3 ms·
These Shapes Are Topologically Equivalent
- js8 4y agoI feel like this is a trick in many wood & string puzzles.
- leto_ii 4y agoI had the same initial impulse, but I think the transformation 1 -> 2 can't work irl. especially if those are compact solids. Edit: 2->3 also seems impossible. you basically move the hole around.
- chmod775 4y agoYou can do it IRL with something like plasticine.
- OscarCunningham 4y agoNo, you can't do this trick with wood and string. Try tracking what would happen to the "string" joining the two loops.
- fwlr 4y agoNot quite, since most topological transformations can’t be done on actual physical objects. (I would love to see some of those wood and string puzzles try to implement some of these style of topological puzzles though, that would be awesome if it was possible.) Most of the wood and string puzzles fall under knot theory specifically, rather than topology. The main trick I’ve seen used in those puzzles is they have knots with an even number of crossings and therefore they can be separated (I believe the formal term might be “they have ambient isotopy to unlinked unknots”, but there’s a lot of subtlety I don’t get, so that’s probably wrong) but the knots are deformed to look like two distinct knots, each with an odd number of crossings (a knot with an odd number of crossings cannot be separated). Our intuition tells us “one inseparable knot plus another inseparable knot equals a single, bigger, still inseparable knot”, but that’s knot the case.
- SetTheorist 4y agoI'm not sure exactly what you're trying to say (I'm not sure what you mean by "separate" here), but your assertions seem incorrect. Knots can have even or odd crossing numbers, that alone will not tell anything about whether it is an "unknot" or not. Also, if you have two non-trivial knots and "join" them (ie each a knot in a circle, cut each circle in one spot and join the circles at the cuts) you will _never_ get an unknot.
- fwlr 4y agoYeah, as I thought, I don’t know knot theory nearly well enough to use their terms properly.
- tmhn2 4y agoThe little cartoon shows they are ambient-isotopic--that they are just topologically equivalent is pretty clear already!
- quadcore 4y agoTopology seems like a field AI could help isnt? Cant AI generate topology equivalent shapes for example?
- deleted 4y ago[deleted]
- ginnungagap 4y agoI do research mostly in general topology and topological dynamics, but I don't really see how AI would help, producing homeomorphic (topologically equivalent) spaces is not really the point. In general even presenting a topological space to a computer is not easy. This can be done for very good spaces, for example those admitting (finite) triangulations, and there are some algorithms there, but my understanding is that you run into undecidability issues very quickly. But for the kind of spaces I work with (which are usually compact metrizable connected, so way better than arbitrary topological spaces) there seems to be no hope to get a computer to work with them.
- orlandpm 4y agoYes! I’m working on this in my PhD research. Here’s a relevant Caltech lecture aimed at a general scientific audience, discussing use of ML to detect equivalent knots. https://m.youtube.com/watch?v=etNErJ1iuno https://m.youtube.com/watch?v=etNErJ1iuno
- seliopou 4y agoThe step from 1 to 2 is dropping a twist. If those are two-dimensional objects I don’t think that’s ok.
- alexmolas 4y agoThese are 3D objects.
- jwestbury 4y agoA pair of jeans with the leg holes sewn together, i.e. where the legs form one continuous tube, is topologically equivalent to a normal, unmolested pair of jeans. You can't trust topologists.
- fwlr 4y agoYou know the joke about how when you kiss someone, you’re temporarily forming a long tube with an anus at each end? A topologist friend heard that joke and immediately offered a conjecture that that shape is topologically equivalent to the shape you’d get if you connect a mouth to an anus. This got the party discussing if there exists a homeomorphism between making out and eating ass. I agree, you really can’t trust topologists.
- krsrhe 4y agoIt doesn’t exist, because there is no homeomorphism that takes a distance of 0 to non 0, so you can’t change the connections.
- vidanay 4y agoTopologically a human is a donut (torus).
- OscarCunningham 4y agoI think humans probably have a variety of tubes at various sizes that complicate the analysis.
- siddMahen 4y agoIt’s interesting to think about how the boundary components of the handcuffs are being transformed; this isotopy sends the inner boundary component of second link in the handcuffs to the outer boundary component!
- pcrh 4y agoNeat! What are the "real world" uses for this kind of mathematics?
- ajkjk 4y agoDo you have some reason to think there are some?
- nh23423fefe 4y agotopology > continuity > calculus
- Shish2k 4y agoYou can tie network and power cables in really annoying ways that make it seem as if you welded the rack together after plugging the cables in and you can’t pull them out without an angle grinder :D