The author knows how to hook you right in the first paragraph:
>> If you’re looking to pick a fight, simply ask your friends, “Is Pluto a planet?” Or “Is a hotdog a sandwich?” Or “How many holes does a straw have?” The first two questions will have them arguing yay or nay, while the third yields claims of two, one and even zero.
> A hollow torus can be cut twice — once around the tube and then along the resulting cylinder — so by this definition, it has two holes.
Fascinating! This idea of an inner hole that is hidden and connected is quite unintuitive to me. I majored in mathematics in college but I wouldn't have reached this conclusion through intuition.
I suspect what was meant was that it's only apparent if you can view the torus from "a bird's eye view". If you were an ant walking on the surface at night, you would never run into a hole in the usual sense. You would be hard pressed to find a way to distinguish the surface from that of a sphere.
As an ant you could easily see that you were not on the surface of an infinite plane simply because you keep coming back to the same spot over and over again. In order to figure out that there was a hole (or more appropriately named as 'handle'), you'd need to leave some thread behind and notice that there was no way to contract various loops to a point.
1. the center hole around which there is a cylindrical ring
2. the whole inside the cylindrical ring, which is hidden and connected.
When we cut along the length of the cylindrical ring, we are effectively creating two edges, just like if we were to cut a circular ring of wire, we would end up with two points at the end.
The two edges then open up, effectively forming another cylinder, which has to be cut again.
Any other object which has only a single hole, would require only a single cut to flatten out.
This discussion confused me until I looked up "torus" and found there are really two different definitions of this word:
(a) a SURFACE generated by a circle rotated about a coplanar axis outside the circle. Examples: an inner tube or a swimming tube. (This is what's meant by "a hollow torus" in the description).
(b) a SOLID generated by a circular lamina rotated about a coplanar axis outside the lamina. Example: a bagel or an annular gasket. I was thinking of this one, and the descriptions made no sense at all.
Now it all makes sense.
Hmm. I thought before this that I knew what a torus was...
So does this mean that a sock has ZERO holes, or HALF a hole? Because if you were to cut the sock to flatten it, your cut would only have to go from the top to the bottom of the sock, but not back up again - it's not a "complete cut."
Note: If you are having trouble visualizing this cut with a sock, replace it with a paper cone.
A cone surface is topologically equivalent to a disk. You don't have to cut it at all in order to flatten it, if you allow as much stretching and bending as you want.
So, 0 holes.
If you want to consider a more restrictive collection of modifications, like, say you want to preserve angles or something inside the shape, then that would be a different question.
Uh, I guess if you had a surface which was generally smooth but possibly has some boundaries and/or corner edge things, and/or corner point things (e.g. sticking two cones together), you could also ask about how many cuts combined with conformal maps [edit : I think conformal maps isn't the thing I'm looking for. Yeah, I think I'm looking for something related to developable surfaces maybe] would be needed to flatten such a thing out? (so this could maybe produce a similar question for which the one or half or whatever cut between the edge of the cone and the center, could be an answer?)
That makes sense, I didn't realize we were allowed to stretch and bend however we want. Seems weird to be allowed to take a 3D object and contort it into 2D.
But it is not a 3D object, it is in fact a 2D object embedded in the 3D space. (So, what you are looking at is the result of an arbitrary “projection” of one into the other - a map, in mathematical terms.)
From the topological standpoint, you do not need to cut the sock for this - it is already “the same as” (homeomorphic to) a piece of the plane surface.
Imagine its made of putty instead of fiber. See if you can squish and stretch its surface in your mind into a sphere or flat shape without tearing it. You'll see the sock is homeomorphic with those shapes and there are 0 holes.
I think this is a pretty good intuitive introduction.
I think there’s a bit of muddling between the fundamental group and first homology group with the article: the algebra of the fundamental group where loops are merged together cares about the order you do each loop and I don’t think it’s intuitive that a + b = b + a for the torus, indeed I found this quite unintuitive when I learned it. Note that the group isn’t always commutative. On the other hand the elements in the homologous group are, to some extent, superpositions of countable many loops and so the group will always be commutative (as far as I know). The direction of those loops matters but the order does not. That said, I still think the article is ok for the level it is aimed at.
Another complaint is that I feel like it can be read in this weird way where topology is all about defining what precisely a hole is. Instead I would say that the things here (Betti numbers, generators of the fundamental group, etc) are merely tools which topologies have found to be convenient in topology and which have been called holes as they roughly correspond to the colloquial meaning of the word.
what follows is a motivation but not a history. It is also biased by the way I learned topology.
For an example, a nice early question in topology is: Why is R^1, the line, different from R^2, the plane. Here different means “not homeomorphic” which has a slightly involved definition. The answer comes in two parts: firstly that if you remove a point from a line you split it in two while that is not true of the plane, and secondly is that connectedness (whether something is split in two) is invariant under homeomorphism (and some boring things about the definition of a homeomorphism). Therefore the two spaces cannot be homemorphic.
Here we have, in some sense used the nature of a hole to answer a question about topology. But here are two questions that it doesn’t answer:
1. Are R^2 and R^3 different?
2. Are R^2, the plane, and S^2, the surface of a sphere, different?
3. If you remove a point from R^2, is that different from removing two points?
Removing single or multiple points doesn’t really help in these. And dealing with lines leads to problems (who’s to say that removing a line from R^2 won’t correspond to removing a plane from R^3, breaking the logic above? The sets of points being removed are uncountably infinite in both cases.) The solution to the first two problems is simply-connectedness: a space is simply connected if you can find a line between any two points and you can, in some formal sense, slide any line between those points into any other line between those two points. R^2 and S^2 are simply connected. But if you remove a point from R^2 it is not simply connected as the line that does a loop around the missing point cannot be slid around into a line that doesn’t do the loop. If you remove a point from S^2 it is still simply connected (if you have a loop around the missing point just slide it to the other side of the sphere where it can be unraveled). The same argument works for the other problem as if you remove a point from R^3 it is still simply connected.
For the last problem these ideas are not sufficient. But following similar techniques leads to the definition of the fundamental group (roughly the algebra of adding up loops discussed in the article) which can solve that problem. The next problem to solve is to show that R^3 and R^4 are different. This can motivate homology (maybe you could get away with removing a line to make R^3 not simply connected but that sounds hard. And it won’t help much with the general case of showing that R^n and R^m are different if n and m are.)
In these cases there are things that might be talked about as holes being measured but the goal isn’t to define what a hole is but rather to answer questions about topology.
Although you are right about the plane and sphere, a much earlier proof of them being not homeomorphic is obtained by simply stating one is compact while the other is not.
21 comments
[ 2.8 ms ] story [ 52.9 ms ] thread>> If you’re looking to pick a fight, simply ask your friends, “Is Pluto a planet?” Or “Is a hotdog a sandwich?” Or “How many holes does a straw have?” The first two questions will have them arguing yay or nay, while the third yields claims of two, one and even zero.
Two, one, or zero? Continues reading...
Fascinating! This idea of an inner hole that is hidden and connected is quite unintuitive to me. I majored in mathematics in college but I wouldn't have reached this conclusion through intuition.
What do you mean by the hole in the torus being "hidden and connected"?
As an ant you could easily see that you were not on the surface of an infinite plane simply because you keep coming back to the same spot over and over again. In order to figure out that there was a hole (or more appropriately named as 'handle'), you'd need to leave some thread behind and notice that there was no way to contract various loops to a point.
In a torus, there are effectively two holes -
1. the center hole around which there is a cylindrical ring
2. the whole inside the cylindrical ring, which is hidden and connected.
When we cut along the length of the cylindrical ring, we are effectively creating two edges, just like if we were to cut a circular ring of wire, we would end up with two points at the end.
The two edges then open up, effectively forming another cylinder, which has to be cut again.
Any other object which has only a single hole, would require only a single cut to flatten out.
I think they mean the “around” holes have a center “inside” the donut.
(a) a SURFACE generated by a circle rotated about a coplanar axis outside the circle. Examples: an inner tube or a swimming tube. (This is what's meant by "a hollow torus" in the description).
(b) a SOLID generated by a circular lamina rotated about a coplanar axis outside the lamina. Example: a bagel or an annular gasket. I was thinking of this one, and the descriptions made no sense at all.
Now it all makes sense.
Hmm. I thought before this that I knew what a torus was...
Note: If you are having trouble visualizing this cut with a sock, replace it with a paper cone.
If you want to consider a more restrictive collection of modifications, like, say you want to preserve angles or something inside the shape, then that would be a different question. Uh, I guess if you had a surface which was generally smooth but possibly has some boundaries and/or corner edge things, and/or corner point things (e.g. sticking two cones together), you could also ask about how many cuts combined with conformal maps [edit : I think conformal maps isn't the thing I'm looking for. Yeah, I think I'm looking for something related to developable surfaces maybe] would be needed to flatten such a thing out? (so this could maybe produce a similar question for which the one or half or whatever cut between the edge of the cone and the center, could be an answer?)
From the topological standpoint, you do not need to cut the sock for this - it is already “the same as” (homeomorphic to) a piece of the plane surface.
Likewise see if you can imagine squishing and stretching a putty coffee mug or a straw into a torus. They both have 1 hole. Like this: https://i.ytimg.com/vi/9NlqYr6-TpA/maxresdefault.jpg
I think there’s a bit of muddling between the fundamental group and first homology group with the article: the algebra of the fundamental group where loops are merged together cares about the order you do each loop and I don’t think it’s intuitive that a + b = b + a for the torus, indeed I found this quite unintuitive when I learned it. Note that the group isn’t always commutative. On the other hand the elements in the homologous group are, to some extent, superpositions of countable many loops and so the group will always be commutative (as far as I know). The direction of those loops matters but the order does not. That said, I still think the article is ok for the level it is aimed at.
Another complaint is that I feel like it can be read in this weird way where topology is all about defining what precisely a hole is. Instead I would say that the things here (Betti numbers, generators of the fundamental group, etc) are merely tools which topologies have found to be convenient in topology and which have been called holes as they roughly correspond to the colloquial meaning of the word.
what follows is a motivation but not a history. It is also biased by the way I learned topology.
For an example, a nice early question in topology is: Why is R^1, the line, different from R^2, the plane. Here different means “not homeomorphic” which has a slightly involved definition. The answer comes in two parts: firstly that if you remove a point from a line you split it in two while that is not true of the plane, and secondly is that connectedness (whether something is split in two) is invariant under homeomorphism (and some boring things about the definition of a homeomorphism). Therefore the two spaces cannot be homemorphic.
Here we have, in some sense used the nature of a hole to answer a question about topology. But here are two questions that it doesn’t answer:
1. Are R^2 and R^3 different?
2. Are R^2, the plane, and S^2, the surface of a sphere, different?
3. If you remove a point from R^2, is that different from removing two points?
Removing single or multiple points doesn’t really help in these. And dealing with lines leads to problems (who’s to say that removing a line from R^2 won’t correspond to removing a plane from R^3, breaking the logic above? The sets of points being removed are uncountably infinite in both cases.) The solution to the first two problems is simply-connectedness: a space is simply connected if you can find a line between any two points and you can, in some formal sense, slide any line between those points into any other line between those two points. R^2 and S^2 are simply connected. But if you remove a point from R^2 it is not simply connected as the line that does a loop around the missing point cannot be slid around into a line that doesn’t do the loop. If you remove a point from S^2 it is still simply connected (if you have a loop around the missing point just slide it to the other side of the sphere where it can be unraveled). The same argument works for the other problem as if you remove a point from R^3 it is still simply connected.
For the last problem these ideas are not sufficient. But following similar techniques leads to the definition of the fundamental group (roughly the algebra of adding up loops discussed in the article) which can solve that problem. The next problem to solve is to show that R^3 and R^4 are different. This can motivate homology (maybe you could get away with removing a line to make R^3 not simply connected but that sounds hard. And it won’t help much with the general case of showing that R^n and R^m are different if n and m are.)
In these cases there are things that might be talked about as holes being measured but the goal isn’t to define what a hole is but rather to answer questions about topology.