Okay, if a torus with a single patch of empty space on its surface is two holes, then this is three holes. But now I’m confused, because I don’t know if that’s true topologically. Otherwise, it’s just a two-torus with a perforation on its 2-surface.
Edit: I’ve been informed by a topologist friend that this has two holes and that the lip of the mug is just a boundary.
Edit 2: So to clarify, that’s a boundary if the lip represents an abrupt transition to either side (i.e. if there’s no “curve” into and out of the mug).
Since that’s less realistic, let’s assume that you can smoothly walk from the interior surface of the mug to the exterior and show that it isn’t topologically a hole for that case either.
I’ve created a continuous loop around the two holes and the candidate “lip hole”. These loops are embedded in the surface of the mug. You can plainly see that you can’t contract the loop to a single point around the handle hole or the donut hole without crossing over the hole.
But now let’s take our blue loop (the lip loop) and trivially contract it down to a point.
Our loop is on the outside around the lip.
Take the loop inside of the lip since this is still a smooth, continous surface. The loop is now on the outer wall of the inner mug.
Now bring the loop down toward the bottom along the wall.
Contract the loop to a single point at the bottom.
In this case, it isn’t even a boundary anymore, and it provably isn’t a hole.
The hole through the center makes a hole around it inside the mug.
Yeah, the hole through the center is one hole (the “donut hole”). The handle is the other (the “handle hole”).
Topologically, the lip of the mug is a boundary, not a hole. This object is of genus 2 given it’s a 2-manifold.
But we can show this assuming the lip isn’t an abrupt boundary and that instead that the outer wall of the inner loop is part of the same side as the outside of the mug (i.e. that you can continuously walk on the surface from the outside past the lip without crossing an abrupt boundary, which more closely matches the physical reality of this mug).
Watch this: imagine a loop around the hole of a donut. Now try to contract that loop down. Without cutting it, you can’t.
Now take a loop around the lip of this coffee mug. You can contract it to a point by bringing it all over the lip of the mug, then inside, along the outer walls, and then finally down to the bottom of the mug. It all meets up there. No hole. There’s the donut hole and the handle hole, but the colloquial “hole” is either a boundary if the lip is flat or not even a boundary if it continuously curves inward.
Now take a loop around the lip of this coffee mug. You can contract it to a point by bringing it all over the lip of the mug, then inside, along the outer walls, and then finally down to the bottom of the mug.
Topologically, that part is not a hole. The liquid basin not being a hole is where the joke in the OP originates, even: a standard coffee mug with a basin for coffee and a handle is homeomorphic (topologically equivalent) to a torus (a donut). And that’s because the handle is the hole, not the basin.
In the case of the OP, ignore the handle for a second just to simplify things. That’s a hollowed-out donut (torus) except that you’ve taken a part of that donut’s surface and cut it out. That area that’s been cut out isn’t topologically a hole; instead, you’ve just created a boundary on the 2-manifold (read: flexible surface).
Topology has rigorous, algebraic definitions under the hood, but that’s what’s going on in this picture topologically: you’ve taken a donut, glued it to another donut, and cut a “hole” (colloquial usage) in the double-donut’s surface, creating a boundary on the surface.
Edit: To hopefully justify this a bit intuitively for the standard coffee mug (not this abomination in the OP), the utilities problem is a classic toy problem in topology that
Tap for spoiler
is unsolvable on a 2D plane
but
Tap for spoiler
can be solved when you embed it into the surface of a coffee mug.
If we ignore the handle, the torus with a surface opening can be deformed continuously into an 8 shape, the lip of the mug is not a hole, but you can go down the lip of the mug, around the tube in the middle, and back out, a fully enclosed path
Okay, if a torus with a single patch of empty space on its surface is two holes, then this is three holes. But now I’m confused, because I don’t know if that’s true topologically. Otherwise, it’s just a two-torus with a perforation on its 2-surface.
Edit: I’ve been informed by a topologist friend that this has two holes and that the lip of the mug is just a boundary.
Edit 2: So to clarify, that’s a boundary if the lip represents an abrupt transition to either side (i.e. if there’s no “curve” into and out of the mug).
Since that’s less realistic, let’s assume that you can smoothly walk from the interior surface of the mug to the exterior and show that it isn’t topologically a hole for that case either.
I’ve created a continuous loop around the two holes and the candidate “lip hole”. These loops are embedded in the surface of the mug. You can plainly see that you can’t contract the loop to a single point around the handle hole or the donut hole without crossing over the hole.
But now let’s take our blue loop (the lip loop) and trivially contract it down to a point.
Our loop is on the outside around the lip.
Take the loop inside of the lip since this is still a smooth, continous surface. The loop is now on the outer wall of the inner mug.
Now bring the loop down toward the bottom along the wall.
Contract the loop to a single point at the bottom.
In this case, it isn’t even a boundary anymore, and it provably isn’t a hole.
It’s three. The hole through the center makes a hole around it inside the mug. Your topologist friend took too quick a glance at this.
Yeah, the hole through the center is one hole (the “donut hole”). The handle is the other (the “handle hole”).
Topologically, the lip of the mug is a boundary, not a hole. This object is of genus 2 given it’s a 2-manifold.
But we can show this assuming the lip isn’t an abrupt boundary and that instead that the outer wall of the inner loop is part of the same side as the outside of the mug (i.e. that you can continuously walk on the surface from the outside past the lip without crossing an abrupt boundary, which more closely matches the physical reality of this mug).
Watch this: imagine a loop around the hole of a donut. Now try to contract that loop down. Without cutting it, you can’t.
Now take a loop around the lip of this coffee mug. You can contract it to a point by bringing it all over the lip of the mug, then inside, along the outer walls, and then finally down to the bottom of the mug. It all meets up there. No hole. There’s the donut hole and the handle hole, but the colloquial “hole” is either a boundary if the lip is flat or not even a boundary if it continuously curves inward.
Nope, donut hole is in the way.
I still don’t get how it’s supposed to be two holes. I mean how is the part where the liquid would be in in this cup not a hole?
Topologically, that part is not a hole. The liquid basin not being a hole is where the joke in the OP originates, even: a standard coffee mug with a basin for coffee and a handle is homeomorphic (topologically equivalent) to a torus (a donut). And that’s because the handle is the hole, not the basin.
In the case of the OP, ignore the handle for a second just to simplify things. That’s a hollowed-out donut (torus) except that you’ve taken a part of that donut’s surface and cut it out. That area that’s been cut out isn’t topologically a hole; instead, you’ve just created a boundary on the 2-manifold (read: flexible surface).
Topology has rigorous, algebraic definitions under the hood, but that’s what’s going on in this picture topologically: you’ve taken a donut, glued it to another donut, and cut a “hole” (colloquial usage) in the double-donut’s surface, creating a boundary on the surface.
Edit: To hopefully justify this a bit intuitively for the standard coffee mug (not this abomination in the OP), the utilities problem is a classic toy problem in topology that
Tap for spoiler
is unsolvable on a 2D plane
but
Tap for spoiler
can be solved when you embed it into the surface of a coffee mug.
If we ignore the handle, the torus with a surface opening can be deformed continuously into an 8 shape, the lip of the mug is not a hole, but you can go down the lip of the mug, around the tube in the middle, and back out, a fully enclosed path
see I figured it was two, being that I’ve seen the transformation of a coffee mug into a torus, I know the lip can be basically “flattened”