divine section' and it is one of the supposed proofs put forwards for the teleological argument for the existence of God (otherwise known as the argument from design).

The golden ratio, also known as phi after the Ancient Greek mathematician who seems to have discovered it and applied it in the construction of the Parthenon, can be found everywhere. Take your fingers for instance, the ratio from the largest bone to the middle bone is phi, so is the ratio between the middle bone and the smallest bone. You also find phi in the arrangement of branches on the stem of a plant, in the growing points in a plant (the distance the shoot grows before it is strong enough to support another branch), in the replication of patterns in leaves, it has been found in the proportions of chemical compounds in crystals. It has even been found as the proportion of drones to the population of bees in a beehive.

The reason it was a composer who lectured on this subject was because phi has been found in Mozart. His piano sonatas which are often conveniently split into two parts can be found to contain phi in the proportion from from one movement to the other. Phi can also be found in the ratio between key changes in Debussy's 'Image, Reflections in Water'. More recently Shostakovich applied the golden ration to his music though from memory it seems to have not produced that pleasant a piece. Leonardo da Vinci incorporated phi into his work, the Mona Lisa being being a prime example of its exercise. The golden ratio, for many, is the key to beauty. Leon Battista Alberti, the fifteenth century Italian architect, believed that beauty was a matter of proportion and that if a body was divided up into 600 parts beauty would be ‘a Harmony of all the Parts, in whatsoever Subject it appears, fitted together with such proportion and connection, that nothing could be added, diminished or altered, but for the worse’. The proportion which he believed would secure a harmony of all the parts was phi.

We humans have an incredible ability to see patterns everywhere, phi really could just be another example of this talent at making order of chaos or it could be one of the keys to unlocking our understanding of the universe. As an aside, Pythagoras was evidently spooked by the discovery of the golden ratio because he worked to keep it secret, its discovery was punishable by death.


I haven't written any especially political entries on this blog partially so that I don't alienate any potential readers but also simply because there are enough political blogs on the net at the moment that adding another doesn't seem a worthwhile exercise. Nevertheless I feel it is important to highlight the joint campaign by the Amnesty International and the Observer Newspaper against censorship on the internet. I've been aware of this campaign for some time now but the reason I've decided to blog on it now is because I was particularly incensed to learn the news last month about Youtube banning videos that mocked the King of Thailand. Lèse majesté laws are of themselves a shocking abuse of power and are the last resort of pathetic despots and egomaniac dictators, but that Youtube acted to support them is particularly atrocious.

You can join the campaign by doing any or all of three things. You can sign the petition at the Irrepressible.info website, you can also publish fragments of censored information onto your website or blog as I have done on the right hand side of this blog, or you can blog on the subject of internet censorship and highlight the campaign. I'm sorry if that sounds preachy.