From: 3blue1brown
Topology, a branch of mathematics, is concerned with the properties of geometric objects that are preserved under continuous deformations, such as stretching, bending, or twisting, but not tearing or gluing [00:00:11]. Early introductions to topology often involve demonstrations like the Möbius strip, highlighting its single-sided nature [00:00:38], or the idea that coffee mugs and donuts are topologically equivalent because they each have one hole [00:00:52]. While these examples can pique interest, they often leave the fundamental question of how topology helps solve concrete mathematical problems unanswered [00:01:03].
The Inscribed Rectangle Problem
A powerful application of topological principles can be seen in the solution to a weaker version of the unsolved “inscribed square problem” [00:01:20]. The inscribed square problem asks whether any closed loop (a continuous line that ends back where it started) always contains four points that form a square [00:01:25]. While this remains unsolved [00:02:03], asking about inscribed rectangles instead of squares yields a beautiful and elegant solution [00:02:13].
The solution shifts focus from individual points on the loop to pairs of points [00:02:28]. A key fact about rectangles is that if you have four points A, B, C, D forming a rectangle, the diagonal pairs (AC and BD) share a common midpoint, and the distance between A and C equals the distance between B and D [00:02:41]. Conversely, if two distinct pairs of points (AC and BD) share a midpoint and have equal distances between them, they form a rectangle [00:02:56]. Therefore, the problem reduces to proving that for any closed loop, it’s always possible to find two distinct pairs of points on that loop that share a midpoint and are the same distance apart [00:03:14].
Defining a Continuous Mapping Function
To demonstrate this, a continuous function is defined that takes in pairs of points on the loop and outputs a single point in 3D space [00:03:37]. If the closed loop is considered to be on the xy-plane in 3D space, for any given pair of points on the loop:
- Their midpoint (m) is identified, located on the xy-plane [00:03:57].
- The distance (d) between them is calculated [00:04:00].
- A new point is plotted exactly d units above m in the z-direction [00:04:06].
Applying this function to all possible pairs of points on the loop generates a surface in 3D space [00:04:14]. This function is continuous, meaning that a slight adjustment to the input pair of points results in only a slight adjustment to the output point in 3D space [00:05:07]. As the pair of points on the loop get closer, the plotted point gets lower, and the midpoint approaches the loop [00:04:38]. When the pair of points coincides (i.e., xx for some point x on the loop), the distance d is zero, and the plotted point lies exactly on the original loop in the xy-plane at point x [00:04:53].
The goal then becomes to show that this continuous function must have a “collision”—two distinct pairs of points that map to the exact same spot in 3D space [00:05:22]. Such a collision would imply shared midpoints and equal distances, thereby confirming an inscribed rectangle [00:05:31]. This means finding an inscribed rectangle is equivalent to showing that this generated surface must intersect itself [00:05:40].
Representing Pairs of Points with Topological Surfaces
To understand this self-intersection, it’s necessary to define a topological space that naturally represents all possible pairs of points on the loop [00:05:51].
Ordered Pairs and the Torus
First, consider ordered pairs of points (where AB is distinct from BA) [00:06:23]. The loop can be “straightened out” by cutting it at one point and deforming it into an interval, for example, from 0 to 1 on a number line [00:06:59]. Every point on the loop corresponds to a unique number on this interval, except for the cut point, which corresponds to both 0 and 1 [00:07:11].
To represent pairs of points, a second interval can be used as a y-axis, forming a 1x1 square [00:07:38]. Each point (x, y) in this square represents an ordered pair of points on the loop [00:07:49]. Since the endpoints 0 and 1 of the interval correspond to the same point on the loop, the edges of this square need to be “glued” [00:08:12]:
- The left edge (x=0) must be glued to the right edge (x=1) [00:08:47].
- The bottom edge (y=0) must be glued to the top edge (y=1) [00:09:00].
Bending the square into a cylinder (gluing left to right) and then gluing the ends of the cylinder (top to bottom) results in a torus (the surface of a doughnut) [00:09:13]. Every point on this torus uniquely corresponds to an ordered pair of points on the loop [00:09:29]. This association is continuous in both directions [00:09:49].
Unordered Pairs and the Möbius Strip
The problem, however, requires understanding unordered pairs (where AB is considered the same as BA) [00:10:09]. In the unit square, this means that any coordinates (x, y) must represent the same pair as (y, x) [00:11:02]. This concept is captured by folding the square diagonally along the line x=y [00:11:11]. The diagonal line itself represents pairs of points where x=y (i.e., xx), which are effectively single points listed twice [00:11:27].
After folding, the remaining edges (the bottom and the right edges) must be glued together according to the original square’s topology [00:11:43]. This complex gluing, which involves a twist, results in a Möbius strip [00:12:09]. Thus, the Möbius strip is the natural topological surface that represents all unordered pairs of points on the loop [00:12:38]. Crucially, the single edge of the Möbius strip corresponds to all pairs of points that look like xx (where the two points coincide) [00:12:44].
The Topological Proof
With the understanding that there is a continuous one-to-one association between unordered pairs of points on the loop and individual points on the Möbius strip [00:13:08], the problem can be solved.
The continuous function defined earlier maps each point on the Möbius strip (representing an unordered pair of points on the loop) to a specific point on the 3D surface [00:13:40]. As established, when the two points of a pair are extremely close, the output of the function is right above the loop itself [00:14:07]. In the extreme case where the points coincide (i.e., xx), the output is exactly on the original loop in the xy-plane [00:14:14].
Since the edge of the Möbius strip represents these xx pairs, when the Möbius strip is mapped onto the 3D surface, its edge must be mapped precisely onto the original loop in the xy-plane [00:14:21].
This leads to the crucial topological argument: given the strange, one-sided nature of the Möbius strip, there is no way to continuously map its edge onto a two-dimensional loop (like the original closed loop in the xy-plane) without forcing the strip to intersect itself [00:14:39]. This fact, while intuitively clear from visualizing the Möbius strip, forms the core of the proof and requires rigorous justification within the field of topology itself [00:15:33].
Because points on the Möbius strip represent pairs of points on the loop, if the strip intersects itself during this mapping, it means that at least two distinct pairs of points on the loop correspond to the exact same output on the 3D surface [00:14:53]. As established, this collision signifies that these two distinct pairs share a common midpoint and are the same distance apart [00:15:02], thereby proving that they form a rectangle inscribed within the original closed loop [00:15:15].
This problem beautifully illustrates how abstract topological concepts, like the Möbius strip and continuous mappings, arise naturally and are essential tools for solving concrete problems [00:16:07].