  "Compactness," or the use of shape as a proxy for fairness, has been a
long-running theme in the scrutiny of electoral districts; badly-shaped
districts are often flagged as examples of the abuse of power known as
gerrymandering. The most popular compactness metrics in the redistricting
literature belong to a class of scores that we call contour-based, making heavy
use of area and perimeter. This entire class of district scores has some common
drawbacks, outlined here. We make the case for discrete shape scores and offer
two promising ideas: a cut score and a spanning tree score. We use recent
United States redistricting history as a source of examples.
  No shape metric can work alone as a seal of fairness, but we argue that
discrete metrics are better aligned both with the grounding of the
redistricting problem in geography and with the computational tools that have
recently gained significant traction in the courtroom.
