3 ms·
Möbius Strips Defy a Link with Infinity
- ericpauley 8y agoThe subtitle is in clear contradiction with the paper, the first sentence of which claims that this was proven for 3 dimensions in 1962.
- mc32 8y agoIt doesn't seem to contradict it in my reading it. (Basically it refuses to be packed infinitely in tame [1962] and now wild embeddings])
- macawfish 8y agoTotally different thing, but reminds me of the Hopf Fibration, which you should really look up if you never have before.
- alanbernstein 8y agoI'm not sure I'm grasping the significance of this, is there an article or video you'd recommend?
- macawfish 8y agoDefinitely this one! https://youtu.be/10sDqSUjXHc https://youtu.be/10sDqSUjXHc Think about how you could fill a 2D plane with infinite parallel lines, one for every single point on a real line. Imagine, for example, running vertical lines through every point on the x-axis, how this would fill the entire plane. The Hopf Fibration is similar. It's a way of filling all of the space on the surface of a 3-sphere (which locally feels like 3D Euclidean space) with interlocking circles. Each of these circles can be mapped a point on an ordinary sphere. Thinking of it in the other direction: for every point on the surface of an ordinary (2-)sphere, you can assign a circle on the surface of a 3-sphere. By taking into consideration all of the points on the 2-sphere, and all of the circles they represent, you can fill the entire surface of the 3-sphere with these interlocking circles!