Is a non orientable 2 manifold in 4 space
Below is my attempt in intepret it (visualize in 4D)
klein bottle rendered in googlesketchup, with corners as it's difficult to make toroid shapes in the software
This is the view that gives me the idea to visualize it in 4D when I drawn that (with curved edgeds of course) on a piece of paper some few weeks ago.
The idea is that you have to compare this projection with the perspective projection of the tesseract.
Then you'll realise the fat "bulb" is actually a normal tube bent foward into 4 space (ana) and then bent upwards. The narrow "nose" area is not "within" the "bulb" but "behind" (kata) the bulb. It is first bent left to form a upside down U in 3D and then bent ana and then downwards to join up with the "bulb"
After I realized this, the intersection in the drawing and the render "disappears".
After understand how the ends connect together in 4D, I manage to simplify the classical projection into a toroid like shape
which looks like this render done by someone (found in a google search)
(P.S. I made that klein torus projection BEFORE I saw this)
and a simple object resembles a moebius shape (Later found similar representations on the internet)
from this moebius view, you'll notice how it has no outside and inside. The twist is done at the circle at the base of the object (which in 4D is just like a line).
So from here you'll understand this
wikipedia wrote:In comparison the Klein bottle is a mobius strip closed into a cylinder.
P.S. I didn't made the klein bagel projection as it is too complicated for googlesketchup to do