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jeans with the ankles joined are topologically identical to jeans without the ankles joined

@flowerenby

we can only acknowledge the discovery of tied ankle jeans, we have no say in whether they exist

@Sorl

there is a little bit of slight of hand. this gif is drawing a false equivalence between different classes of topological holes. if you follow the gif a few times, you can see that the joined legs hole becomes a new pant leg, which isn't totally fair. but topology doesn't differentiate between those things, so the statement still stands as far as the brutal math goes. lol

@baronnarcveldt I didn't see it as slight of hand, I understood watching it how a new hole was made

I have no clue of the point of topology since the effect is oversimplification of three dimensional shapes, like in what way is it useful to think of my bathrobe as identical to a pair of pants or a bucket with a handle identical to a coffee mug

@Sorl

oh, it's not useful in that way. using jeans and donuts etc as examples is simply an attempt to get the concept across to us. the actual use of topological math is either in calculation using sets of information that behave under similar rules, or in places where the topology is actually at its simplified form like how molecules interact with each other on that atomic level

@baronnarcveldt jeans with the jankles joined are jopologically jidentical to jeans without the jankles joined

@baronnarcveldt That's whack but "identical" is pushing it. This is a homotopy equivalence, but not a homeomorphism. If you consider the boundary of each space (as in manifolds with boundary), the normal jeans give you three circles, and the ones with the ankles joined give you a single circle. This proves they are not homeomorphic.

@jay

it sounds like you're coming at this with prior knowledge and I have limited experience with topology, but I don't think your point is consistent with the conclusion of the video i got this from. the holes that go with either the pant legs or the punctured torus don't map to each other, but i would need a more specific demonstration to believe they're not in fact homeomorphic

youtu.be/ymF1bp-qrjU

@baronnarcveldt Ehh I was being pedantic, to really settle it you would have to decide e.g. if you are modelling the pants as a surface or as a volume. If it's as a surface, then no, they are not homeomorphic, because they have different boundaries.

Basically, two spaces are homeomorphic if there is a one-to-one correspondance between the points of either space which preserves open sets. (This in turn means it preserves any property that can be defined with only "topology words"). This is pretty strict: a point, a line segment, and a disc are all distinct by this definition, even though none of them have any "holes".

Homotopy is a weaker form of equivalence, and is basically the "bending and stretching" thing. Intuitively, homotopy preserves "global" structure, like the number of holes, but it doesn't preserve "local" structure like dimension, or whether something is on a boundary.

@baronnarcveldt I normally wouldn't care too much about this, but the fact that there are multiple notions of equivalence in topology can be confusing to people learning, and the one that's closer to meaning "identical" is definitely homeomorphism.

math stuff 

@jay

hmm you got me. I don't see the distinction you're making about local structure. It's got to be somewhere between the argument that you can bend and stretch them to a 2 holed disc and that they're literally not the same object, but I don't see where you're drawing the line. what is the meaning of boundary in this sense?

math stuff 

@baronnarcveldt My argument isn't just that this particular map isn't a homeomorphism, but that there is *no* homeomorphism between the two spaces. To prove this, I just need to find some "purely topological" property that one space has but not the other. The one that stuck out to me first was the boundary.

If we're deciding to model the pants as surfaces, that means that both spaces are "2-manifolds with boundary". That means that around *most* points on the space, there is some small open neighborhood which is homeomorphic to a plane. The set of points which do *not* have this property are called the boundary. These are the points on the "edge" of the surface.

On the original pants, the boundary is three disjoint circles: one for the belt and two for the ankles. On the pants with the ankles sewn together, the boundary is only one circle. One circle and three circles are not homeomorphic, since e.g. one circle is connected.

math stuff 

@jay

Ah I guess I wasn't aware how those features were relevant. The big thing I know about topology is all the stretching reduction stuff. I'm just a hobbyist at this, so I appreciate you explaining it to me. thank you :)

math stuff 

@baronnarcveldt @jay In the video this comes from, Matt Parker explicitly mentions that there is one step that is arguable, which is the sort of twisting motion to get from the two right-angled rings to the sheet with two holes. He said that since real fabric has nonzero thickness, he's made the decision to treat the pants as a volume, so "real pants" have this homeomorphism even if they wouldn't when modeled as a flat surface.

@baronnarcveldt @breakfastgolem All I learned from topology is that just about everything can be a donut and the most likely shape that the universe itself is is also a donut

I appreciate that topology is the Homer Simpson of math and just wants everything to be tasty donuts. Mmm, 🍩

@baronnarcveldt was this made for matt parker? i remember him asking if someone could make this animation for him

@baronnarcveldt
this is gonna be one of those facts that will rattle around inside my brain probably until I die

@baronnarcveldt Amazon review: "These jeans may be the newest fashion trend, and they say they're topologically identical to my other jeans that fit. But after many tries, I haven't been able to get them on! Save your money. Do not buy!"

@baronnarcveldt If someone shows you a topology animation where anything gets collapsed to a line or point, they're pulling one over on you.

(Specifically, a pair of jeans. That's what they're pulling over your head.)

Same goes for anything that gets crimped or torn, which *might* be happening in this video as well, although it's already setting up a lie by the time it shrinks a 2-surface down to a 1-surface.

@baronnarcveldt This starts out as a 3-hole, 0-handle surface. Joining the ankles results in a 1-hole, 1-handle surface. (That's obviously an illegal deformation, but it's acknowledged as such, so that's OK.)

But *no legal deformation* will ever change that 1-hole, 1-handle status. So it's just a matter of looking for where in the animation you're being tricked.

@varx

I kind of agree with you in that my use of the word "identical" was inappropriate, but on the level of the number of holes, these are all legal deformations. I've already hashed this out a few times. as for being tricked, I'm the one doing the tricking by posting this without the context for laughs. but it was presented with all the appropriate conceits in a mathematical discussion :) thanks for keeping me honest

@baronnarcveldt I get that you posted it for laughs, but it also sounds like you think you can use legal deformations to change the number of holes or handles a surface has, which isn't true.

@varx

No, I understand the deformations don't change the number of holes, but the animations demonstrates that both configurations have 2 total holes, albeit in different ways. Again, I admit identical is the wrong word, but there is a similarity

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