The grid example contains two distinct moves, and neither one needs a strange outline. Cracking spreads the smaller party thin: horizontal bands put four voters of one party and six of the other into every district, so the six-voter party wins all five times [6]. Packing does the reverse, concentrating almost all of the larger party's voters into two districts it wins overwhelmingly while losing the remaining three narrowly, which is how the same 50 voters hand three of five seats to the party holding 20 votes [7]. Nobody's ballot changes. Across the three plans the larger party's delegation ranges from two seats to five out of five [7].
Every one of those plans can be drawn with straight lines. The proportional map is five vertical strips [5]. The sweep is five horizontal bands [6]. The packing map works as two full avenues that are wholly one party, plus the remaining three avenues cut into three bands holding roughly 3 voters of one party and 7 of the other apiece [6]. A compactness score rates all three about equally well. What shape measures is a specific crude technique, the one that got Elbridge Gerry's district compared to a salamander in the early 19th century [10], not whether lines were chosen for advantage.
The gap between votes and seats does leave a trace. Scientific American's New York case from 2012 gives Democrats 58 percent of the vote and 21 of 27 seats [8], which is 77.8 percent of the delegation, 19.8 points above vote share [1]. Multiply 27 by 0.58 and you get 15.7 seats, so 21 sits about five above proportional, the figure the article names [3]. Pennsylvania that year runs the other direction: 51 percent of the vote against 27.8 percent of the seats, a 23.2-point deficit [9][2].
Those gaps establish a mismatch against a chosen baseline. Intent is a separate question, and so is whether a fairer map was available. Proportionality is the yardstick in both comparisons, and it is a chosen standard rather than a measured one; the article's own first map shows a plan can land on it, but nothing here reports what a neutral map would have produced in either state [8][9]. That is roughly where the law sits too. The Supreme Court ruled intentional gerrymandering illegal in 1986 and, according to Scientific American, has barely touched a district since, which the article attributes to the difficulty of setting rules for fair districting [13].
Two bookkeeping notes on the example itself: they change nothing in the underlying geometry. Its party labels do not close: the setup lists 20 blue and 30 red voters [4], while the horizontal districts are described as four red and six blue each [6], which requires the reverse totals [4]. And five vertical districts of ten voters each need five avenues crossed by ten streets, not the ten avenues and five streets the text describes [3][5]. The geometry survives both slips, which is why the parties are identified here by their vote counts.
So the arithmetic earns less than the phrase "the math holds up" implies. It can show that a delegation is skewed against a stated baseline, and it can rule out the idea that ugly boundaries are the only signature worth looking for. On the record in this account, courts have relied on other grounds since 1986.
