This open problem taught me what topology is

The core breakthrough is a topology-driven route to a classic geometric claim: every closed continuous loop in the plane contains a non-degenerate inscribed rectangle. The proof turns the rectangle-finding problem into a collision problem on a carefully constructed surface, then uses a key impossibility about Möbius strips and Klein bottles in three dimensions to force that collision.

The argument starts by reframing what it means to have a rectangle on the loop. Instead of hunting for four points that form a rectangle directly, it looks for two distinct pairs of points whose connecting segments share the same midpoint and the same length. Those conditions force the four endpoints to be the vertices of a rectangle. To organize all possibilities, the proof packages each ordered pair of loop points into a single point in three-dimensional space: the x–y coordinates come from the midpoint of the pair, and the z-coordinate encodes the distance between the points. Because the midpoint and distance change continuously as the chosen pair moves along the loop, this “pair-to-3D” assignment is continuous.

Now comes the geometric heart of the method. The set of all outputs of this continuous assignment forms a complicated surface in 3D—described as something like a Frank Gehry design—whose self-intersections correspond exactly to the desired rectangle condition: two different pairs of points mapping to the same 3D point means they share midpoint and distance, hence form a rectangle. For a circle, the surface’s many-to-one behavior concentrates at the top of a dome; for an ellipse, that collision spreads into a line. Either way, the proof needs a general reason that self-intersection cannot be avoided for an arbitrary loop.

To force that reason, the proof builds a second surface that represents unordered pairs of points. Ordered pairs (a,b) and (b,a) are treated as distinct, but rectangles require the unordered version. Folding the unit square along its diagonal identifies (x,y) with (y,x), and the resulting quotient space becomes a Möbius strip. Crucially, the “edge” of this Möbius strip corresponds to degenerate pairs where both points coincide (x,x), and those must land back on the original loop in the plane.

A continuous mapping from the Möbius strip onto the earlier 3D surface is then unavoidable because each unordered pair determines a unique point on the rectangle-encoding surface. If the Möbius strip could be embedded in 3D without self-intersection while keeping its boundary confined to the plane, the rectangle collision might be avoided. But Möbius strips cannot be realized in 3D with their boundary staying in a plane without forcing self-intersection—an obstruction tied to the broader fact that non-orientable closed surfaces (like the Klein bottle that emerges from gluing two Möbius strips along their boundaries) cannot be represented in 3D without intersecting themselves.

That impossibility forces the rectangle-encoding surface to self-intersect. The self-intersection means two distinct unordered pairs share the same midpoint and distance, so the loop must contain a non-degenerate inscribed rectangle.

Finally, the discussion explains why the original unsolved problem—inscribed squares—remains open. Squares require not just equal midpoint and length, but also a 90-degree angle between the corresponding segments. For smooth curves, angle behavior is well-controlled, and mathematicians Joshua Green and Andrew Lobb proved in 2020 that smooth curves admit inscribed squares (and even rectangles of every aspect ratio). The difficulty persists for rough curves, where tangent directions can fail to behave continuously, leaving the square problem unresolved in full generality.