Compactness - this common redistricting rule requires district boundaries to be fairly regular, without extensive tendrils that might indicate unreasonable groupings to gain political favor. Although most states do not have precise methods of quantifying compactness, some commonly used measures are the Polsby-Popper score, Reock degree, Convex Hull method, and perimeter length.
Contiguity - this redistricing rule requires all portions of the district to be physically adjacent. That is, all points within a district should be reachable without having to cross the district boundary.
Gerrymandering - the intentional manipulation of electoral district boundaries for personal or partisan gain. Though most often talked about in terms of partisan outcomes, racial gerrymandering can be employed to dilute or suppress the voices of certain demographic groups either by clustering those groups together to limit the number of seats they can get (‘packing’) or by separating them into many districts to dilute their votes (‘cracking’).
Markov Chain Monte Carlo - a random sampling method relying on a random walk (Markov Chain) to traverse the state space. This method is used to sample districting plans for a distribution that is ‘tuned’ to satisfy a state’s legal criteria. New plans are generated by making iterative change to a given initial plan, while still conforming to the state’s redistricting rules.
Neutral Map - maps created without partisan bias, while accounting for the specific redistricting rules of the state
Population Balance - most commonly accepted rule requiring districts to be relatively equal in population to adhere to the ‘one person, one vote’ principle. The variation allowed depends on state specific requirements, and may differ based on the district type (i.e. congressional districts versus state house districts can have different thresholds)
Redistricting - the process of reapportioning congressional seats to ensure proportional representation across states due to population shifts. Using U.S. census data collected every 10 years, all 50 states must redraw the geographic boundaries of their congressional districts. The rules regulating this process can differ on a state by state basis, although there are common federal rules (e.g. the Voting Rights Act) that govern the process nationwide. Additionally, each state has discretion over who controls the process - some states elect to have their state legislatures draw new district boundaries themselves, whereas other states refer the process to a committee of their choosing. Since district boundaries have so much power to shape the dynamics of political representation and influence the outcomes of future elections, this is always a contentious and politically charged process, which creates incentives for gerrymandering.