3 ms·

Real material objects can't self-intersect. Self-intersection is the only trick this uses that is nonexistent in the real world.

• pdonis 6y ago
  How is self-intersection allowed in the math? Isn't that inconsistent with maintaining the topology?
  • riffraff 6y ago
    Exactly my question. Also why isn't the same allowed in 2d for the circle at the beginning?
    • lloeki 6y ago
      It’s not the same. The article goes to that in the second visualisation, and shows that the loop/curl strategy produces a singularity of sorts, making the process non-continuous (two points for the circle, for the sphere it’s a whole circle). Once that is shown it goes on to show it going continuously, even though matter passes through other matter.
    • caconym_ 6y ago
      > why isn't the same allowed in 2d for the circle at the beginning? It is. The picture illustrating the impending cusps is after the two sides of the circle have passed through each other.
  • amelius 6y ago
    Probably because topology isn't concerned with absolute coordinates. It's all about local connectivity.
  • panic 6y ago
    If you just look at the set of points at each instant in time, the self-intersections would produce discontinuities. So instead of deforming a raw set of points in 3D space, topologists deform a function which takes a point on the sphere and returns a point in 3D space: https://en.wikipedia.org/wiki/Homotopy https://en.wikipedia.org/wiki/Homotopy Mechanically, you get this deformation by adding another parameter to the function between spaces. In Go-ish pseudo-code, say at each instant of time you have a function // lon ranges from -180 to 180 // lat ranges from -90 to 90 func eversion_t(lon float, lat float) (x float, y float, z float) { // return the xyz point corresponding to the lon/lat } which maps lon/lat points on the sphere to 3D space. Then the homotopy is a single function, parameterized by a t parameter // t ranges from 0 to 1 inclusive // lon ranges from -180 to 180 // lat ranges from -90 to 90 func eversion(t float, lon float, lat float) (x float, y float, z float) { // return the xyz point corresponding to the lat/lon at time t // see https://arxiv.org/pdf/1711.10466.pdf for the implementation of this function } where this combined function is required to be continuous. By the way, the "push the ends of the sphere through each other" function is a perfectly valid homotopy. There's no topological way to talk about "creasing" -- you need derivatives for that. In particular, the eversion function is required to be an immersion (https://en.wikipedia.org/wiki/Immersion_(mathematics) https://en.wikipedia.org/wiki/Immersion_(mathematics)) at each point in time, which is an additional constraint beyond just being a homotopy.
    • shezi 6y ago
      I don't even think you need to go this far. Since it's differential topology, everything is defined locally anyway, so there must just be environments of nonzero size around every point that don't intersect themselves.
• pbhjpbhj 6y ago
  B-E condensates? Or as someone else said a fluid could model it is you had control over it.