Reading an interesting paper on urban form, and the author writes: "City compactness can be measured simply using urban spatial form or morphology: the more concentrated the built-up area, the more compact the city is". Which gets me thinking: How is Compactness different from Density?
Planners have moved away from Density, to some extent, on two fronts. First, the Perils of Average Density, and second that density has no rhetorical value---to harp on density "offers no hope to places that are already built at low densities and unlikely to change". For the built environment is a durable place, and changes slowly. Sometimes very slowly. Roads vanish but rarely, and once the development pattern is 'set', very little can be done to change it.
But what is 'Compactness"? The word has many distinct meanings, but it fundamentally refers to a measure of area. And therein may lay the rhetorical difference--for density, it's a matter of how much stuff is in a given unit of area. For compactness, it's a matter of what unit of area is needed to contain all the stuff. It's a difference of emphasis--the amount of stuff versus the size of the container. What is 'held constant' in each measure is different. For density, it is the unit of area. For compactness, it is the amount of stuff.
Take two hypothetical cities, with the same population. Pi-town is round, constrained within a circumferential wall. Spiderville stretches out in all directions along major roadways. Yet they have the same land area, and thus the same average density. But their compactness is very different. The length of Spiderville's town boundary is much much longer than the Pi-town boundary.
For transportation, Compactness is a much more important measure then Density. Compactness is also a measure of the distance between any random pair of points within the city.
This also brings up the interesting topic of fractal density--a city whose edges are yet more scalloped, more indented, then Spiderville, with little tentacles of development reaching out from the urbanized area.