Every chord slid to the middle

The most area a fence can hold tells the story of Jakob Steiner’s proofs that the circle encloses the most area for its perimeter, and of the gap Karl Weierstrass found in them: each of Steiner’s arguments shows that a shape that is not a circle can be improved, and none shows that a best shape exists. Of Steiner’s improvements the most powerful is the one that essay mentions only in passing — an operation that takes any shape and returns a more symmetric one with the same area and no more boundary. It is called Steiner symmetrisation, and it is worth drawing as an operation in its own right, because repeated it becomes a machine for making circles, and combined with the right existence argument it becomes a proof.

One symmetrisation, chord by chord #

Choose a line. Every line perpendicular to it crosses the shape in a chord — or, for a shape with dents, in several pieces whose lengths can be added. Slide each chord along its own perpendicular, keeping its length, until the chosen line cuts it exactly in half. The chords, restacked, form a new shape: the Steiner symmetrisation of the old one about that line.

The figure takes a lopsided blob with a dent and symmetrises it about the horizontal: four chords are marked in both shapes, the same lengths, re-centred. The new shape is symmetric about the line by construction. Its area is the same, because area is the sum of the chords’ lengths — Cavalieri’s principle, the fact that slices of equal length make regions of equal area however they are slid — and no chord’s length changed. Its boundary is shorter, $7.729$ against $7.908$. That last fact is the only one that needs an argument.

Two more things happen that the figure shows without announcing. The symmetrised shape is vertically convex — every vertical chord is a single piece, even where the original crossed the dent in two — because re-centring stacks the pieces into one. And its outline is smoother where the original was lopsided: a bulge on one side and a hollow on the other, at the same position along the line, average out into two moderate bulges. Both are ways of saying that the operation throws away information about where the chords were and keeps only how long they were, and everything it throws away was only making the boundary longer.

Why the boundary cannot grow #

Look at a thin slice of the shape between two nearby chords, and suppose for the moment that each chord is a single piece.

Over a short run of width $\delta$ the slice has a top edge rising at some slope $a$ and a bottom edge at slope $b$, so the chord’s length changes at rate $a - b$. After symmetrisation the chord is centred, and its top and bottom edges move symmetrically: the top rises at $(a - b)/2$ and the bottom falls at the same rate. The length of a short edge of width $\delta$ and slope $s$ is $\delta\sqrt{1 + s^2}$, and $\sqrt{1 + s^2}$ is a convex function of $s$ — its graph bends upward, like the curve of wall against sides in the essay on equal rooms. The average of its values at $a$ and at $-b$ is at least its value at their average, $(a - b)/2$. So the two new edges are together no longer than the two old ones, slice by slice, and adding over all the slices, the whole boundary is no longer.

Equality holds only when $a = -b$ in every slice — when the shape was already symmetric about a line parallel to the chosen one. So symmetrisation strictly shortens the boundary of any shape that is not already symmetric in that direction. For shapes whose chords break into several pieces the argument needs one more observation: stacking the pieces into a single centred chord removes the boundary between the pieces, which only helps.

Doing it again, and again #

One symmetrisation makes a shape symmetric about one line. The next, about a different line, makes it symmetric about the new line and generally destroys the old symmetry — but it keeps the area and shortens the boundary again. The question is where a sequence of them leads.

Turning the line by one radian each time — an angle whose multiples never repeat, so the lines never cycle — the blob is still dented after one step, becomes rounded and roughly triangular by the tenth, and by the sixtieth is visually a disc. The isoperimetric ratio $4\pi A/L^2$, which is one exactly for a circle and less for everything else, climbs from $0.712$ to $0.991$. The area stays fixed throughout, to within the three parts in a thousand that sampling the chords at 361 places loses over sixty rounds, so all of the change is the boundary shortening towards the circle’s $2\sqrt{\pi A}$.

Constantin Carathéodory and Eduard Study proved in 1909 that a suitable sequence of symmetrisations carries any convex shape to a disc, and later work widened the class of sequences that do it. The convergence is not fast. Most of the ratio’s gain happens in the first ten steps — from $0.712$ to $0.946$ — and the last half-per-cent takes fifty more, because once a shape is nearly round a symmetrisation about a random line can only correct its small asymmetries a little at a time.

Two lines are not enough #

The choice of lines matters, and the next figure shows how badly it can go.

Symmetrise alternately about the horizontal and the vertical, and after two steps the shape is symmetric about both — a four-lobed cushion, drawn beside the plot. Symmetrising it again about either line changes nothing, since each chord is already centred. The sequence has reached a fixed point that is not a disc, and the ratio sits at $0.787$ for ever, the small drift being the sampling’s rounding.

Every finite set of directions has the same problem: a shape symmetric about all of them is left unchanged by all of them, and plenty of such shapes are not discs — a regular polygon with those symmetries is one. A sequence of symmetrisations reaches the disc only if its directions keep visiting new angles, densely enough that no non-circular shape can be symmetric about all of them at once. Turning by a radian does that; alternating two directions does not.

The stalled cushion is a useful warning about what monotone processes prove. Its boundary never grows, its area never changes, and it has stopped — and it is not the answer. A monotone quantity that has stopped improving has reached a point the operation cannot improve, which is a statement about the operation as much as about the shape. Only an operation rich enough that the disc is its only fixed point turns “cannot be improved” into “is the circle”.

A triangle made equilateral #

Symmetrisation also proves smaller theorems, and one of them is the $n$-sided isoperimetric inequality used in the essay on equal rooms — for three and four sides.

Symmetrise a triangle about the perpendicular to one of its sides: the chords are parallel to that side, their lengths fall off linearly from the base to the apex, and re-centring them gives an isosceles triangle with the same base and height. The area is kept, the perimeter shrinks, and the result is still a triangle. Doing it about each side in turn, the triangle is pushed towards equilateral — the figure’s perimeters fall from $7.397$ to $7.166$ in five steps, against the equilateral triangle’s $7.165$. So among triangles of a given area the equilateral has the least perimeter, and the proof never compares it with anything: it improves every other triangle into it.

Quadrilaterals can be handled the same way with a good choice of lines, pushing them to squares. From five sides on, the method breaks. A pentagon symmetrised about almost any line comes out with eight corners, because every vertex at a different height along the chords makes its own corner on both sides of the line. The operation preserves area and shortens boundary for every shape, but it only preserves the number of sides for triangles and quadrilaterals, and a proof about $n$-gons that works by symmetrising can only work where symmetrisation keeps $n$.

How symmetrisation closes Steiner’s gap #

Steiner used symmetrisation as one of his improvement arguments: if a shape is not symmetric about some line, symmetrising it about that line gives a better shape, so the best shape, if there is one, is symmetric about every line — and only the disc is. Weierstrass’s objection applies in full: the argument proves the disc is the only possible winner, not that there is a winner.

The repair adds an existence argument that Steiner did not have. Among convex shapes of a given area inside some large bounded region, there is one of least perimeter. That is a compactness statement — Wilhelm Blaschke proved in 1916 that any sequence of convex shapes in a bounded region has a subsequence converging to a convex shape, and perimeter and area behave continuously along it — and it supplies what the improvement arguments lacked: a minimiser, known to exist before anything is said about its shape. Now Steiner’s argument applies to it. The minimiser cannot be improved by symmetrisation, so it must already be symmetric about every line, so it is a disc. The same proof works in any dimension, with the ball in place of the disc.

The division of labour is typical of how such questions are settled. The improvement — here, symmetrisation — identifies the answer. A separate, non-constructive argument — here, compactness — guarantees there is an answer to identify. Neither alone proves anything. The Fourier proof of Hurwitz avoids the division by computing directly, which is why it works so neatly in the plane and does not generalise as readily, while symmetrisation with compactness generalises to every dimension.

Symmetrisation in physics #

George Pólya and Gábor Szegő made symmetrisation into a general tool in their 1951 book on isoperimetric inequalities in mathematical physics. The idea is that many physical quantities attached to a shape behave like perimeter under symmetrisation — they can only move one way. The lowest frequency of a drum of given area is one: symmetrising the drum lowers its lowest note, so the circular drum has the lowest note of all, which is the theorem of Georg Faber and Edgar Krahn. The electrical capacity of a conductor, the torsional stiffness of a beam of given cross-section, the rate at which heat escapes from a region — each is monotone under symmetrisation, and each is therefore extremal for the round shape. Pólya and Szegő’s book is largely a catalogue of such quantities, with symmetrisation as the single method behind most of its theorems.

The same limitation appears there as for polygons. Pólya and Szegő proved that among triangles of given area the equilateral drum has the lowest note, and among quadrilaterals the square, precisely because symmetrisation keeps triangles and quadrilaterals. For pentagons it does not, and the polygonal drum problem is open from five sides up.

The same move with no shapes in it #

Symmetrisation is a geometric operation, and its logic — an operation that keeps one quantity fixed, never makes another worse, and ends at the symmetric object — has been carried into parts of mathematics with no geometry at all.

In a sphere of many dimensions the operation pushes mass towards a cap, and Lévy’s isoperimetric inequality on the sphere, which says the cap is the set of given size whose neighbourhood grows least, is proved by symmetrising towards it. In a graph, Alexander Zykov’s proof of Turán’s theorem takes any graph with no complete subgraph of a given size and repeatedly replaces one vertex by a copy of another, an operation that only ever adds edges and never creates the forbidden subgraph; the process ends at the balanced complete multipartite graph, the symmetric extremal answer, exactly as the chords end at a disc. In families of sets, shifting moves elements towards the smallest labels without destroying the property being studied, and it proves results about the largest family whose members all meet — except where the extremal answer is itself lopsided, and then shifting destroys it, just as symmetrising destroys a pentagon.

The pattern is the same in every setting. The operation earns its use by being monotone: it never makes the thing being optimised worse. It identifies the answer by having a fixed point that is symmetric. And it proves nothing about existence, which has to come from somewhere else — compactness for shapes, finiteness for graphs and families.

Higher dimensions, and why chords are the right thing to slide #

In three dimensions the chords perpendicular to a plane are slid until the plane bisects them, and every step of the argument survives: volume is kept by Cavalieri’s principle, surface area never grows by the same convexity of the length of a sloped piece, now a sloped patch, and repeated symmetrisation in well-spread directions carries any convex body towards the ball. That is how the isoperimetric inequality in three dimensions — among all solids of given volume the ball has least surface — is most often proved, and in $n$ dimensions the same operation works unchanged, which is the main reason it is preferred over proofs that compute.

The choice to slide chords, rather than reflect or rotate pieces of the shape, is what makes the operation keep the area exactly. Reflection, the device of the half-circle against a wall, also keeps area and length, but it only helps when there is a mirror line with a dent on one side. Sliding chords always helps, because it is the unique operation that makes the shape symmetric while leaving every one of its cross-sections alone, and so it can be applied about any line whatever the shape looks like.

What the pictures cannot show #

Exactness. The symmetrised shapes are drawn from 361 sampled chords, so every area and perimeter reported is an approximation; the exact statements — area equal, perimeter no larger — are the convexity argument’s. The triangle figure is the exception: there the symmetrisation is computed exactly at the vertices’ levels, and the areas agree to the last digit.

The limit. The turning sequence reaches a ratio of $0.991$ at sixty steps; that it tends to one, and not to some slightly smaller number, is Carathéodory and Study’s theorem, not something sixty steps can show.

Shapes with several pieces per chord. The blob’s chords near the dent cross the shape in two pieces, and the symmetrisation adds their lengths into one centred chord. The figure shows that this works; the argument that it can only shorten the boundary is stated in words, since the boundary between the pieces simply disappears.

The existence argument. Compactness appears nowhere in the figures. It is a statement about all convex shapes at once, and no drawing of any particular sequence of shapes shows that a limit exists.

Still open: a tool that keeps five sides #

The polygonal drum question is where symmetrisation’s one weakness has been most expensive. The conjecture that the regular $n$-gon has the lowest note among $n$-gons of given area is supported by every computation and, as a general statement for five or more sides, unproved, and the obstruction is exactly the eight-cornered pentagon in the figure: the only general tool that decreases the lowest note does not keep the number of sides, and nobody has found a replacement that does.

Several substitutes have been tried — symmetrisations that move vertices rather than chords, continuous rearrangements, polarisation about a moving line — and each either fails to preserve the polygon or fails to decrease the quantity. Computer-assisted work on pentagons and hexagons in the 2020s — checking that the regular polygon is a local minimum and bounding every other shape numerically — has come close for those two cases, and a general argument for every $n$ is missing. The question is, in effect, whether there is a version of the operation drawn on this page that respects corners.