The Lines, Not the Votes

In a country that elects one representative per district, an election has two moving parts: how people vote, and where the boundaries fall. We obsess over the first. The second is quieter, drawn years in advance in rooms most people never see — and it can overrule the first completely.

Below is the whole argument in one object. The electorate is fixed: ten red voters, fifteen blue, locked in place. Your job is to carve them into five districts of five, each one connected (no flying islands). Whoever has the most voters inside a district wins its single seat. Draw one map and the blue 60% majority takes three seats out of five — the proportional result. Draw another and the red 40% minority takes three. Nothing changed but the lines.

Drag across the grid to paint cells into the selected district. Each district must end up with exactly five cells, all connected. The seat tally updates as you go.

If you reached red 3 — blue 2, you just gerrymandered an election. The 40% party holds the majority of the legislature. No fraud, no rigged machines, no voter turned away — every vote was counted exactly once, honestly. The lever was geometry.

The votes are the same in every map. What moved was the boundary, and the boundary is what won.

Two techniques: packing and cracking

There are exactly two ways to waste your opponents' votes, and a real map uses both at once. Packing stuffs them into a handful of districts they win by enormous margins — every vote past the 50%-plus-one they needed is wasted. Cracking spreads the rest so thinly that they are the majority in none — every one of those votes is wasted too. In the grid above, the "five rows" map cracks the blue edge: each of the top three rows is 3-red, 2-blue, so blue's voters there are scattered into losing positions. The "pack the red" map does the opposite — it seals all ten red voters into two districts, so red wins those two overwhelmingly and loses the other three, landing exactly on the proportional 2-3.

Same voters. One map cracks, the other packs, and the seat count swings by a full seat in a five-seat house — a 20-point swing — without a single person changing their vote.

How rigged is a map? The efficiency gap

"It looks unfair" is not a standard a court can use. In 2015 two scholars, Nicholas Stephanopoulos and Eric McGhee, proposed a number that measures the asymmetry directly: the efficiency gap. It counts every party's wasted votes — votes for a loser, plus a winner's surplus past the bare majority — and takes the difference as a share of all votes cast. A map that wastes both parties' votes equally has a gap of zero. The more one-sided the wasting, the larger the gap.

The instrument below builds a five-district state vote by vote and computes the gap live, so you can watch a perfectly efficient gerrymander against a balanced map. The arithmetic is exactly the published definition; the worked numbers are re-derived in the offline verifier.

This is not a thought experiment

The grid above is a toy, but the mechanism is exactly the one that draws real legislatures. In 2012, Republicans won about 49% of the two-party vote for the U.S. House in Pennsylvania and roughly 72% of its seats (13 of 18). In 2018, Wisconsin Democrats won a majority of the statewide Assembly vote and took 36 of 99 seats. The votes were not in question. The maps were.

Wisconsin's map was the one challenged with the efficiency gap as evidence, in Gill v. Whitford (2018). The Supreme Court did not rule on whether the gap is a good test — it sent the case back on standing grounds (the plaintiffs, the Court held unanimously, had not yet shown a concrete district-by-district injury). A year later, in Rucho v. Common Cause (2019), the Court closed the federal door entirely: by 5 to 4 it held that partisan-gerrymandering claims are a non-justiciable political question — beyond the power of federal courts to decide, however unfair the map. The arithmetic is settled; the remedy is not.

The honest limit: there is no perfectly fair line

It is tempting to answer gerrymandering with "just make the seats proportional to the votes." But proportionality is not a property a district map can always have — sometimes it is arithmetically impossible, before any politics enters. Take four voters, three for one party and one for the other, to be split into two equal districts. The only seat outcomes available are 2-0 or 1-1 — never the 1.5-0.5 that 75% would require. With single-member districts and equal population, the seat share is forced onto a coarse grid the vote share rarely lands on. The verifier checks this as a negative result: it searches every split and confirms proportionality cannot be reached.

So "fair" has to mean something subtler than "proportional" — symmetry between the parties, compactness, respect for communities, a small efficiency gap — and mathematicians and courts genuinely disagree about which. That disagreement is real and unsettled, and this page does not pretend to resolve it. What it does settle, and shows you with your own hands, is the thing underneath all of it: that the line, not the vote, can decide the seat.

One last fact, because the name is older than you'd think

The word is from 1812. Elbridge Gerry, then governor of Massachusetts, signed a redrawing of the state's senate districts so contorted that one district was said to resemble a salamander. The Boston Gazette ran a cartoon of it in March 1812 and welded the governor's name to the beast: Gerry-mander. (Gerry pronounced his name with a hard G, like "Gary"; the word that outlived him drifted to a soft one.) The technique is older than the United States is old enough to have forgotten — and it has never needed a single fraudulent vote.