Klein Bottle self intersecting in $\mathbb{R^3}$ and not in $\mathbb{R^4}$

Question

The Klein bottle is a surface that has an oval of self-intersection when it is shown in 3-space. It can live in 4-space with no self-intersection. How?

I'm having a hard time approaching how to solve this. Any ideas or advice would be greatly appreciated.

Answer 1

Consider the surface $S$ of a sphere of radius $r$ in $3$-dimensional space. Let $C$ be its center and let $P$ be a point of $\mathbb{R}^3$ whose distance to $C$ is greater than $r$. Can we go from $C$ to $P$ along a continuous path in $\mathbb{R}^3$ which does not intersect $S$? No, we can't.

But in $\mathbb{R}^4$ we can. To simplify, suppose that $r=1$, that $C=(0,0,0)$ and that $P=(2,0,0)$. The sphere is$$S=\left\{(x,y,z)\in\mathbb{R}^3\,\middle|\,x^2+y^2+z^2=1\right\}.$$ Now we will add a fourth coordinate $t$. Now $C=(0,0,0,0)$, $P=(2,0,0,0)$, and$$S=\left\{(x,y,z,0)\in\mathbb{R}^4\,\middle|\,x^2+y^2+z^2=1\right\}.$$Now, can we go from $C$ to $P$ along a continuous path in $\mathbb{R}^3$ which does not intersect $S$? Yes, we can! First, we go from $(0,0,0,0)$ to $(0,0,0,1)$ through the points of the form $(0,0,0,t)$ and then we go from $(0,0,0,1)$ to $(1,0,0,0)$ through the points of the form $(t,0,0,1-t)$ (in both cases with $t\in[0,1]$).

When we add an extra dimension, we have more room to move around. What happens with the Klein bottle is similar: in $3$ dimensions it must have self-intersections, but in $4$ dimensions we can avoid them.

Answer 2

It might be helpful to think in lower dimensions for an analogy. For example, consider a trefoil knot:

This object lives in three dimensions with no self intersections. However, if you try to smoosh it down (this is a technical term) into two dimensions, you get something like this:

Notice that this has three self-intersections. Something similar is going on with the Klein bottle. It is a native of four-dimensional space. When you try to look at it in three dimensional space (i.e. if you look at projections of the Klein bottle into $\mathbb{R}^3$), it self-intersects.

NB: Both images are blatantly stolen from Wikipedia (with some minor editing of the second).