urban

Yurtle the Turtle had nothing on this!

September 21, 2009 in New York, Technology

The Burj Dubai is by far the tallest building in the world, despite being unfinished. However, I find it difficult to grasp just how massive it is. A recent Gizmodo post came close to capturing its immense height (see the image below) but still, a true sense of scale is absent.

The trouble is that I have no concept of relative height when I’m looking at those images; yes, the tower looms over other buildings that I know deep down would be considered immense in their own right, but they might as well be townhouses. They provide no context because I have nothing tangible to compare them to. Meanwhile, silhouette comparisons such as this one convince me of the Burj Dubai’s height, but do little to impress any grand sense of scale:

What I need is a comparison that marries the abstraction of the silhouette with the concrete grounding of the actual photos. Once again quoting Tom Lehrer, I have a modest example here…

Turning to Google Earth, I mocked up the views from two popular Manhattan observation decks – the Empire State Building and the Rock (that’s Rockefeller Center for the non-30 Rock fans New Yorkers among you). Then, I raised the viewpoint to 2,690 feet – the height of the Burj Dubai’s hypothetical observation deck. The result is an impossible view of Manhattan which instantly captures the building’s enormous scale by putting its height in a familiar context. If you are unfamiliar with the New York cityscape, then these examples may be as abstract as the actual Dubai pictures are to me; however this is an experiment well worth repeating in your own urban backyard.

Note: Clicking the following images will launch an image gallery in a lightbox. The first image will show the view from an existing New York observation deck. Clicking the right side of that image will load the next image, which shows the same view from the top of the Burj Dubai. Click outside the lightbox to close it. Note that all the images below will load, so you can click through all five viewpoints without leaving the lightbox.

Note also: The effect is much more dramatic in Google Earth, which supplies smooth transitions between the viewpoints – like taking an elevator up the spire. But I’m having trouble embedding the 3D view here, so I hope these images suffice…

First up, the view from the ESB looking north toward Central Park. The real view is impressive but the Burj Dubai can practically see upstate:

Next, a similar view – the ESB looking northeast into midtown and across the East River. The Burj Dubai view makes the surrounding buildings look tiny:

Another view familiar to tourists – the ESB looking south toward the Financial District. From the Burj Dubai, you could see clear across New York Harbor and out into the Atlantic:

Turning now to the Rock, here’s a similar view to the south, including the Empire State Building and Chrysler Building. The Burj Dubai towers over these New York giants:

Finally, here’s another view north, this time from the Top of the Rock. The difference is unbelievable:

I hope that these visual comparisons give some greater meaning to how incredibly tall the Burj Dubai is by supplying a familiar context for its height. In a final push for perspective, we are all familiar with this iconic view of downtown Manhattan:

Typically, a helicopter would be used to capture an image from such height. But in this case – you guessed it – all you’d have to do is take the elevator. Yurtle the Turtle had nothing on this!

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Urban mathematics

May 20, 2009 in Math

Zipf’s law is another mathematical phenomenon not entirely unrelated to Benford’s law (in fact, some think that Benford is a special case of Zipf). (Aside, it’s funny how after you discuss something, it seems to pop up everywhere – Kahneman and Tversky would have a lot to say on that, I’m sure.) Zipf’s law is used to describe datasets in which an item’s occurrence is inversely related to its rank. For example, the most frequently observed element occurs X times. The second-most frequently observed element X/2 times, the third ranking element X/3 times, etc.

Zipf, a linguist at Harvard, first noticed the pattern which would bear his name while studying word frequencies in texts. For a time, it was thought that Zipf’s observation was a mathematical representation of some underlying language process. However, if text is randomly generated by uniformly picking from 26 letters and a space, the resulting corpus also exhibits Zipf’s law, suggesting it is more of a statistical artifact than evidence of a deeper semantic process.

In today’s The Wild Side blog of the NYT, guest blogger Steven Strogatz, a professor of applied math at Cornell, discusses Zipf’s law in relation to city populations. Strogatz states that Zipf actually discovered his pattern while looking at urban populations, which surprised me since I was familiar with the word-frequency story – it turns out both are true: he presented the words and then the cities as two examples of the pattern. There is some doubt that the fact that city sizes follow Zipf’s law actually means anything, just as it appears to be a statistical artifact in linguistics. Other statistical distributions provide a good fit, but Zipf’s law is “nice” because of its utter simplicity.

The remainder of the blog is on other urban mathematics, including evidence of economies of scale, before connecting such observations back to biology and cellular processes (the usual focus of the blog – gotta keep the readers satisfied!). Strogatz draws an interesting connection between the mechanism for transporting nutrients through a cell and the transportation infrastructure of a city, both of which exhibit similar economies of scale with exponents close to 0.75. It turns out there is a mathematical justification for why that would be so.

As someone fascinated by urban development, I especially appreciated the essay, especially for the philosophical lens it borrows:

These numerical coincidences seem to be telling us something profound. It appears that Aristotle’s metaphor of a city as a living thing is more than merely poetic. There may be deep laws of collective organization at work here, the same laws for aggregates of people and cells.

The numerology above would seem totally fortuitous if we hadn’t viewed cities and organisms through the lens of mathematics. By abstracting away nearly all the details involved in powering a mouse or a city, math exposes their underlying unity. In that way (and with apologies to Picasso), math is the lie that makes us realize the truth.

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