Who cares about topology? (Inscribed rectangle problem)

The video begins with an introduction to topology and the excitement around sharing mathematical concepts, particularly those related to shapes and their properties. The speaker reminisces about their childhood experiences with math and how topology, often demonstrated through a mobius strip or comparing a coffee mug to a donut, was used to spark interest. However, these demonstrations left the speaker questioning the practical applications of such math. The video aims to address this by discussing an unsolved problem known as the inscribed square problem, which asks whether every closed loop can have at least one inscribed square. The speaker then introduces a weaker version of the problem, inscribed rectangles, for which there is a beautiful solution involving a shift in focus from individual points to pairs of points on the loop.

The speaker explains a method to find inscribed rectangles by defining a function that takes pairs of points on a loop and outputs a point in 3D space, encoding the midpoint and distance information. The loop is considered to lie on the xy-plane, and for each pair of points, the midpoint is identified on the plane, with the distance between the points represented as height in the z-direction. This results in a surface that closely follows the loop. The function is continuous, meaning small changes in the input pair of points result in only slight changes in the output point. The goal is to demonstrate that this function has a collision, where two distinct pairs of points map to the same 3D point, implying the existence of an inscribed rectangle. The speaker then delves into the concept of ordered and unordered pairs of points and how they can be represented using a two-dimensional coordinate plane, leading to the idea of a torus representing all pairs of points on the loop.

The speaker continues the exploration of pairs of points by transitioning from ordered pairs, which consider the sequence of points, to unordered pairs, where the sequence does not matter. The challenge lies in representing unordered pairs in a unique way, which leads to the concept of a Möbius strip. The speaker describes a process of transforming a unit square into a Möbius strip by gluing its edges in a specific manner, with the diagonal fold being crucial for representing pairs where the points are the same. The Möbius strip is shown to have a continuous one-to-one association with unordered pairs of points on the loop, which is key to solving the inscribed rectangle problem. The speaker emphasizes the significance of the Möbius strip in understanding the problem and how its properties lead to the proof that an inscribed rectangle must exist for any closed loop.

The video concludes with the speaker presenting the proof for the existence of an inscribed rectangle in any closed loop. By leveraging the continuous association between the Möbius strip and the 3D surface defined by the pairs of points, the speaker shows that the Möbius strip must intersect itself when mapped onto the surface, indicating the presence of two distinct pairs of points with the same midpoint and distance apart, thus forming a rectangle. The speaker reflects on the intuitive nature of the proof when considering the Möbius strip and acknowledges that a rigorous proof would require a deep understanding of topology. The speaker highlights the practical applications of topology and how it provides insights into solving concrete problems, encouraging appreciation for the field and its definitions.