An essential question for democracy is how to rigorously determine the likelihood that a congressional map has been gerrymandered. A number of state constitutions require districting plans to be “compact,” yet no technical legal definition of compactness exists in this context, leaving us to contend with the oft-cited sentiment that “you know it when you see it.” In this talk, I will introduce a novel compactness measure based on a geometric structure known as the medial axis. This skeleton-like structure has been shown to have strong ties to the science of how the human brain perceives and processes complex shapes, thus offering a mathematically rigorous version of “the eye test.” I will explain the construction of this metric in detail, and then compare it to a recent machine-learning-based compactness metric introduced by Kaufman, King, and Komisarchik (2021). Specifically, in this work we examine the performance of our measure and theirs in several case studies, including two states whose districting plans were especially contentious and the entire 2016 congressional district map. This is joint work with Ellen Gasparovic at Union College, and Jason D’Amico and Mushan Zhong, who were undergraduates at Union College at the time of their contributions.