The numbers game – Degrees of separation and mathematical quirks

Materials World magazine
1 Aug 2007

My article in the April 2007 issue of Materials World on the average number of acquaintances between any two random individuals has stimulated more interest than anything I have written for a long time. I have had numerous e-mails and telephone calls from friends telling me of encounters with apparent strangers who amazingly find they have a mutual acquaintance.

I am writing this on the Island of Arran where I am hoping to observe dolphins and sharks in the Firth of Clyde, but am seeing only nuclear submarines bound presumably for Faslane or Coulport (do we really still need nuclear submarines?). I have just met a man who knows someone who lives just a few doors from my Gloucestershire home. I must be careful, this is becoming an obsession.

Call for interdisciplinarianism as IOM3 has grown

I have been asked why I think that a survey of acquaintance links within IOM3 might be valuable. Our Institute has grown by a process of mergers and accretions. Firstly, the Institute of Metals merged with the Iron and Steel Institute, and then the combined body joined with the Institution of Metallurgists. This was so far merely a metallurgical coalescence, but then the Plastics and Rubber Institute (already having united two separate societies into one) and the Institute of Ceramics joined us, and we have many mining experts and geologists in our ranks as well.

These developments are welcome, but there is a loss of intimacy - everyone who is somebody no longer knows everyone else. If we conducted a survey to determine numbers of intermediaries between individuals within and across the discipline boundaries, the information could spur more interdisciplinarianism. The results could even be published in our Interdisciplinary Science Reviews journal.

Mathematical oddities

When I told a mathematically-minded friend that so few links separate anyone on Earth, he was not in the least surprised. He pointed out that if an individual had a thousand acquaintances (which is quite possible) and each of these also knew a thousand people, none of whom knew any of the first thousand (which is the less plausible assumption), and each of the second thousand also knew a thousand people, then we are already up to a billion, between each of whom there are only three intermediaries. Taking the world population as six billion, the number of intermediaries, n, can be deduced from the formula 1,000n = six billion, ie n = 3.3. As this is such a small number, my friend said he would not be taken aback if the actual number is less than 10.

I have to admit that mathematical quirks and oddities have always interested me. I find it fascinating, for example, that in an assembly of only 23 people, the odds on two of them having the same birthday is greater than evens. I can remember an article by the late Auberon Waugh in the Oldie where he said that in church, where he ought to have been thinking of higher things, it suddenly occurred to him that 17.5% of 25 was exactly the same as 25% of 17.5, so all he had to do was divide 17.5 by four. This isn't quantum mechanics, but I have found this simple rule useful. And did you know that the ratio of successive terms in a Fibonacci sequence approaches nearer and nearer to the number 1.618034, which also happens to be the ratio of the golden section, so beloved by the classical Greeks?

Just one more, before I get too excited, a favourite of Hans Bethe - the difference between two successive squares is always an odd number and equal to the sum of the numbers being squared. So there you are - try it out.


Further information:

Link to history of the IOM3 page
'Material Matters', April 2007