Abstract:
When you first heard about the Klein Bottle you might have been confounded by the fact that there is no good depiction of it. Why can’t people just draw a proper picture of the thing? One that does not self intersect? It turns out that it is impossible to “draw the Klein Bottle“ in \mathbb R ^3 without self intersections, and this is a math fun fact that people often throw around but that is rarely proven (even in lectures on topology). In this Colloquium I want to close this gap by providing an elementary self-contained proof that the Klein bottle (and many more objects) cannot be “drawn in 3-d space“ without self intersections.