Would it be manifold if we split it into just the cone?
Max
Yes - this shape could also be represented by two manifolds (top and bottom), but then we lose the connectivity at their intersection.
RyuK
Is there some sort of mathematical definition that can classify shapes as a manifold or not?
hmm
@RyuK I think the definition given on the previous slide can be made mathematically precise. See https://en.wikipedia.org/wiki/Manifold
Informal: If you zoom in far enough at any point, it looks like a plane
Precise: Every point has a neighborhood (this is the zoom in far enough part) that is homeomorphic to (looks like/can be transformed into without ripping or cutting) Euclidean space of dimension n (a plane, since n = 2 for our purposes).
OillyNoodle
In lecture prof. used a piece of paper as an illustration - if we can somehow approximate the surface using that paper, than it is manifold
Would it be manifold if we split it into just the cone?
Yes - this shape could also be represented by two manifolds (top and bottom), but then we lose the connectivity at their intersection.
Is there some sort of mathematical definition that can classify shapes as a manifold or not?
@RyuK I think the definition given on the previous slide can be made mathematically precise. See https://en.wikipedia.org/wiki/Manifold
Informal: If you zoom in far enough at any point, it looks like a plane
Precise: Every point has a neighborhood (this is the zoom in far enough part) that is homeomorphic to (looks like/can be transformed into without ripping or cutting) Euclidean space of dimension n (a plane, since n = 2 for our purposes).
In lecture prof. used a piece of paper as an illustration - if we can somehow approximate the surface using that paper, than it is manifold