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Why hasn't this been achieved in the real world?I strongly feel they need a mathematical trick inexistant in the real world, either a tear or a supplementary di
by The_rationalist 6y ago
Why hasn't this been achieved in the real world?I strongly feel they need a mathematical trick inexistant in the real world, either a tear or a supplementary dimension.
- sideshowb 6y agoI was wondering if there's some real process that uses the "twist" trick, even if it's not inverting a sphere. Packing a pop-up tent maybe? Always seems difficult enough..!
- andreareina 6y agoThe stowable shades people use to cover car windshields and windows use a twist to fold them up without having to crimp/crease the stiff wire on the exterior that gives it its support.
- om2 6y agoReal material objects can't self-intersect. Self-intersection is the only trick this uses that is nonexistent in the real world.
- pdonis 6y agoHow is self-intersection allowed in the math? Isn't that inconsistent with maintaining the topology?
- riffraff 6y agoExactly my question. Also why isn't the same allowed in 2d for the circle at the beginning?
- lloeki 6y agoIt’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 agoProbably because topology isn't concerned with absolute coordinates. It's all about local connectivity.
- panic 6y agoIf 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 agoI 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 agoB-E condensates? Or as someone else said a fluid could model it is you had control over it.
- Arrrlex 6y agoIn the real world you usually can't pass a material through itself.
- The_rationalist 6y agoI wonder if that couldn't be possible with colored gaz or liquid (magnetic "liquids" that hold together). That would be quite fascinating to see IRL.
- throwaway_pdp09 6y agoThe vid says the surfaces are allowed to pass through each other. I don't see how that corresponds to anything in the real world.
- amelius 6y agoFrom the article: > Note that unlike physical objects, self-intersection is permitted.