3Blue1Brown - This open problem taught me what topology is
The video delves into the inscribed square problem, originally posed by Otto Toeplitz in 1911, which questions whether every closed continuous loop has an inscribed square. The discussion begins with a simpler problem of finding inscribed rectangles, using Herbert Vaughan's proof. The approach involves mapping pairs of points on a loop to a three-dimensional space, focusing on their midpoints and distances. This mapping is continuous, meaning small changes in input lead to small changes in output, and the goal is to find a collision where two pairs map to the same point, indicating a rectangle. The video explains how this mapping creates a complex surface in 3D space, which can be analyzed for self-intersections to find inscribed rectangles. The concept of a Mobius strip is introduced as a natural representation of unordered pairs of points on a loop, leading to the conclusion that embedding a Mobius strip in 3D without self-intersection is impossible, thus proving the existence of inscribed rectangles. The video also touches on the unsolved problem of inscribed squares, suggesting that considering angles and embedding in higher dimensions might offer solutions. The discussion highlights the role of topology in problem-solving, showing how seemingly abstract shapes like Mobius strips and Klein bottles are practical tools in mathematical proofs.
Key Points:
- The inscribed square problem asks if every closed loop has an inscribed square, unsolved since 1911.
- Herbert Vaughan's proof for inscribed rectangles uses continuous mapping of point pairs to 3D space.
- A Mobius strip represents unordered pairs of points on a loop, crucial for proving inscribed rectangles.
- Embedding a Mobius strip in 3D without self-intersection is impossible, ensuring rectangle existence.
- Topology uses abstract shapes like Mobius strips as practical tools for logical deduction and proofs.
Details:
1. 🔍 The Inscribed Square Problem
1.1. Historical Context and Problem Definition
1.2. Mathematical Implications and Related Concepts
2. 📚 Revisiting the Rectangle Proof
- The video is a second edition of an earlier video on the same proof, motivated by new research and interesting connections.
- The proof discussed is about any closed loop having an inscribed rectangle, though it lacks practical applications.
- Engaging with challenging puzzles like this proof can sharpen problem-solving instincts for practical applications.
- The proof provides a deeper understanding of topology, beyond common classroom activities like creating a Mobius strip.
- Topology is often misunderstood as just bizarre shapes or rubber sheet geometry, but it has deeper problem-solving implications.
3. 🔄 Mapping Pairs to 3D Space
- The process involves reframing the problem of finding rectangles in a closed loop by identifying two pairs of points with the same midpoint and length.
- Mapping pairs of points on a loop to a 3D space involves considering the midpoint and distance between points as coordinates in this space.
- The mapping is continuous, meaning small changes in input result in small changes in output, which is crucial for identifying inscribed rectangles.
- The concept of self-intersection in the 3D mapping indicates the presence of inscribed rectangles, as different pairs of points map to the same 3D point.
- For a circle, the mapping results in a dome-like surface with infinite inscribed rectangles, all having midpoints at the circle's center.
- The surface formed by mapping is not a function graph but a set of all possible outputs, representing complex relationships between pairs of points.
4. 🔗 Torus and Möbius Strip Connections
- The association of points on a loop with numbers between 0 and 1 creates a continuous mapping, except for the endpoints 0 and 1, which map to the same point on the loop.
- Pairs of points on the loop can be represented as points in a unit square, with x and y coordinates corresponding to each point on the loop.
- The mapping between the unit square and pairs of points on the loop is continuous, but requires gluing edges to account for the equivalence of 0 and 1.
- By gluing the edges of the unit square, a torus shape is formed, representing all possible pairs of points on the loop.
- The torus allows for a continuous mapping where each point on the torus corresponds to a unique pair of points on the loop, and vice versa.
- For unordered pairs of points, where (a,b) is equivalent to (b,a), the Möbius strip is a more suitable representation.
- The Möbius strip is formed by folding the unit square along its diagonal and introducing a half twist, representing unordered pairs of points on the loop.
- The Möbius strip maintains a continuous relationship between points on the strip and pairs on the loop, with small changes on one side corresponding to small changes on the other.
5. 🔍 Möbius Strip and 3D Embedding
- The Möbius strip, a one-sided surface with a boundary, can be represented by unordered pairs of points, allowing a continuous mapping onto a 3D surface.
- It is impossible to embed a Möbius strip in 3D without self-intersection if its edge is confined to a plane, as this would require two distinct points to map to the same surface point.
- Dan Asimov's construction demonstrates a Möbius strip with a circular boundary, challenging the notion of planar confinement without intersection.
- Reflecting a Möbius strip surface and gluing it forms a Klein bottle, a non-orientable surface that cannot exist in 3D without self-intersection, illustrating the complexity of such embeddings.
- The construction involves pairs of points with identical midpoints and distances, forming a rectangle, which is crucial for proving the impossibility of certain 3D embeddings.
6. 🔎 The Challenge of Inscribed Squares
- The problem involves proving that any loop has an inscribed square, not just a rectangle, which has been a longstanding mathematical challenge.
- A strategic approach involves analyzing the angle of line segments to find two segments that share a midpoint, length, and differ by 90 degrees, forming a square.
- In 2020, Joshua Greene and Andrew Lobb extended results for smooth curves, demonstrating that not only can you find a square, but rectangles of every aspect ratio, which was a significant advancement.
- Their innovative work involves embedding Mobius strips and Klein bottles into four-dimensional space, a complex yet effective method for addressing the problem.
- Smooth curves provide well-defined tangent lines and clean limiting behavior, facilitating the solution to the problem.
- The inscribed square problem's complexity is heightened by the lack of limiting behavior for angles in rough curves like fractals, making it a challenging area of study.
7. 🔮 Topology's Broader Implications
- Mathematicians study shapes like Mobius strips and Klein bottles not just for their bizarreness but for problem-solving applications.
- Mobius strips are not limited to one surface; they represent an abstract idea, such as unordered pairs of points on a loop, applicable in describing musical intervals.
- Topology is about understanding continuous associations between shapes and what is possible under those associations.
- Famous topological shapes represent large families of shapes with similar behavior under continuous maps.
- Constraints and impossibilities in topology are crucial for logical proofs and mathematical progress.