The largest sofa round a corner

Anyone who has carried furniture up a stairwell knows the question. A hallway of width 1 turns a right angle. A rigid flat shape has to go round the corner, without lifting it. What is the largest area it can have? Leo Moser put the question in print in 1966, in a collection of problems, and it is usually known as the moving sofa problem.

It belongs with the other great extremal problems about area, and there is a sharp contrast with them. The fence that encloses the most area has a proof that a circle wins, and has done since antiquity. For a sofa, the winning shape has no such simplicity. The constraint is not a single perimeter but a motion, and every point of the shape must obey every wall at every moment of the motion. It took about sixty years to go from the question to a proof of the answer, and the answer is not a round number. It is a constant, 2.2195…, defined by a system of equations and with a boundary in eighteen pieces.

The lesson of how it was found is one that recurs in this kind of mathematics. For a long time the only tools were hand constructions that kept getting better and were not known to be right, and each small improvement required much more elaborate geometry than the previous one.

Three shapes in order #

The first shape anyone tries is a unit square. It goes along the first arm, and it does not rotate: it simply slides sideways down the second arm. Its area is 1. Any shape needs the width to pass through the first arm, so its height in that direction can be at most 1. It is tempting to think the answer is little more than this. But a shape can also rotate, and that gains a lot.

Next, take a half-disc of radius 1. Push it along the first arm with its flat side against the outer wall until the centre of the flat side reaches the inner corner. Now it can rotate about that corner, since its curved edge just clears the walls, until its flat side lies along the other wall. It can then continue along the second arm. Its area is $\pi/2 = 1.5708$. That is already more than half as large again, and the shape is a lot larger than a square would suggest.

The third step belongs to John Hammersley, in 1968. Cut the half-disc along its diameter into two quarter-discs, and pull them apart by a distance $2r$, joining them by a rectangle of length $2r$ and height 1. A plain rectangle in between would hit the inner corner as it rotates. So remove a half-disc of radius $r$ from the bottom of the rectangle, and the inner corner can then sweep through that notch.

The area of the resulting shape is that of the two quarters, plus the rectangle, less the half-disc:

$$A(r) = \frac{\pi}{2} + 2r - \frac{\pi r^2}{2}.$$

This is a downward parabola in $r$, so its maximum takes one line: $2 - \pi r = 0$, giving $r = 2/\pi$ and area $\pi/2 + 2/\pi = 2.2074$. Hammersley actually conjectured that this might be optimal. It is a shape like a telephone handset, with an elegant value. The raster check drawn from this family agrees with the formula at three radii, $0.3$, $0.5$ and $0.85$. It is natural that everyone thought this might be the end of the matter, as the regular polygons seemed to be for the longest fence of diameter one, and it nearly is. It turns out to be only half a per cent short.

A shape is a motion #

The essential simplification is to look at the problem from the point of view of the sofa. The hallway is now the thing that moves. It rotates by 90° in the course of the journey and its inner corner traces some path $p(\theta)$ in the sofa’s frame. Given that path, any shape that makes the journey must be inside the hallway at every stage; so it is contained in the intersection of all those hallways. And the intersection itself makes the journey. So a sofa is determined by the path.

Precisely, a point $s$ in the sofa’s frame stays in the hallway at angle $\theta$ if, after moving its coordinates by $p(\theta)$ and rotating them through $\theta$, it satisfies $x' \le 1$, $y' \le 1$, and not both of $x' < 0$ and $y' < 0$. Those are the walls of an L-shaped region whose inner corner is at the origin. So the largest sofa problem becomes an optimisation over curves $p(\theta)$, with the objective the area of an intersection of infinitely many L-shapes.

Hammersley’s path is a semicircle of radius $2/\pi$, written as $p(\theta) = (2/\pi)(-\cos 2\theta, \sin 2\theta)$. A semicircle is the first guess because it has the right symmetry, and it gives the inner corner a constant speed. Once it is posed this way, a computer can evaluate the area for any path by testing a grid of points against a large number of L-shapes. So it can look for better paths directly. Doing that here is a small experiment that leaves out everything that made the real proof difficult, but it does reproduce the phenomenon.

A search over paths #

Write the path as a short Fourier series, with the symmetry that makes the turn look the same forwards and backwards, and with four terms in each coordinate, eight numbers in all. Start from the semicircle, change the numbers at random, and keep any change that increases the area.

The result is a path differing from the semicircle by amounts barely visible in the figure, with a mid-turn height of 0.642 against the semicircle’s 0.637. Raster measurements at 360 angles report 2.2095 for Hammersley’s shape and 2.2195 for the shape from the new path. The exact value of Hammersley’s is 2.2074, so the raster has an excess of 0.002, because it counts each grid cell whose centre survives at those 360 angles and no more. Allowing for that, the new shape has area about 2.217. That beats the semicircle by 0.010 and it is a few thousandths short of Gerver’s optimum, using only eight parameters.

The shape in the opening figure shows what changed. The notch is no longer a semicircle. The upper boundary is no longer a straight line in the middle, and the sides are slightly different from quarter-discs. Nothing obvious suggests how the true optimum would look. To find that needs an actual theory.

Gerver’s eighteen pieces #

Joseph Gerver found that theory in 1992. He assumed a sofa that is locally optimal and asked what conditions it must satisfy: at each stage of the motion, some parts of the shape’s boundary are pressed against walls. As the motion varies slightly, the area changes by a certain integral over those contact points, and this must vanish. That gives a condition of balance on the curve, in the form of equations connecting the contact points at each angle. He solved them. The solution is not a single formula. The boundary falls into eighteen analytic pieces, some arcs of circles and some involutes and other curves of a kind produced by a rotating contact, and the pieces are joined at points determined by four angle parameters that must solve a system of equations. The total comes to 2.2195.

There is no closed form for this number. Its value depends on four solutions of a transcendental system, and that makes it quite different from Hammersley’s $\pi/2 + 2/\pi$. The shape has the appearance of the one in the opening figure: nearly the same, only a little different. The extra 0.012 of area over Hammersley’s is less than a part in a hundred.

Dan Romik put Gerver’s construction on a clean footing in 2018, deriving the six ordinary differential equations that the shape must satisfy and solving them explicitly. Romik also studied the ambidextrous version, a shape that must be able to turn both left and right. Its answer is a beautiful shape with an area of 1.64495, which is close to, but not equal to, $\pi^2/6 = 1.64493$, a coincidence good to four digits. It was a warning, in a problem full of numerical temptations, that a value matching a famous constant to a few digits proves nothing.

The upper bound #

Gerver’s shape is a lower bound on the optimum and it was conjectured but not known to be best. Upper bounds are much harder. For a long time the best statement was $2\sqrt 2 = 2.8284$, from a simple argument with the two arms alone, an estimate that Hammersley gave and nobody improved for fifty years.

Yoav Kallus and Romik in 2018 had the idea that the figure illustrates. Checking only a finite list of angles gives a shape that may not really turn, and its area is an overestimate of whatever the true shape can do. The figure shows it for one motion: the area is 2.52 with two angles, 2.36 with four, 2.26 with twelve, and it comes down towards the exact value as more are added. They took the same idea and made it work for every motion at once. With a finite list of angles, the path enters only through finitely many positions, a finite-dimensional search. A computer can then explore that space by subdividing, with exact bounds, and prove that no choice of positions can exceed a given area. That gave $2.37$, the first substantial improvement since Hammersley, and it demonstrated that a computer proof could close the gap to some extent. But it is a difficult computation that gets far slower with more angles, and it cannot be pushed to 2.2195.

In late 2024 Jineon Baek posted a proof of more than a hundred pages that Gerver’s sofa is optimal. The key was to find a different quantity that bounds the area for any path, and whose maximum is easier to analyse. It is not the area itself: the area of a sofa as a function of its path is not a concave function in any convenient parameter, so a local maximum need not be a global one. Baek found a larger quantity, in terms of mixed areas of convex bodies, that is concave on the appropriate space and equals the area at Gerver’s shape. Since a concave function’s local optimum is global, the proof could then conclude. When this was written, it was circulating as a preprint and had been carefully read by specialists. If it holds up as expected, the sixty-year-old problem is closed.

Related questions of motion #

The sofa is one of a family of problems in which the constraint is a motion, and the extremal shapes they give are unusual. The Kakeya needle problem asks how little area is needed to turn a segment of length one around completely. It looks like the reverse question: one asks how small a room can be for a fixed object, and the other how large an object can be for a fixed room. The first answer is astonishing, because the area can be as small as desired, by back-and-forth moves. For the sofa no such trick helps: moving back and forth does nothing, since a sofa that turns needs the intersection over all its positions, and repeating a position adds nothing.

Half a circle against a wall asks for the greatest area a curve of fixed length can enclose against a straight wall, and it reflects the same half-disc in a static way. The half-disc is the starting shape here, and then it turns out not to be the answer, because the walls of a hallway give a new freedom. A similar thing happens in the largest polygon of diameter one, where the regular polygon is not the winner despite every expectation. That situation is quite general: symmetric shapes are often excellent, and it is rarely safe to assume they are optimal. Here the shape has the symmetry of a reflection in its centre line, one of the assumptions used by the optimisation here and proved legitimate in Baek’s argument. Beyond that symmetry, there is nothing round about it.

Steiner’s method of symmetrising a shape is powerful for the isoperimetric problem, because it preserves area and does not increase perimeter. For the sofa it fails completely. Sliding the chords of a sofa to the middle gives a shape that can have the same area, but will not, in general, make the same turn. That is the most direct way to see why a century of methods for optimal shape did not solve this problem: a constraint expressed through motion is not preserved by any of the usual moves.

The hardest part of the opaque wall for a square was similar in spirit, since there too a finite object has to deal with an infinite family of positions, and the gap between the easy bound and the best construction stayed open for far longer than anyone expected. Close to a known answer, a shape’s area can be controlled by how far it is from the winner, as nearly the most means nearly round shows for the fence; for the sofa, no such stability result is known, since the winner has only just been identified.

And a three-dimensional analogue is open: what is the largest solid that can be carried around a corner in a hallway with a square cross-section? Solids can rotate about several axes, and nobody has good bounds.

What the computation here does and does not establish #

The search here finds a path. It does not establish that its area is a true value for any shape, only that a raster with 360 angles and a grid of 126,000 cells measures it as such, and this is calibrated by the known exact value for Hammersley’s shape. The correction used for the second shape assumes the same excess, and it is only an estimate. It says nothing about the optimum, and with more Fourier terms and more angles the same search would reach closer to Gerver’s value without ever proving anything.

All statements about Gerver’s eighteen pieces, Kallus and Romik’s bound, Romik’s ambidextrous shape and Baek’s proof are taken from the published papers and were not reproduced here. The drawing of the upper bound as a finite list of angles is an illustration of their idea on one fixed motion, not of their proof, which searches over all motions. The drawing of the motion assumes a particular parameterisation, with one fixed outer wall. The value 2.2195 itself is quoted to the digits given in the literature, and the raster here has neither the resolution nor the accuracy to confirm its last digits.

Still open: other corners and other solids #

If Baek’s proof stands, the first question is closed and others remain. What is the largest shape for a corner of a different angle, or a hallway of varying width? Gerver’s analysis assumes a right angle, and for other angles the solution changes its structure. A hallway that is wider beyond the corner than before it changes the answer again. Some numerical work has been done, but the nature of the solution as a function of the angle is not settled. Then there is the three-dimensional problem: for a sofa moved through a rectangular corridor with a right-angled bend, only rough bounds are known and even the shape of the problem, which rotation axis to use, or whether more than one is needed, has not been worked out.

There is one further natural question, which has attracted little work. How much better can a sofa be if it may be deformed? A shape allowed to flex along hinges is the problem faced when moving a real sofa-bed, and it has a very different geometry. Nobody seems to have settled what such a shape should be for a single hinge.

Half a per cent and sixty years #

A shape that can turn the corner is an intersection of moving hallways, determined by the path the inner corner traces in the shape’s own frame. A semicircle for that path gives Hammersley’s $\pi/2 + 2/\pi = 2.2074$ by one line of calculus. A slight change in the path gives a shape near 2.217, and the true optimum, Gerver’s 2.2195, has a boundary in eighteen pieces and no formula. Upper bounds came from checking finitely many angles, which can only overstate the area. The last step was a quantity larger than the area and easier to maximise, a proof of the kind that isoperimetric problems usually do not need.

The contrast with a fence is what makes the problem interesting. A fence constrains a shape once. A hallway constrains it at every angle, and the sum of all those constraints is a shape with no symmetry beyond a reflection, a number with no formula, and half a per cent that took thirty years to justify.