Tuesday, December 04, 2007

At 8am Helsinki is a 1.77-Dimensional

And worse, at Sunday noon its dimensions decrease to 1.45. No wonder I occasionally feel a little confused about our geography...

Ok, maybe I need to explain a little. Assume we are given distances between random pairs of points in a hypercube of unknown number of dimensions and extent. A little calculus shows that if the coordinates of the points are uniformly distributed along each axis, the square of their distance has an expectation of a2 N / 6 and variance of 7 a4 N / 180 where N is the number of dimensions and a the edge length of the hypercube. I hereafter make a lofty assumption that these formulas apply also to fractional dimensions, at least for some such definition.

According to YTV there are 6913 bus stops or other kind of access points for regional public transit in Helsinki and its satellite municipalities Espoo, Vantaa, Kerava, Kauniainen and Kirkkonummi combined. I wrote a script which picks two points at random and queries the optimum public transport solution between them from a web service generously provided by YTV. Due to the nature of the data provided by the web service, I define distance not by meters but by duration. More specifically, how many minutes before 8am must one enter the starting point in order to reach the end point by 8am on a working day's morning.

Based on some 20000 distance samples, the mean distance is 54.4 minutes with a whopping maximum of 275 minutes. No wonder people prefer cars in those more rural areas of the region. But from the statistics of the squares of the distances and solving the equations for a and N shows that the regional public transport system behaves like a 1.77-dimensional hypercube with edge length of 109 minutes of travelling.

On weekends public transport becomes less frequent and a larger portion of traffic goes to and from various local centers instead of offering short-cuts around them. This increases the mean travelling time by almost 10 minutes to 64 minutes when aiming to arrive on Sunday noon. In our hypercubistic terms, this decreases dimensionality to 1.45 and increases edge length to 142 minutes.

The really curious observation is the significance of high-speed rail traffic: refraining from subways and trains and using only busses, trams, ferries and feet doesn't increase the mean traveling time more than six minutes, but more importantly, it breaks off clusters reachable from others only through very long walks or infrequently running night-time busses. Consequently the variance of traveling time is larger than a homogenous cube-like space would allow. Mechanical use of the equations would in fact describe Helsinki without fast rails as a hypercube with 0.39 dimensions, i.e. fewer dimensions as a straight line!

Of course, the true reason for rails is not in fact so much in their speed (compared to an express bus) than in their capacity. And they indeed do add another dimension to our lives, or 1.38 to be more precise.


Anonymous Anonymous said...

This is ingenuine stuff. I don't know, if you update anymore and read this comments, but this is quite smart, in fact. I quess just about every geek, or most of us, have sometimes thought about the geometry/topology of a town. Never ever occured to me to use these formulae, anyhow.

8:55 PM  
Blogger cessu said...

Thank you for the compliments, mc. I have much more stuff to write about that I have time to write, but hopefully I'll be less busy from now on.

For hypercubes my use of the term "dimension" actually coincides with that of spectral dimensions, which you can read more in the literature, including wikipedia.

2:57 PM  

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