Embedding a Torus (John Nash) - Numberphile
Why Mathematics Stack Exchange recommends it
Mathematics Stack Exchange users link it when discussing flat tori, isometric embeddings in 3-space, and smooth fractal-like surfaces.
Recommended by 0 people across 0 different months, 2017 to 2024.
What it is
A Numberphile video in which James Grime, with some input from Edward Crane, explains John Nash's result on isometric embeddings using the flat torus. It shows how a flat torus can be embedded in 3D space with a corrugated, fractal-like surface that keeps distances intact. It also links to the Hevea project's visualisations.
What people said
Cites AlgorithmsX · 2017-04 · Mathematics Stack Exchange · on “Modelling the meeting of two random walks”
Instead of defining them independently from each other, define one's position in terms of her absolute location and defining the second person by his location relative to the first person, like the difference between world and local coordinates. His relative position would be described in a coordinate system with the origin representing the first person. As an example, her position could be $(2,4)$, which is her absolute position, and his relative position could be $(3,3)$, which means he is 3 units to the right of her and 3 units above her. Then, you would need to adjust his relative random walk. For instance, if both people can go one space in 4 directions, and both move one space up, in h
Cites emacs drives me nuts · 2021-09 · Mathematics Stack Exchange · on “If we "wrap" coordinates on plane we get torus surface, what we get when wrap coordinates in cube?”
what we get, if we go to 3d, "wrap around" coordinates in cube? It's called a 3-torus [LINK] . The 2-torus is already a bit tricky, because the 2-torus from computer screens is flat, whereas the usual depictions of 2-tori in 3-space are not flat. You can embed a flat 2-torus isometrically $^1$ (i.e. angles and distances on the torus remain the same) in Euclidean 4-space, but that's not easy to imagine. With the 3-torus it's even harder, but shares some similarities with the 2-torus: it's finite and has no borders. And it can be assigned a geometry that's flat everywhere. A good way to get a better understanding is imagining how one moves around in such a space. And you can have different typ
Cites user284619 · 2024-05 · Mathematics Stack Exchange · on “Examples of smooth fractals”
Does this Hévéa Project post [LINK] and links therein lead to such an example? (found in Embedding whole flat 2-Torus in ℝ3 [LINK] , also in Math Overflow [LINK] ) or perhaps here in the Numberphile video "Embedding a Torus (John Nash) - Numberphile" [this video] ?