3 ms·

The surface.

catnaroek 11y ago
The surface itself is the Klein bottle, so I would expect its boundary to be some 1-dimensional object, in the same way the boundary of a disk is the circle surrounding it.
- Manishearth 11y ago
  A Mobius strip is a twisted non-orientable 2D surface that exists in 3 dimensions, with a 1D edge. A Klein bottle is a twisted non-orientable 3D volume that exists in 4 dimensions, with a 2D "edge" surface.
  - GFK_of_xmaspast 11y ago
    You get a Klein bottle from Möbius strip by identifying the boundary 1-cell with itself (plus a twist, whatever). It is not a "3d volume"; there is no 3-cell involved.
- GFK_of_xmaspast 11y ago
  From wiki: "Whereas a Möbius strip is a surface with boundary, a Klein bottle has no boundary (for comparison, a sphere is an orientable surface with no boundary)." (https://en.wikipedia.org/wiki/Klein_bottle https://en.wikipedia.org/wiki/Klein_bottle)
  - catnaroek 11y ago
    Then, could you explain what Pxtl meant by “a Klein bottle is a 4d moebius strip”?
    - GFK_of_xmaspast 11y ago
      Sure. One standard way to construct both a Möbius strip and a Klein bottle (and other classic manifolds) is to take a square, and glue some edges together. For a Möbius strip, you glue together the left and right (say) edges, such that the upper part of one connects to the bottom part of the other (that is, put a twist in the square before you glue). For a Klein bottle, you additionally glue together the top and bottom edges, but don't twist them. This is what it means to say a Klein bottle is "like a Möbius strip" or "two strips glued together" or similar things. The "4d" comes in because if you want to do this with a physical object, you need four spatial dimensions unless you're ok with it passing thru itself. Here's a good picture : http://web.ornl.gov/sci/ortep/topology/topo5.gif http://web.ornl.gov/sci/ortep/topology/topo5.gif The arrows indicate which edges to glue together, and how to line them up. See also bmm6o's comment here: https://news.ycombinator.com/item?id=11196493 https://news.ycombinator.com/item?id=11196493
      - catnaroek 11y ago
        Thanks!
deleted 11y ago
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The surface.

The surface itself is the Klein bottle, so I would expect its boundary to be some 1-dimensional object, in the same way the boundary of a disk is the circle surrounding it.

Manishearth 11y ago
A Mobius strip is a twisted non-orientable 2D surface that exists in 3 dimensions, with a 1D edge. A Klein bottle is a twisted non-orientable 3D volume that exists in 4 dimensions, with a 2D "edge" surface.
- GFK_of_xmaspast 11y ago
  You get a Klein bottle from Möbius strip by identifying the boundary 1-cell with itself (plus a twist, whatever). It is not a "3d volume"; there is no 3-cell involved.
GFK_of_xmaspast 11y ago
From wiki: "Whereas a Möbius strip is a surface with boundary, a Klein bottle has no boundary (for comparison, a sphere is an orientable surface with no boundary)." (https://en.wikipedia.org/wiki/Klein_bottle https://en.wikipedia.org/wiki/Klein_bottle)
- catnaroek 11y ago
  Then, could you explain what Pxtl meant by “a Klein bottle is a 4d moebius strip”?
  - GFK_of_xmaspast 11y ago
    Sure. One standard way to construct both a Möbius strip and a Klein bottle (and other classic manifolds) is to take a square, and glue some edges together. For a Möbius strip, you glue together the left and right (say) edges, such that the upper part of one connects to the bottom part of the other (that is, put a twist in the square before you glue). For a Klein bottle, you additionally glue together the top and bottom edges, but don't twist them. This is what it means to say a Klein bottle is "like a Möbius strip" or "two strips glued together" or similar things. The "4d" comes in because if you want to do this with a physical object, you need four spatial dimensions unless you're ok with it passing thru itself. Here's a good picture : http://web.ornl.gov/sci/ortep/topology/topo5.gif http://web.ornl.gov/sci/ortep/topology/topo5.gif The arrows indicate which edges to glue together, and how to line them up. See also bmm6o's comment here: https://news.ycombinator.com/item?id=11196493 https://news.ycombinator.com/item?id=11196493
    - catnaroek 11y ago
      Thanks!

deleted 11y ago

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