A paper from 2011 on measuring compactness for purposes of legislative redistricting is publicly posted here: * A Single Compactness Measure for Legislative Redistricting Some of the paper is highly mathematical (the proof that an entire class of compactness measures ranks any two redistricting plans exactly the same). However, most of the paper is very understandable, especially the counterexamples showing why most of the proposed measures of compactness can be used for gerrymandering because they do not reliably measure compactness.