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There is a good YouTube about this [1]: Pop a balloon and slowly let the air out. You can stretch the resulting fabric into a simple shape with no holes. The ba

There is a good YouTube about this [1]: Pop a balloon and slowly let the air out. You can stretch the resulting fabric into a simple shape with no holes. The balloon therefore had -1 holes. They then go on to talk about 2D and 3D holes and Euler Characteristic etc. [1] https://www.youtube.com/watch?v=ymF1bp-qrjU https://www.youtube.com/watch?v=ymF1bp-qrjU

Jensson 5y ago
The pants example is wrong. Pants are equivalent to a disk with two holes, when you circle the two holes in a disk you get 3 separated areas you cannot move between. Topologically valid transformations of surfaces doesn't alter this property. The pants with connected legs however lets you circle two holes without separating any areas at all, you can move between any two points on the pants without crossing the lines just by circling the 8 they collapsed the connected pants into. So a punctured torus surface is not topologically equivalent to a pants surface. The problem with their transformation was that they twisted the surface. You aren't allowed to do that. It works if you see the pants as a volume though, but they specifically talked about surfaces. Similarly a möbius strip is different from a regular loop as a surface, but not as a volume. The twist here is similar.
- quickthrower2 5y ago
  Thanks for that I was a bit bothered by the shorts transform but thought it was my not understanding something. All the animations require the twist to just kinda disappear.
  - Dylan16807 5y ago
    Some of the animations, not all.
- Dylan16807 5y ago
  He talks about that at the 29 minute mark. Rough quote: Technically, if these trousers were made from infinitely thin fabric, then this shape is not equal to this shape because of the twist problem. However, fabric is not infinitely thin [...] so I'm prepared to say a real pair of trousers are homeomorphic to the same pair with the ends of the legs stuck together.
yazaddaruvala 5y ago
I also really like the Vsauce video essay about holes. https://youtu.be/egEraZP9yXQ https://youtu.be/egEraZP9yXQ
gnu8 5y ago
Is the opening of the balloon no longer a hole because it is tied into a knot?

Jensson 5y ago

The pants example is wrong. Pants are equivalent to a disk with two holes, when you circle the two holes in a disk you get 3 separated areas you cannot move between. Topologically valid transformations of surfaces doesn't alter this property. The pants with connected legs however lets you circle two holes without separating any areas at all, you can move between any two points on the pants without crossing the lines just by circling the 8 they collapsed the connected pants into. So a punctured torus surface is not topologically equivalent to a pants surface. The problem with their transformation was that they twisted the surface. You aren't allowed to do that. It works if you see the pants as a volume though, but they specifically talked about surfaces. Similarly a möbius strip is different from a regular loop as a surface, but not as a volume. The twist here is similar.

quickthrower2 5y ago
Thanks for that I was a bit bothered by the shorts transform but thought it was my not understanding something. All the animations require the twist to just kinda disappear.
- Dylan16807 5y ago
  Some of the animations, not all.
Dylan16807 5y ago
He talks about that at the 29 minute mark. Rough quote: Technically, if these trousers were made from infinitely thin fabric, then this shape is not equal to this shape because of the twist problem. However, fabric is not infinitely thin [...] so I'm prepared to say a real pair of trousers are homeomorphic to the same pair with the ends of the legs stuck together.

yazaddaruvala 5y ago

I also really like the Vsauce video essay about holes. https://youtu.be/egEraZP9yXQ https://youtu.be/egEraZP9yXQ

gnu8 5y ago

Is the opening of the balloon no longer a hole because it is tied into a knot?