Zipf’s Law – modelling the megalopolis

Paul Raven @ 25-05-2009

Taipei urban skylineMore statistical sensawunda in the urban environment. Remember us mentioning that guy who suggested that cities can be considered  as super-organisms? Well, a mathematician chap called Stephen Strogatz dropped into the New York Times blogs to talk about Zipf’s Law and other statistical phenomena that surround our urban environments:

The mathematics of cities was launched in 1949 when George Zipf, a linguist working at Harvard, reported a striking regularity in the size distribution of cities. He noticed that if you tabulate the biggest cities in a given country and rank them according to their populations, the largest city is always about twice as big as the second largest, and three times as big as the third largest, and so on. In other words, the population of a city is, to a good approximation, inversely proportional to its rank. Why this should be true, no one knows.


For instance, if one city is 10 times as populous as another one, does it need 10 times as many gas stations? No. Bigger cities have more gas stations than smaller ones (of course), but not nearly in direct proportion to their size. The number of gas stations grows only in proportion to the 0.77 power of population. The crucial thing is that 0.77 is less than 1. This implies that the bigger a city is, the fewer gas stations it has per person. Put simply, bigger cities enjoy economies of scale. In this sense, bigger is greener.

The same pattern holds for other measures of infrastructure. Whether you measure miles of roadway or length of electrical cables, you find that all of these also decrease, per person, as city size increases. And all show an exponent between 0.7 and 0.9.

Now comes the spooky part. The same law is true for living things. That is, if you mentally replace cities by organisms and city size by body weight, the mathematical pattern remains the same.

It looks as if there’s a lot of things that mathematical analysis could tell us about the cities we live in. The question is, are these properties inherently emergent, or could we design our urban environments more effectively and adjust some of those efficiency values in the process? [image by tylerdurden1]

The power of autism… and the danger of selecting against it

Paul Raven @ 08-01-2009

maths calculationsNew Scientist has a fascinating interview with record-breaking autist savant Daniel Tammet, where – among other stuff – he discusses the way in which he’s able to perform what seems to us like mathematical magic:

You wouldn’t use a word like “giraffe” without understanding what the words “neck” or “tall” or “animal” mean. Words only make sense when they are in this web of interconnected meaning and I have the same thing with numbers. Numbers belong to a web. When somebody gives me a number, I immediately visualise it and how it relates to other numbers. I also see the patterns those relationships produce and manipulate them in my head to arrive at a solution, if it’s a sum, or to identify if there is a prime.

Somehow I suspect that when Tammet talks of visualising numbers, he doesn’t see little piles and stacks of units like I do! Strange to think that what seems to us to be the only way to do something is not only one of many ways, but a horribly inefficient one as well. [image by Robert Scarth]

Meanwhile, a British autism expert warns that screening for autism in utero could deprive us of geniuses like physicist Paul Dirac:

… if this test led to some kind of prenatal treatment, such as the use of drugs to block the effect of testosterone which is already medically possible, would this be desirable? Caution is needed before scientists embrace prenatal testing so that we do not inadvertently repeat the history of eugenics or inadvertently ‘cure’ not just autism but the associated talents that are not in need of treatment.

Then again, maybe we’ll just be able to use drugs to induce autism as necessary in the near future, whether for personal gain or for society at large…

Has anyone here read David Zindell’s Requiem for Homo Sapiens series, by the way? It features a sort of art cult that voluntarily lives in a pseudo-autistic state, among many other weird ideas. Superb set of books, but a real head-f*ck.

Climate change explained through probability and risk: It doesn’t matter if it exists, we should act anyway

Tomas Martin @ 22-01-2008


Craven has created a series of fun, educating videos that should be watched by all.Science teacher Greg Craven posted a video entitled ‘The Scariest Video you’ll ever see’ on Youtube in June 2007. The ten minute video garnered over 7000 replies including many criticisms from global warming sceptics. Craven decided to rebut these criticisms. He spent four months of his spare time researching data on the debate, ticking off each criticism that had been made. He then released “How It All Ends”, another ten minute video but this time with an ‘expansion pack’ of videos going into each of his arguments in exhaustive detail.


Interestingly, much of the content of the six-hour, 44 part series is not devoted to proving whether global warming is happening or not, or whether man is causing it or not. He looks instead at the four main outcomes: global warming exists and we do something, it exists and we don’t do something, it doesn’t exist and we do nothing or it doesn’t exist and we do something. He concluded the costs of doing nothing far outweigh the cost of doing something, so it makes sense to take action even if we don’t know whether global warming is happening or not.

A site has also started up devoted to the videos, where the forum members critique and find responses to each new criticism as it comes through on Youtube. The efforts of these people to encourage reasoned debate is heartening. Many of the arguments against combating climate change revolve around the fact that science doesn’t agree 100% with the precise outcome. Well, science never will agree, not totally, especially with oil industry-paid advocates in the mix. But even without more and more evidence leaning towards the ‘we need to do something camp’, the logical thing to do is to take action, even if it turns out we didn’t need to. There’s also a great interview with Kim Stanley Robinson at BLDGBLOG about this.

Wormholes a ‘possibility’

Tomas Martin @ 15-10-2007

‘metamaterial’ wormholes will make beams of radiation invisibleA group of mathematicians who previously worked on possible cloaking devices have found the same theory could be equally applied to create things resembling wormholes. The team uses mathematical theory to create ‘metamaterials’ that can bend and curve electromagnetic fields – like bending light to make something appear invisible.

The wormholes they describe aren’t quite the instantaneous transportation portals described by Star Trek or Valve’s new computer game. The light still travels through the metamaterial tube but isn’t detectable outside it, by sight or other methods. Uses for this idea include endoscopic surgery and 3D televisions where all but the end tips of many beams of light are hidden by the wormhole, giving the appearance of a floating image.

[image and link via ScienceDaily, image courtesy of Rochester University]

Evolving better bridges

Paul Raven @ 04-08-2007

In light of the recent and tragic bridge collapse in Mississippi, mathematics uber-geek Stephen Wolfram has been doing some thinking about how evolutionary computing could be used to design stronger bridge structures. It looks like strength doesn’t always correlate to regularity of patterns. [BoingBoing]

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