@baronnarcveldt this is not allowed

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

@baronnarcveldt hey crystal what the fuck

it's a jorus (jean torus)

@baronnarcveldt hmm don't like this

@baronnarcveldt topology is fucked up

@baronnarcveldt >:-(

@baronnarcveldt I see

@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

@baronnarcveldt ah okay, see that sounds useful

@baronnarcveldt I want to eat the jonut (jean donut)

at last somebody gets it!

@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.

math stuff

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

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 I love topology

@baronnarcveldt I've been staring at this for a long time

@baronnarcveldt jisomorphic (jeans isomorphic)

@baronnarcveldt fucked up

@Aleums i love it lol

@baronnarcveldt djeans (doughnut jeans)

@baronnarcveldt I appreciate good topology shitpost!

@baronnarcveldt if a snake wore jeans would they wear it like this or--

yeah actually i clipped this from his video

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

run don't walk from the homeotropic pants

@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.

Sid 🎃@InternetEh@dads.cool@baronnarcveldt hypnotic