In the next video he talks about how a Klein bottle is made, and how, contrary to a bottle, it has no edge.
I'm not sure I understand why a bottle "has to" have an edge? Surely it's possible to make a bottle with no edge? For example, if one takes a sphere and progressively turns it into a bowl (by punching into it), and then makes the bowl deeper, it still has just one surface and no edge, no?
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.
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?
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.
Topology is an interesting area of study.
Can you do the same with your sphere/donut/bowl?
The abstract object doesn't, the 3d manifestation of it does.
It's just sealing the local minimum. (Maximum? I don't really know what the sign should be for this. Probably maximum, for enclosed volume, since I'm not sure if it's reasonable to talk about the topology having a minimum in this context...)
I don't think I can reason clearly about the 4 dimensional object, part of pointing out that a 3d manifestation does not have the same properties is me trying to do that.
http://www.amazon.com/Shape-Space-Chapman-Applied-Mathematic...
he says that a bootle has an edge (~3'40).
A sphere does not have an "inner surface", it has just one surface. So does a potato. If you go from sphere to potato to bowl, when does an edge appear and does it have to appear, is my question.
The inside and outside of a bottle is just by convention. Usually you would say something is inside the bottle if it is past the opening. (As he says in the video.)
So your example with the bowl-shaped "sphere" is correct. There's no clear limit when a bottle has an inside and an outside.