## 19 October 2007

### The chain of cities

From radical cartography -- a map of Europe with cities that are near each other (within 150 km) and with populations of more than 100,000 connected by lines. In some places (southern England, the Netherlands, Germany, northern Italy) the lines are very dense; in others they are not. The graph has one large connected component, but there are smaller components -- in Scotland, southern Italy, and the coasts of Spain -- which are separate from the main one.

I suspect you'd get similar-looking maps at smaller scales (that is, if you reduced both the population threshold and the distance threshold). I've heard that the number of cities of population at least N scales like 1/N. To get a similar-looking map with smaller cities you'd want the number of cities "near" each one to be the same. So, for example, say you looked at cities of size 25,000 or greater; there should be four times as many of those "near" a given point as cities of size 100,000 or greater. If we then shorten the distances over which we're willing to draw lines by a factor of two, then we should get the same average degree in the resulting graph. What I'm saying is that I suspect the graph of cities of size 25,000 or greater within 75 km of each other has similar qualitative features. (Of course, at some point you run into the problem of how to define distinct "cities", by which I mean centers of population; should distinct neighborhoods of the same legal municipality be counted differently?)

I'd be interested to see a similar map for the United States (since I'm more familiar with the settlement patterns in the U.S. than in Europe), but not interested enough to make it.

In particular, I see a lot of things that look like the fragments of triangular lattices, with the triangles involved being roughly equilateral. (This seems most visible in Poland.) Now, if you imagine cities as circles -- which isn't a horrible approximation, because you don't expect to see two cities too close to each other -- and you pack them in as tightly as possible on the plane, you should see exactly a hexagonal lattice. Of course, cities aren't all equivalent, and you can't just slide them around arbitrarily...

This makes me wonder -- are there any cities which have street plans that are triangular lattices?