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Inscribed square problem

10 min read

From: 3blue1brown

The inscribed square problem poses a question: does every closed continuous loop necessarily contain an inscribed square? [00:00:18] This means finding four points on the loop that form the vertices of a square [00:00:11]. A “closed continuous curve” or “loop” is defined as any squiggle that can be drawn on paper without lifting the pen, ending where it starts [00:00:03]. As of the transcript’s creation, nobody in the world knows the answer to this question [00:00:00].

The question was originally posed by Otto Toeplitz in 1911 [00:00:27]. A simpler, but related, version of the problem asks whether every closed continuous loop necessarily has an inscribed rectangle [00:00:38]. The argument for the inscribed rectangle problem is attributed to Herbert Vaughan [00:00:46]. This proof is considered a “favorite piece of math” by the video’s creator [00:00:34].

Motivation and Connection to Topology

While there are no known practical applications for proving the existence of an inscribed rectangle, engaging with such challenging puzzles can sharpen problem-solving instincts applicable to other areas [00:01:47].

The proof for the inscribed rectangle problem provides a fundamental understanding of topology [00:02:02]. Topology is often misrepresented in recreational math as the study of bizarre shapes like the Möbius strip [00:02:10] or as “rubber sheet geometry” where shapes are considered the same if one can be deformed into another without tearing [00:02:35]. However, these notions do not fully capture what topology is truly about, which is the study of continuous associations between things and what is or is not possible under those associations [00:02:43][00:26:22]. The proof demonstrates how seemingly bizarre shapes and their properties become tools for logic and deduction [00:02:55].

Proof of the Inscribed Rectangle Theorem

The proof for the inscribed rectangle theorem involves reframing the question and constructing geometric representations.

Reframing the Question

Instead of searching for four points forming a rectangle, the problem is reframed as searching for two distinct pairs of points on the loop [00:03:14]. These two pairs must satisfy two conditions:

  1. They have the same midpoint [00:03:21].
  2. They have the same length [00:03:25].

If these conditions are met, the four endpoints of these two distinct line segments will necessarily form a rectangle [00:03:41]. This is a straightforward geometry exercise to prove [00:03:49].

Mapping Pairs of Points to 3D Space

For any pair of points on a given loop, two pieces of data are important:

  1. Midpoint coordinates (x, y): The two-dimensional coordinates of the midpoint of the segment connecting the pair [00:04:04].
  2. Distance (d): The distance between the two points in the pair [00:04:12].

These three numbers (x, y, d) can be packaged together as a single point in a three-dimensional space [00:04:17]. The x and y coordinates represent the midpoint’s location on the plane where the loop sits, and the z-coordinate (or height) represents the distance between the points [00:04:39].

This process creates a continuous mapping from pairs of points on the loop to points in three-dimensional space [00:04:56]. Continuity means that slightly wiggling the input pair of points only slightly wiggles the output point in 3D space, with no sudden jumps [00:05:07].

The search for an inscribed rectangle now amounts to finding a “coincidence” where two distinct pairs of points on the loop map to the same output point in 3D space [00:05:16]. This means they would share the same midpoint and distance, forming a rectangle [00:05:28].

The “Wild Surface” of Outputs

Collecting all possible outputs of this mapping from all possible pairs of points on the loop forms a complex, “wild surface” in 3D space [00:06:02]. Self-intersections within this surface correspond to inscribed rectangles [00:07:06].

  • Cross-sections: Near the base (xy-plane), cross-sections of this surface approximately resemble the loop itself [00:06:25].
  • Circle Example: For a circular loop, the surface forms a dome shape [00:07:43]. While it may not visually appear to self-intersect, a circle has infinitely many inscribed rectangles, all sharing the circle’s center as their midpoint and the diameter as their diagonal length [00:08:03]. This means infinitely many pairs of points map to a single point at the top of the dome, representing a “collision” or self-intersection in the sense of the mapping [00:08:17].
  • Ellipse Example: Squishing a circle into an ellipse causes this single point of intersection to become a vertical line of self-intersections [00:08:34].

It is important to note that this surface is not a function graph, but rather the set of all possible outputs from the mapping [00:09:02].

The “Gun on the Wall”: Pairs of Identical Points

A crucial detail, referred to as “the gun on the wall,” is what happens when the two points in a pair are infinitesimally close or even the same point on the loop (e.g., x, x) [00:09:49]. In this extreme case, the distance between them is zero, so the output point in 3D space will have a z-coordinate of zero [00:10:00]. The midpoint will be the point itself. Therefore, all such pairs (x, x) map directly onto the original loop lying on the xy-plane [00:10:10].

Constructing a Second Surface: The Möbius Strip

To prove that the “wild surface” must self-intersect, another natural surface is constructed that also represents pairs of points on the loop.

  1. Coordinate System on the Loop: Assign each point on the loop a number between 0 and 1 [00:10:52]. This is like snipping the loop and flattening it onto the unit interval, where 0 and 1 map to the same point [00:10:58].

  2. Unit Square for Pairs: For a pair of points (P1, P2) on the loop, they can be represented by coordinates (x, y) in a unit square, where x is the coordinate for P1 and y for P2 [00:11:31].

  3. Gluing for a Torus (Ordered Pairs): To account for the loop’s continuity (0 and 1 being the same point), edges of the unit square are “glued” together:

    • The left (x=0) and right (x=1) vertical edges are glued, forming a tube [00:12:13].
    • The bottom (y=0) and top (y=1) horizontal edges are then glued, curling the tube into a donut shape, or a torus [00:12:32]. This torus continuously and bijectively represents all ordered pairs of points on the loop [00:13:22].

  4. From Torus to Möbius Strip (Unordered Pairs): For the inscribed rectangle problem, the order of the points in a pair doesn’t matter (a,b is the same as b,a) [00:13:56]. In the unit square, this means points (x, y) and (y, x) are equivalent. This is represented by folding the square along its main diagonal (from (0,0) to (1,1)) [00:14:40].

    • The diagonal itself (where x=y) represents pairs of identical points (x, x) [00:15:00].
    • After folding, the remaining edges need to be glued. This process, involving cutting and re-gluing with a half-twist, naturally results in a Möbius strip [00:15:21][00:15:54].

This Möbius strip continuously and bijectively represents all possible unordered pairs of points on a loop [00:16:10]. The edge of this Möbius strip corresponds precisely to the diagonal line where x=y, representing pairs of identical points (x, x) [00:16:30].

The Topological Argument

  1. Mapping the Möbius Strip to the Wild Surface: Since both the Möbius strip and the “wild surface” represent unordered pairs of points on the loop via continuous bijections, there must be a continuous function from the Möbius strip onto the “wild surface” in 3D space [00:17:05].
  2. The Crucial Constraint: The edge of the Möbius strip (representing pairs like x,x) must land on the original loop, which lies on the xy-plane (z=0) [00:17:42]. Additionally, by its construction, the “wild surface” must always exist above the xy-plane (distance d > 0) for distinct points [00:19:31].

The proof then relies on the claim that it is impossible to embed a Möbius strip into 3D space such that its edge is confined to the xy-plane and its interior is strictly above the plane, without the strip self-intersecting [00:19:41].

  • Dan Asimov’s Counterexample: Initially, it might seem impossible to embed a Möbius strip with its edge on a plane at all without self-intersection. However, mathematician Dan Asimov demonstrated a construction where a Möbius strip’s boundary lies on a circle in a plane, without self-intersection [00:18:40]. This embedding, however, has parts of the strip going both above and below the plane [00:19:27]. This refinement is critical for the proof.

  1. Connection to the Klein Bottle: Consider the “wild surface” together with its reflection underneath the xy-plane [00:20:05]. This combined surface is topologically equivalent to what you get when you glue the edges of two Möbius strips together [00:20:18]. This composite surface is famously known as a Klein bottle [00:21:31].

    • A Klein bottle is a “celebrity shape” in math because it has no clear interior or exterior [00:21:34].
    • A fundamental property of the Klein bottle is that it is impossible to properly represent it in three dimensions without the surface intersecting itself [00:21:49]. This is a general fact about any closed, non-orientable surface [00:22:01].

  2. Conclusion: Because the “wild surface” combined with its reflection forms a Klein bottle, and Klein bottles cannot be embedded in 3D without self-intersection, the original “wild surface” must also self-intersect [00:22:12]. This self-intersection means there are two distinct pairs of points on the loop that map to the same point in 3D space (same midpoint and distance), thus forming a rectangle [00:22:23].

The Unsolved Inscribed Square Problem Revisited

The inscribed square problem is harder because it requires a specific 90-degree difference in angle between the two diagonals, in addition to sharing a midpoint and length [00:23:01]. This additional piece of information might lead one to consider embeddings into four dimensions [00:23:14].

Smooth Curves

In 2020, mathematicians Joshua Green and Andrew Lobb proved a significant extension of this result for smooth curves [00:23:26]. Smooth curves are those where derivatives can be taken as many times as desired, meaning every point has a well-defined tangent line [00:24:23]. For smooth curves, it was already known that an inscribed square could always be found [00:24:40]. Green and Lobb showed that for smooth curves, rectangles of every possible aspect ratio can be found [00:24:47]. Their paper involves embedding Möbius strips and Klein bottles into four-dimensional space [00:24:51].

Rough Curves

What makes the general inscribed square problem (for all closed continuous curves) unsolved is the case of “rough curves,” like fractals [00:25:05]. For rough curves, the angle of a line segment connecting two points might not have a clean limiting behavior as the points get closer, making the topological argument more complex or inapplicable [00:25:02].

Takeaways

  • Purpose of Abstract Math: Mathematicians study seemingly bizarre shapes like Möbius strips and Klein bottles not for their oddity, but because they are tools for solving complex problems and provide logical constraints essential for proofs [00:25:22].
  • Topological Space: A Möbius strip is not just one specific physical shape; it represents an “infinite family of shapes” or a “topological space,” all connected by a certain notion of equivalence [00:25:47][00:26:03]. Examples include the familiar twisted strip, Asimov’s snail-shell shape, or even the abstract idea of unordered pairs of points on a loop [00:25:55].
  • Topology Defined: Topology fundamentally involves understanding continuous associations between different mathematical objects and determining what is or is not possible under those associations [00:26:29]. Constraints and impossibilities revealed through topological reasoning are crucial for mathematical progress [00:26:53].

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