What does Robert Pershing Walden, the tallest man to have ever lived on record, have to do with density of cities?

If you understand why there are not many people nine feet tall, you understand why big drops in density can be devastating for city finances.

For every inch that Walden grew in one dimension, his blood vessels had to grow in 3 dimensions, to reach every point in his body. As height goes up linearly, the length of blood vessels goes up exponentially, and at some point, the vessels get so long it is no longer be viable for the heart to service them all.

The same thing happens with the length of roads as density drops. Here’s a diagram that lays out the basic idea:

For many in the public, it may seem intuitive that homes separated by 200 meters won’t cost dramatically less to service than homes separated by 300 meters. But for this idealized block, while density drops by about only one half, the length of road per household doubles. That means that every service a city provides each house—water, road maintenance, snow clearing, etc.—will cost twice as much in the second scenario.

The reality doesn’t track this ideal quite so directly. One tricky way to avoid adding roads as density goes down is to abandon the walkable grid and have the fewest possible intersections, as we discuss in another blog post.

But the basic pattern certainly does show up in the data. Muller *et al* find that as the density of whole cities goes down, the length of roads, water, and sewer pipes all go up exponentially.

We wanted to know if the same pattern holds *within* cities. If compare the road length per resident for different neighbourhoods within the same city, will it go up exponentially as density goes down?

The answer is yes, and in a remarkably similar way. We looked at nine cities, but here are four to give a flavour.

The underlying pattern is remarkably consistent, like an underlying law geometric principle shaping cities. Victoria has no census tracts on the extreme low-end of density, and Oshawa has almost no high-density census tracts, but all their other census tracts nonetheless fall on the curve exactly where we would expect them.

Here are the trend lines for all nine cities, combined:

These results do not reveal enormous differences between reasonable-density and high-density areas. They do show, however, that small changes in density at the extreme low-end can have a huge impact on costs. Think exurban communities with two-to-three-acre lots. As density drops below 25 people per hectare, communities can start to three, four or five times longer roads per person than other suburban neighbourhoods.

It may seem intuitive to think that going from 47 to 46 people per hectare will have the same cost impact as dropping from 14 to 13 people per hectare. In both cases, the density is dropping by one, right?

That idea would be mistaken. The following diagram is based on the average trend line for the nine cities we analyzed.

Losing one person from 33 people per hectare only adds about 10cm of road length per person. Dropping from 18 to 17, however, adds 30cm. In other words, the rate that costs go up with a single person drop is three times faster. As density reaches low levels, the rate costs increase accelerates quickly.

Not all drops in density are equal. The extreme low-end of low-density suburban neighbourhoods can cost many times more than other neighbourhoods.