Fig. S1.

The last state from each of the above runs of the chain (perimeter, $L^1$, $L^2$, and $L^{\mathrm{\infty }}$, respectively). Note that the $L^{\mathrm{\infty }}$ districting is quite ugly; with this notion of validity, every district among the 18 is allowed to be as noncompact as the worst district in the current Pennsylvania districting. The perimeter constraint produces a districting that appears clean at a large scale but allows rather messy city districts, because they contribute only moderately to the perimeter anyway. The $L^1$ and $L^2$ constraints are more balanced. Note that none of these districtings should be expected to be geometrically “nicer” than the current districting of Pennsylvania. Indeed, the point of our Markov chain framework is to compare the present districting of Pennsylvania with other “just as bad” districtings to observe that, even among this set, the present districting is atypical.