This open problem taught me what topology is

Source: This open problem taught me what topology is, 3Blue1Brown, 27:25, uploaded 2024-12-24, category Education, Watch Later position 642.

3Blue1Brown begins with a question that nobody knows how to answer in full. Take any closed continuous curve, which is a squiggle that returns to its starting point without lifting the pen. Must four points on that loop always form the vertices of a square? Otto Toeplitz posed the question in 1911. It is now known as the inscribed square problem, or the square peg problem.

The video takes a nearby problem whose answer is known. Every such loop contains the vertices of a rectangle. Herbert Vaughan’s proof reaches that result through a Möbius strip and a Klein bottle. The shapes appear because they organise the possible pairs of points on the loop, so topology becomes a method for deduction rather than a catalogue of strange objects.

The rectangle problem and its first reformulation

The rectangle question has a small but useful change of language. Instead of looking for four points that make a rectangle, look for two distinct pairs of points whose connecting segments have the same midpoint and the same length. Two segments with a shared centre and equal lengths have endpoints that form a rectangle. The diagonals of a rectangle have exactly these properties.

This is a search through every pair of points on the loop. Each pair supplies the two coordinates of its midpoint and the distance between its endpoints. Those three numbers can be treated as the coordinates of a point in three-dimensional space. A pair whose midpoint is $(x,y)$ and whose length is $d$ becomes the point $(x,y,d)$, placed above the midpoint at height $d$.

The resulting map is continuous. A small movement of the two input points produces a small movement of the midpoint and the distance, so the output does not jump. A rectangle appears when two distinct pairs map to the same point. Their equal outputs say that the midpoints and lengths agree.

The set of all outputs forms a surface in three-dimensional space. It can look like a complicated folded sheet, so cross-sections make it easier to read. Near the base plane, pairs of points that sit close together produce cross-sections that resemble the original loop. At a higher level, two different pairs with the same midpoint and length show up as a self-intersection of the surface. In a particular example, a whole curve of such intersections represents a continuous family of inscribed rectangles.

The circle gives a useful warning about the word “self-intersection”. Every rectangle inscribed in a circle has the same midpoint, the centre of the circle, and the same diagonal length, the diameter. Infinitely many pairs therefore map to one point at the top of the dome that represents the circle. An ellipse turns that single concentration into a vertical line of intersections. What matters is the collision of two different pairs, even when the picture does not look like one sheet passing through another.

The surface is also not a function graph in the ordinary sense. A graph in three dimensions takes two input numbers and returns one output number. Here the input is a pair of points on the loop, while the output is a full point in three-dimensional space. The surface collects every output of that map.

The boundary that has to stay on the loop

The argument plants one constraint early. A pair consisting of a point with itself, written $(x,x)$, has zero length and a midpoint at $x$. Its image therefore lies on the original loop in the base $xy$-plane. The boundary of the output surface is the loop itself.

3Blue1Brown calls this a “gun hanging on the wall”, borrowing Anton Chekhov’s advice that an object introduced early in a story should become useful later. The mathematical role is precise. Any topological surface that represents all pairs must send its boundary to the loop in the base plane. The eventual obstruction depends on that boundary condition.

To see what sort of surface represents the pairs, give the loop an internal coordinate from 0 to 1. This is like cutting the loop at one point and flattening it onto a unit interval. The endpoints 0 and 1 still describe the same point on the original loop. A pair of loop points can then be represented by a point $(x,y)$ in a unit square.

If the order matters, the square’s left and right edges must be glued because $0$ and $1$ refer to the same first point. The bottom and top edges must be glued for the same reason for the second point. Gluing one pair of edges makes a tube, and curling that tube around to join the remaining edges makes a torus. The torus gives a continuous two-way correspondence between its points and ordered pairs of points on the loop.

The torus contains a trivial collision. The ordered pair $(a,b)$ and the ordered pair $(b,a)$ have the same midpoint and length. They would give a rectangle with zero width, since the two segments are the same segment. The proof needs pairs without an order, so these two descriptions must become one.

Unordered pairs and the Möbius strip

In the unit square, identifying $(x,y)$ with $(y,x)$ means folding the square along its diagonal. The diagonal itself records pairs of the form $(x,x)$, so it becomes the boundary that must land on the original loop. After the fold, the edge identifications have reversed orientations. Cutting along the other diagonal makes the remaining gluing visible, and joining the two reversed edges requires a half-twist. The result is a Möbius strip.

The Möbius strip is therefore a geometric model for every unordered pair of points on the loop. Each point of the strip corresponds to one pair, and each pair gives one point of the strip. The association works continuously in both directions. A small change on the strip gives a small change in the pair, and a small change in the pair gives a small change on the strip.

Its single boundary is the red edge that came from the diagonal of the unit square. That edge represents pairs of the form $(x,x)$, which means that the boundary has to map to the original loop in the $xy$-plane. The midpoint-and-length construction therefore gives a continuous map from a Möbius strip into the output surface in three-dimensional space.

If this map had no self-intersections, it would place a Möbius strip in three dimensions with its boundary confined to the plane and its interior strictly above that plane. Such a crossing would give two distinct points of the Möbius strip with the same image. Those points would represent two distinct unordered pairs with the same midpoint and length, which would give the desired rectangle.

The first version of that geometric claim is too strong. Dan Asimov showed a way to embed a Möbius strip in three dimensions with its boundary on a plane, even with the boundary equal to a circle. The interior of his example passes both above and below the circle. The output surface from the loop-pair construction cannot do that because its height records a distance and is therefore non-negative. The condition that matters is more specific: the interior must stay strictly above the boundary plane.

Reflect the output surface below the plane and glue it to the original along the loop. This glues two Möbius strips along their common boundary. Cutting and rejoining the resulting diagram gives a tube whose two circular ends have opposite orientations. Passing one end through the tube to join them produces a Klein bottle.

A Klein bottle cannot embed in ordinary three-dimensional space without intersecting itself. The intersection is usually visible in physical models of the shape, although the crossing belongs to the representation rather than to the abstract surface. The reflected pair of Möbius strips therefore has to intersect. Since the two halves meet only on the original boundary, the relevant intersection gives two distinct pairs of points in the loop with the same midpoint and length. Their four endpoints form a rectangle.

This is the part of the proof where a topological impossibility does the work of a geometric construction. The Klein bottle is useful because its failure to embed in three dimensions forces a collision in the surface built from the original loop. Meyerson’s 1981 account in Topology Proceedings attributes this rectangle argument to Herbert Vaughan and describes the same unordered-pair map in terms of a Möbius strip and a projective-plane contradiction.

Why squares remain harder

The rectangle proof records three numbers for each pair: the two coordinates of the midpoint and the length of the segment. A square needs one more condition. The two equal-length segments must also have diagonals at right angles. Recording the angle adds a fourth number, which suggests embedding the Möbius strip in four-dimensional space.

Joshua Greene and Andrew Lobb followed that instinct in their 2020 paper. For every smooth Jordan curve and every chosen rectangle shape, they proved that a similar rectangle has all four vertices on the curve. Their proof maps unordered pairs into four-dimensional symplectic space. Rotating the associated Möbius strip changes the angle coordinate whilst preserving the midpoint and distance coordinates. If the rotated strip avoided the original, the two could be combined into a non-intersecting Lagrangian Klein bottle. Shevchishin’s theorem rules out that object in the relevant symplectic space, so every rotation produces an intersection and every aspect ratio occurs.

The smoothness condition explains why this result does not settle the original square problem. As the two points in a pair approach one another, the segment joining them approaches the curve’s tangent line when the curve is smooth. The angle therefore has a clean limiting value at the boundary of the Möbius strip. A rough curve, such as a fractal, may have no tangent and no limiting angle. The extra coordinate then loses the boundary behaviour that made the topological obstruction work. The square peg problem remains open for the general class of rough continuous loops described at the beginning.

A working sense of topology

The familiar introduction to topology often starts with the Möbius-strip exercise or with rubber-sheet geometry. A half-twist in a strip of paper produces a surface with one side, and smooth deformations preserve some properties even as the exact shape changes. Those examples can leave the practical question unanswered: how do strange surfaces help solve a problem?

Here the answer is visible in the order of the construction. The problem asks for four points with a relation between their midpoints, lengths, and angles. Pairs of points become a space. The constraints on those pairs become a boundary. The spaces are glued according to the equivalences the problem already contains. Their possible and impossible embeddings then force the coincidence that the original geometry was looking for.

A Möbius strip is therefore not one particular strip-with-a-twist. It is also the space of unordered pairs of points on a loop, and the same kind of space can describe every possible musical interval. A topological space is better understood here as a family of forms linked by a notion of continuous equivalence. Topology studies the continuous associations between things and what those associations allow or rule out.

Limits

This note follows the complete English caption track, the video’s chapters, its description, and the three substantive references linked there. The video gives an animated explanation rather than a formal proof of the non-embedding results. Its statement that closed non-orientable surfaces must intersect in three dimensions is used as an intuition and then connected to the Klein-bottle obstruction. The exact scope of the rectangle result is more carefully stated in the cited literature for continuous Jordan or simple closed curves, whilst Greene and Lobb’s all-aspect-ratio theorem assumes a smooth Jordan curve.

The video’s discussion of Green and Lobb compresses a proof that uses Lagrangian and symplectic geometry into the geometric idea of adding an angle coordinate. The linked paper supplies the formal construction and the dependence on Shevchishin’s theorem. The Quanta account supplies historical context around Greene and Lobb’s result, including Cole Hugelmeyer’s earlier four-dimensional approach and the one-third-of-aspect-ratios result. Those details belong to the references rather than to claims established by the captions alone.

Further reading / references

• Mark D. Meyerson, “Balancing Acts”, Topology Proceedings 6 (1981), pp. 59–75. Its discussion of planar closed curves presents the rectangle proof attributed to Herbert Vaughan and lists Vaughan’s 1977 lecture, “Rectangles and simple closed curves”.
• Joshua Evan Greene and Andrew Lobb, “The Rectangular Peg Problem”, arXiv:2005.09193 (2020). The paper proves the all-rectangle-shapes result for smooth Jordan curves using symplectic geometry and the non-embedding of a Lagrangian Klein bottle.
• Kevin Hartnett, “New Geometric Perspective Cracks Old Problem About Rectangles”, Quanta Magazine, 2020. The article reconstructs the history from the square peg problem through Vaughan, Hugelmeyer, and Greene and Lobb.