Thread by @divbyzero

It's common to hear the following facts about Klein bottles:
1) To avoid self-intersections, it must live in at least 4-dimensional space.
2) Like a Möbius band, it's a one-sided surface.
The problem is, a Klein bottle living in 4-dimensional space is NOT one-sided! Why not? 1/13

The familiar picture of a Klein bottle is (using technical language) an "immersion" in 3-dimensional space. Everything is fine except where the bottle passes through itself—it intersects itself along a circle. In this immersion, it is not hard to see why it is called 2/13

one-sided. If we were to paint the surface, and we required a consistency of paint color everywhere, even as we pass through the self-intersection, we would not be able to do so using two colors. The entire surface would be the same color.

It's possible to prove that such 3/13

self-intersections are unavoidable in 3 dimensions. However, if we had 4 dimensions at our disposal, we could construct it without self-intersections (a so-called "embedding"). Intuitively, it could look almost like the picture in 3-d but where the self-intersection is about 4/13

to happen the surface "hops over itself" in the fourth dimension. We can visualize this by analogy: if we had two intersecting lines in the plane, we could avoid the intersection if we were allowed, even very briefly, to hop over the other line in the third dimension. 5/13

So, where are we? It makes sense to say that a Klein bottle immersed 3-dimensional space is one sided, and we can embed a Klein bottle in 4-dimensional space. The problem is that a Klein bottle in 4-dimensional space is not one-sided! In fact, the term "sides" doesn't have 6/13

any meaning! Let's think of a simpler example. A closed curve in the plane (a circle for example) is two sided. If you were standing at a point on the curve you could point (math version: give a "normal vector") in two directions. These are the two "sides." 7/13

However, imagine that curve lived in 3-dimensional space—like a rubber band held between your fingers. There's no inside or outside of the curve. At a point on the curve there are infinitely many directions for the normal vector to point, not just two. 8/13

A simple way to describe this is that a curve is locally 1 dimensional, and so in 2 dimensions, the normal vectors span a 2-1=1 dimensional space—the normal vectors span a line. However, in 3-dimensional space the normal vectors span a 3-1=2 dimensional plane. 9/13

Now let's talk about surfaces—objects that are locally 2-dimensional. If they live in 3-dimensional space, then at each point there two directions for a normal vector to point. This is true because 3-2=1. So we can talk about "sides." A sphere and a torus are 2-sided and a 10/13

Möbius band is 1-sided. What about the Klein bottle?Since the Klein bottle is a 2-dimensional surface and 3-2=1 the set of normal directions at a point on the Klein bottle in 3-dimensional space is 1-dimensional. So we can have the conversation about "sides" for an 11/13

immersed Klein bottle. But since 4-2=2, at a point on the Klein bottle in 4-dimensional space, the normal vectors span a 2-dimensional space. Just like the curve in 3-dimensional space, we can't even talk about "sides." So, in 4-d the Klein bottle is NOT one sided and not 12/13

two sided! It doesn't even have "sides"!
FYI I talk about this and more in (shameless plug) my book Euler's Gem, and that is where these pictures come from. 13/13