The largest shape that turns a corner in a unit-wide hallway: Gerver guessed it in 1992, and Jineon Baek proved him right in 2024
What is the largest shape you can carry round a right-angled corner in a hallway one unit wide? Leo Moser asked in 1966, Joseph Gerver proposed an answer in 1992, and Jineon Baek proved him right in 2024.
Watch a sofa make the turn, then choose your own bite radius and compare the measured area with Hammersley’s formula.
The bite is what makes room for the inner corner: as the sofa turns, the corner traces exactly that half-circle. Wider blocks gain area linearly but lose it quadratically to the bite, which is why the best r is 2/π.
The obvious candidate is a half-disc of radius 1. It pivots on the inner corner and turns with no trouble, for an area of π/2 ≈ 1.571. A unit square does worse, with area 1.
In 1968 John Hammersley did much better by cutting the half-disc in two and pushing the halves apart. Fill the gap with a block of width 2r and height 1, then bite out a half-disc of radius r from the bottom of the block so the inner corner has room. The area is 2r + π/2 − πr²/2. The block adds area in proportion to r and the bite removes it in proportion to r², so there is a best radius, r = 2/π, giving 2/π + π/2 ≈ 2.2074.
Hammersley also proved that no sofa can have area more than 2√2 ≈ 2.83. For decades the truth was known only to lie somewhere between those two numbers.
Hold the sofa still and turn the hallway around it. Drag the angle and watch which part of the plane each hallway position allows.
Hammersley’s corner runs along a half-circle, and by Thales’ theorem its two inner walls then always pass through the ends of the diameter. That is why the bite is a half-disc.
Change your point of view and the problem becomes much cleaner. In the sofa’s own frame the hallway moves: it rotates through 90° while its inner corner travels along some path. A sofa survives the move exactly when it lies inside every position of the hallway. So for any path, the best sofa is the intersection of all those hallway positions, and the problem becomes choosing the path that leaves the most behind.
For Hammersley’s sofa the inner corner runs along a semicircle of radius 2/π. By Thales’ theorem, a right angle with its corner on a semicircle has its arms through the ends of the diameter, so the two inner walls always pass through the same two points. That is why the bite is exactly a half-disc.
This formulation also turns the problem into something a computer can attack. In 2018 Yoav Kallus and Dan Romik checked the hallway at finitely many angles, which gives a region that contains every possible sofa, and used it to prove that no sofa has area more than 2.37. They also showed that any sofa beating Gerver’s would have to rotate through at least 81.2°.
Start from Hammersley’s sofa and let a blind search nudge the corner’s path. It climbs toward Gerver’s area and stops below it.
Areas are measured by slicing the sofa into thin strips and checking the hallway at 90 angles, with a correction for the slivers between them; they are good to about three decimal places. Gerver’s sofa is bounded by eighteen curve pieces, and a blind search will not reproduce all of them.
In 1992 Joseph Gerver found a better sofa by writing down conditions that a best sofa has to satisfy, roughly that it cannot be improved by any small change to the path. His sofa has area 2.21953…, only about half a percent more than Hammersley’s. Its boundary is made of eighteen separate curve pieces, and the motion that carries it is more elaborate than a simple pivot.
Proving that nothing beats it took another thirty-two years. In November 2024 Jineon Baek posted a long proof (arXiv:2411.19826). Most people had expected that any proof would need a large computer search like Kallus and Romik’s. Baek’s does not. He first narrows the possible maximal sofas to those satisfying a few structural conditions, including one he calls injectivity, and then bounds the area of every such sofa by a quantity that Gerver’s sofa maximises.
The proof has been submitted for publication and is still under review, but experts who have studied it broadly accept it. The demo’s search is far cruder. It cannot find Gerver’s eighteen pieces exactly, but it should never beat Gerver’s area, and it does not.
Spot a mistake or have an idea?