4 ms·

> 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 ide

> 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.

vecter 6y ago
> This idea of an inner hole that is hidden and connected is quite unintuitive to me. What do you mean by the hole in the torus being "hidden and connected"?
- pontus 6y ago
  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.
- thunderbong 6y ago
  This is how I understood the comment - 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.
- AnHonestComment 6y ago
  There’s two holes: places you can draw a non-contractile circle, which can’t be turned into each other — “circumference” and “around”. I think they mean the “around” holes have a center “inside” the donut.
- ripe 6y ago
  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...
hammock 6y ago
If a torus's "hidden" hole is the one we are thinking of, does that also mean a sphere has 1 hole, a similarly "hidden" one?

vecter 6y ago

> This idea of an inner hole that is hidden and connected is quite unintuitive to me. What do you mean by the hole in the torus being "hidden and connected"?

pontus 6y ago
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.
thunderbong 6y ago
This is how I understood the comment - 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.
AnHonestComment 6y ago
There’s two holes: places you can draw a non-contractile circle, which can’t be turned into each other — “circumference” and “around”. I think they mean the “around” holes have a center “inside” the donut.
ripe 6y ago
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...

hammock 6y ago

If a torus's "hidden" hole is the one we are thinking of, does that also mean a sphere has 1 hole, a similarly "hidden" one?