/sci/ - Science & Math

>>11491860
The 3D glass object you see people represent the Klein bottle with is not the object as mathematically defined.

The Klein bottle is defined as a manifold and as such has a tangent plane at each point.
Wheres the glass object you see has an ringlike self-intersection where, for all points of this intersection, you cannot define what exactly the tangent space should (it's not unique, there's two tangent planes). That is to say, the mathematical Klein bottle doesn't have any self-intersection in any embedding and wouldn't have a self-intersection in a faithful representation. As can be proven, the embedding can't be in R^3, we can proof that we can only coursely represent it (but not really) with that intersection.

>>11491860
The 3D glass object you see people represent the bottle with is not the object as mathematically defined. The Klein bottle is defined as a manifold and as such has a tangent plane at each point. Wheres the glass object has this intersection where, for points at the intersection, you can't define what exactly the tangent space is (it's not unique). That is to say, the Klein bottle doesn't wouldn't have any self-intersection in any embedding. (The embedding can't be in R^3, because there we can proof that we can only represent it with that intersection.)