3 ms·

The rim of your bowl is the edge: it's a point where the curvature changes suddenly. But that's not the important thing: the important thing is that not only is

The rim of your bowl is the edge: it's a point where the curvature changes suddenly. But that's not the important thing: the important thing is that not only is there only no edge, but only one surface. Let's talk topology for a bit. A manifold can be thought of as essentially a flat and infinitely thin sheet. If you have a spherical manifold (and your bowl would be a spherical manifold), it has two surfaces: one on the inside and another on the outside. Deforming the manifold doesn't change this. The thing about a Klein bottle (and a Moebius strip) is that it only has one continuous surface, the difference between a Moebius strip and a Klein bottle is that the latter is a single surface with no edge whereas the former is a single surface with one edge, unlike the sphere (or your bowl), which has two surfaces and no edge. So what's important here is the number of sides it has, not really that it has no edge. The lack of an edge is really only what separates a Moebius strip from Klein bottle.

bambax 11y ago
But with an ordinary bottle, the edge is the point (if I may), because if a bottle had no edge, couldn't it be said that has just one ordinary surface? So in fact, what I understand so far is: - a (hollow) sphere has two surfaces, one inside and one outside, and you can't connect one to the other (you can't walk from one to the other) - a potato, or a bowl also have the same properties: two distinct surfaces - a bottle is not a special kind of bowl, it's more of a broken sphere (an egg the top of which has been removed) so that both surfaces (in and out) are accessible (but disjoint) - to make a true bottle out of a bowl one needs to somehow "collapse" the inner and outer surfaces (therefore creating an edge) - if you make a bottle that has a hollow space between the inside and the outside (think Thermos), then you actually have three surfaces Is this correct?
- talideon 11y ago
  I think the confusion here is that you're taking what he's saying in terms other than those of topology. In topology, you deal with manifolds. Manifolds have no thickness, but they do have area. In topology, there's no such thing as a 'filled' sphere: a sphere in topology has no edges and an inside surface and an outside surface, much like a football. There are two ways you can make a 'bottle': you can deform a sphere, which gives you a 'bottle' much like a vaccuum flask; or you can deform a planar manifold (which is like a sheet of paper). The former will have two surfaces and no edge, whereas the latter will have two surfaces and one edge, that being the rim. A broken sphere would be a planar manifold. A vaccuum flask is simply a deformed sphere in topological terms, and thus only has two surfaces, the inside and the outside.
  - bambax 11y ago
    Much helpful, thanks!!
    - talideon 11y ago
      One quick clarifying note: the reason he said in the video that bottles have an edge is because he was treating it as a planar manifold. Topology is an interesting area of study.
  - david-given 11y ago
    ...a topologist is someone who can't figure out whether to dip their doughnut into their coffee mug or vice versa.
- escherplex 11y ago
  Yeah but last time I heard, a Klein bottle being only a one sided surface can only be rendered in 4D space with the 'passing through' funnel not breaking or intersecting the curved surface and this can't be rendered in 3D space. Unless recently they've decided to relax that restriction. Nice model though and worth getting but I wonder how best to mount it in a display case.
  - talideon 11y ago
    Yeah, the 3D klein bottle is to a real 4D klein bottle what a circle is to a sphere.
    - escherplex 11y ago
      IE, cross-sections or projections. Would be interesting if there were a way to render a Klein bottle using techniques similar to those used when 'impossible triangles' are modeled with boards and 2x4s in fields which then seem to work when viewed from a certain angle. But then what would non-intersecting intersecting lines look like. Sounds like double-think.

bambax 11y ago

But with an ordinary bottle, the edge is the point (if I may), because if a bottle had no edge, couldn't it be said that has just one ordinary surface? So in fact, what I understand so far is: - a (hollow) sphere has two surfaces, one inside and one outside, and you can't connect one to the other (you can't walk from one to the other) - a potato, or a bowl also have the same properties: two distinct surfaces - a bottle is not a special kind of bowl, it's more of a broken sphere (an egg the top of which has been removed) so that both surfaces (in and out) are accessible (but disjoint) - to make a true bottle out of a bowl one needs to somehow "collapse" the inner and outer surfaces (therefore creating an edge) - if you make a bottle that has a hollow space between the inside and the outside (think Thermos), then you actually have three surfaces Is this correct?

talideon 11y ago
I think the confusion here is that you're taking what he's saying in terms other than those of topology. In topology, you deal with manifolds. Manifolds have no thickness, but they do have area. In topology, there's no such thing as a 'filled' sphere: a sphere in topology has no edges and an inside surface and an outside surface, much like a football. There are two ways you can make a 'bottle': you can deform a sphere, which gives you a 'bottle' much like a vaccuum flask; or you can deform a planar manifold (which is like a sheet of paper). The former will have two surfaces and no edge, whereas the latter will have two surfaces and one edge, that being the rim. A broken sphere would be a planar manifold. A vaccuum flask is simply a deformed sphere in topological terms, and thus only has two surfaces, the inside and the outside.
- bambax 11y ago
  Much helpful, thanks!!
  - talideon 11y ago
    One quick clarifying note: the reason he said in the video that bottles have an edge is because he was treating it as a planar manifold. Topology is an interesting area of study.
- david-given 11y ago
  ...a topologist is someone who can't figure out whether to dip their doughnut into their coffee mug or vice versa.
escherplex 11y ago
Yeah but last time I heard, a Klein bottle being only a one sided surface can only be rendered in 4D space with the 'passing through' funnel not breaking or intersecting the curved surface and this can't be rendered in 3D space. Unless recently they've decided to relax that restriction. Nice model though and worth getting but I wonder how best to mount it in a display case.
- talideon 11y ago
  Yeah, the 3D klein bottle is to a real 4D klein bottle what a circle is to a sphere.
  - escherplex 11y ago
    IE, cross-sections or projections. Would be interesting if there were a way to render a Klein bottle using techniques similar to those used when 'impossible triangles' are modeled with boards and 2x4s in fields which then seem to work when viewed from a certain angle. But then what would non-intersecting intersecting lines look like. Sounds like double-think.