What does it mean to "cut" something with a sphere? Since a sphere is a three-dimensional object, does this entail taking a chunk out of whatever we're cutting?

brian-stone

@lawrenceyan Essentially, the result of "cutting" a surface with a sphere is simply the set of points in the surface that are within the sphere. In the case of a manifold, we can always choose a sphere small enough such that there are no points on the surface for which cutting does not yield a disk, whatever that means (based on the slide, I'd guess it means there are no fragments of the cut surface that are connected only by a point, or that are connected to more than one fragment by the same face).

ziyaointl

I wonder what classes should one take if one want's to understand the rigorous mathematical definition of a manifold and how it is equivalent to this sphere-cutting definition.

IsabelCDaniels

Is someone able to explain what the "with border" is trying to explain in this image?

FLinesse

@IsabelCDaniels I was also confused but I found this slide from an equivalent graphics course from CMU that's pretty helpful

What exactly does "yield a disk" entail? For example, in the pyramid example, isn't that more a 3-d cut of a sphere?

chsean

Maybe the with border label means that surfaces whose boundaries look like the surfaces on the left are allowed to be considered manifold even though cutting with a sphere may not yield a disk?

SofieHerbeck

@et-yao my math-major roommate has explained to me that topology doesn't care about distances, so my understanding of what the whole 'yield a disk' idea entails is that we're looking at situations where a cut gives something that could be considered a disk if you just looked at it in a 2D way, removing the depth-distance that makes it look less like a disk. In the pyramid example you cited, looking at the cut shape from above would look like a disk, without the added depth of the diagram on this slide, so it is manifold. For the non-manifold examples, there is no way in which you could look at the cut shapes and see them as a disk, because they're missing chunks.

hfan9

I haven't taken a topology class, but from what I know you're right about "distances" not mattering per se. I usually imagine these objects as made out of play-doh, and topologically equivalent if you can mold it without closing or creating new holes. For the pyramid, you kinda just slightly tilt the flaps up and you'll have a disk; with the borders, it's just that some of the disk is cut off. For the non-manifolds, the issue is that you have a 2D object arranged as flaps. You aren't allowed to mold the flaps together (like how you can't fill in holes), and you can't squish it without breaking the 2D part.

What does it mean to "cut" something with a sphere? Since a sphere is a three-dimensional object, does this entail taking a chunk out of whatever we're cutting?

@lawrenceyan Essentially, the result of "cutting" a surface with a sphere is simply the set of points in the surface that are within the sphere. In the case of a manifold, we can always choose a sphere small enough such that there are no points on the surface for which cutting does not yield a disk, whatever that means (based on the slide, I'd guess it means there are no fragments of the cut surface that are connected only by a point, or that are connected to more than one fragment by the same face).

I wonder what classes should one take if one want's to understand the rigorous mathematical definition of a manifold and how it is equivalent to this sphere-cutting definition.

Is someone able to explain what the "with border" is trying to explain in this image?

@IsabelCDaniels I was also confused but I found this slide from an equivalent graphics course from CMU that's pretty helpful

http://15462.courses.cs.cmu.edu/fall2018/lecture/meshes/slide_013

What exactly does "yield a disk" entail? For example, in the pyramid example, isn't that more a 3-d cut of a sphere?

Maybe the with border label means that surfaces whose boundaries look like the surfaces on the left are allowed to be considered manifold even though cutting with a sphere may not yield a disk?

@et-yao my math-major roommate has explained to me that topology doesn't care about distances, so my understanding of what the whole 'yield a disk' idea entails is that we're looking at situations where a cut gives something that could be considered a disk if you just looked at it in a 2D way, removing the depth-distance that makes it look less like a disk. In the pyramid example you cited, looking at the cut shape from above would look like a disk, without the added depth of the diagram on this slide, so it is manifold. For the non-manifold examples, there is no way in which you could look at the cut shapes and see them as a disk, because they're missing chunks.

I haven't taken a topology class, but from what I know you're right about "distances" not mattering per se. I usually imagine these objects as made out of play-doh, and topologically equivalent if you can mold it without closing or creating new holes. For the pyramid, you kinda just slightly tilt the flaps up and you'll have a disk; with the borders, it's just that some of the disk is cut off. For the non-manifolds, the issue is that you have a 2D object arranged as flaps. You aren't allowed to mold the flaps together (like how you can't fill in holes), and you can't squish it without breaking the 2D part.