Every Loop Holds a Rectangle

A Square on a Loop

Draw a closed line in the plane that does not cross itself. Are there always four points on it that are the corners of a square?

Otto Toeplitz raised the question in 1911 at the annual meeting of the Swiss Society of Natural Sciences, and said he had a proof only for convex curves. That proof was never published.

In 1913 Emch proved it for smooth convex curves; in 1929 Schnirelman proved it for curves smooth enough. For every continuous closed curve the question is still open.

The brass loop on the slate has three inscribed squares. A square need not lie inside the loop; its four corners on the line are enough.

One Step Back: Rectangles

Ask for a rectangle instead, of any shape, and the answer is known: every closed curve that does not cross itself has four points forming a rectangle.

The proof is H. Vaughan’s. According to Meyerson he gave it in a lecture around 1977, and Meyerson published it in a paper in 1981. It needs only that the curve is continuous, not that it is smooth.

A curve holds more than one rectangle. The rectangle in red thread slides along the wire, changing its shape as it goes, with its four corners always on the line.

Two Diagonals

The two diagonals of a rectangle have the same length and bisect each other. Conversely, two segments with the same midpoint and the same length have four ends that are the corners of a rectangle.

So what we are looking for is two pairs of points on the line. Each pair is joined by a chord; the two chords share a midpoint and have the same length.

Every pair of points gives two things: where its chord’s midpoint lies in the plane, and how long the chord is. The question becomes whether two different pairs agree in both.

Lifting Every Pair

For every pair of points on the line, stand a rod on the midpoint of their chord, half as tall as the chord is long, with a bead on top.

The beads of all the pairs make a surface. When the two points meet the chord has length 0, so the surface’s edge lies exactly on the brass loop.

Hold one point and send the other once round the loop, and the bead traces an arch: seen from above, it is the whole loop shrunk by half towards the held point. The 40 fine wires on the model are the arches of 40 points on the line.

Two different pairs on the same bead make a rectangle.

The Surface Passes Through Itself

Put another way, rectangles are where this surface passes through itself. The red line marks where it does; every point on it belongs to two pairs at once.

As the bead runs along the red line, the four threads reach down to four points of the wire that are always a rectangle. The angle between the diagonals keeps changing.

What is left to see is why the surface must pass through itself.

The Pairs Are a Möbius Strip

Ordered pairs (s, t) fill a square, with s and t each going once round the curve. The curve closes up, so 0 and 1 are the same point.

But (s, t) and (t, s) are the same pair. Fold along the diagonal and keep one half, a triangle. On the diagonal s = t the two points coincide; that edge is the surface’s edge.

The triangle’s two short sides are the same pairs too: {0, e} and {e, 1}. Cut along the other diagonal, turn one half over and sew those two sides together, and you have a square of cloth. The two sides still open point in opposite directions.

Give it a half twist and join them: a Möbius strip. So all the unordered pairs make a Möbius strip whose edge is the whole curve. Sew that edge to the wire, and the cloth covers the surface exactly.

Why It Must Pass Through

If the surface never passed through itself, the Möbius strip would sit cleanly in the half of space above the slate, with its edge on the brass loop.

Now add the piece of plane the loop encloses. A Möbius strip and a disc make a closed surface with no edge, the projective plane.

The projective plane cannot be put into three-dimensional space without passing through itself. So the strip must pass through itself: two different pairs whose chords share a midpoint and a length, the two diagonals of a rectangle.

That is Vaughan’s proof. Another version joins the strip to its reflection in the slate, which makes a Klein bottle; that cannot sit in space without crossing itself either.

Every Shape of Rectangle

Vaughan’s proof promises a rectangle but says nothing of its shape. Can the shape be chosen, say 2 by 1?

In 2020 Greene and Lobb proved that for every smooth closed curve and every rectangle, the curve has four points forming a rectangle of that shape. The proof rests on a theorem of Shevchishin: the Klein bottle has no smooth Lagrangian embedding in C².

Before that, Hugelmeyer had proved the case of aspect ratio √3 (2018), and then at least a third of all aspect ratios (2019).

On the model the rectangle stops at diagonal angles of 15°, 30°, 45°, 60°, 75° and 90°; its aspect ratio is the tangent of half the angle. For continuous curves that are not smooth, the question of a chosen shape is still open.

The Square Is Still Open

A square is a rectangle whose diagonals are at right angles. Where the angle on the red line reaches 90° there is an inscribed square. This loop has three.

A typical smooth curve has an odd number of inscribed squares. Deform an ellipse into the curve and squares only appear or vanish in pairs; an ellipse has exactly one. An odd number is not 0, so every smooth curve has an inscribed square.

The trouble is curves that bend everywhere. Approximate one by smooth curves, and the squares found on them may shrink, until in the limit they are a single point and no longer a square.

Other cases settled include piecewise analytic curves (Emch, 1916) and locally monotone curves (Stromquist, 1989). Toeplitz’s question itself is still open.