Inscribed Rectangles, Möbius Strips and the Square-Peg Problem

Grant Sanderson introduces the inscribed-square problem: must every simple closed continuous planar curve contain four distinct points forming a square? Otto Toeplitz posed the question in 1911; the general case remains open.

The lesson instead develops Herbert Vaughan’s proof for the easier rectangle question. It uses topology as a problem-solving tool, leading naturally to a Möbius strip and a Klein bottle.

Sanderson contrasts this with popular descriptions of topology as merely strange surfaces or rubber-sheet geometry. His aim is to show how continuous mappings and impossibility results can support a logical deduction.

A rectangle has diagonals with the same midpoint and length. Conversely, two distinct segments with a common midpoint and equal nonzero length have endpoints forming a rectangle. This reframes the search for four vertices as a search for two pairs.

For each pair on the loop, record the midpoint coordinates (x, y) and the distance d between the points. These three numbers define a point (x, y, d) in three-dimensional space.

This map is continuous: moving the two input points slightly changes their midpoint and distance slightly. If two distinct unordered pairs map to the same output, they have the same midpoint and distance and yield the desired rectangle.

All outputs together form a surface above the original plane. For a typical illustrated loop, a self-intersection in that surface represents different point pairs sharing an output.

For a circle, the image looks like a dome. Many different diameter pairs map to its top: every such pair shares the center and the same length. A collision need not visually resemble two sheets crossing.

Pairs whose two points approach one another map close to the original loop: their distance approaches zero and their midpoint approaches the curve. Identical pairs (x, x) map directly onto the loop in the plane.

To understand the domain of the map, parameterize the simple loop by a coordinate from 0 to 1, with the endpoints identified. An ordered pair of loop points is then represented by a point in a square.

Opposite edges of that square represent the same ordered pairs because 0 and 1 are the same loop point. Gluing both pairs of opposite edges gives a torus, a continuous representation of all ordered pairs.

Order does not matter for a diagonal: (a, b) and (b, a) describe the same segment. Treating them as distinct would give a trivial collision rather than a nondegenerate rectangle.

Identify square coordinates (x, y) with (y, x), equivalent to folding along the diagonal. After tracking the remaining edge identifications and their orientation, the resulting space of unordered pairs is a Möbius strip.

The boundary of this strip consists of repeated pairs (x, x). Under the midpoint-distance map, that boundary lies on the original curve in the plane, while all pairs of distinct points map strictly above the plane.

A tempting claim is that a Möbius strip cannot be embedded in three dimensions with its edge in a plane. Sanderson warns that this claim alone is false: Dan Asimov constructed an embedding whose boundary is a circle but whose interior goes both above and below the plane.

The missing condition is essential. In the rectangle construction, the interior must remain strictly on one side of the plane; with that added restriction, an embedding without a collision is impossible.

Reflect the hypothetical embedded strip across the plane and glue the two strips along their common boundary. Gluing two Möbius strips in this way gives a Klein bottle.

A Klein bottle is a closed non-orientable surface and cannot be embedded in ordinary three-dimensional space without self-intersection. Thus the hypothesized one-sided embedding cannot exist.

The contradiction forces distinct unordered point pairs to share the same midpoint and distance, yielding an inscribed rectangle. The argument illustrates how the topology of a configuration space constrains a geometric problem.

For a square, equal diagonals and a shared midpoint are not enough: the diagonals must also meet at right angles. Tracking the direction adds information and leads toward a higher-dimensional construction.

Sanderson cites Joshua Green and Andrew Lobb’s work on smooth curves: they proved the existence of inscribed rectangles of every aspect ratio in that setting. This does not resolve the general square problem for all simple continuous curves.

Smoothness provides controlled tangent directions as two points merge. For rough curves, such as some fractal curves, that directional limit may fail, making the general problem harder.

The broader lesson is that a Möbius strip is not one particular paper model: different surfaces, and the abstract space of unordered point pairs on a loop, can share the same topology. Continuous relationships and their constraints make these objects useful in proofs.