From ‘I’ (a point: 0-D {nothingness, emptiness}) to ‘Other’ (a line: 1-D {linearity, bivalence}) to ‘I  Other’ (a plane: 2-D {possibly 3-valued}) to ‘Community’ (a cube: 3-D {many-valued}) to ‘Cosmos’ (a hypercube: 4:D {potentially  - valued})

The Klein bottle is an unorientable surface. It can be constructed by gluing together the two ends of a cylindrical tube by protruding one end through the tube and connecting it with the other end (while simultaneously inflating the tube at this second end). The resulting picture looks something like this:

However, the result is not a true picture of the Klein bottle, since it depicts a self- intersection which isn't really there (in other words, there should be no discontinuity; the surface should be continuous throughout). The Klein bottle, in contrast to its limited 3-dimensional, can easily be realized in 4-dimensional space: one lifts up the narrow part of the tube in the direction of the 4-th coordinate axis just as it is about to pass through the thick part of the tube, then drops it back down into 3-dimensional space inside the thick part of the tube.

However there is no need to go through the mental contortions of visualizing the Klein bottle in 4-dimensional space, if we adopt the intrinsic point of view we developed for dealing with the Möbius strip. We do not attempt to physically realize the gluing described above, but rather think of it as an abstract gluing, imagining how the resulting space would look to a 2-dimensional crab swimming within the surface of the Klein bottle. This leads us to the following convenient model of the Klein bottle:

To relate this to our previous description of the Klein bottle, note that the gluing instructions tell us to glue the top and bottom edges of the rectangle. The result is a cylindrical tube with the left and right edges forming the two circular ends of the tube. The gluing instructions then tell us to glue the two ends of the tube with a twist. Note also that in creating the Möbius- band the gluing instructions tells to glue all four corners of the rectangle into a single point.

Our friend Ms Triangle, is, when navigating along the band, Either Outside or Inside, or Both Inside and Outside, or Neither Inside nor Outside, however we wish to define her. Or, we might define her in another manner, as…

The containing-contained-uncontained is from our view of the strip from a 3-D viewpoint. For the flatlander, her trajectory is 1-D, whereas we perceive 2-D surfaces. Her discontinuous point at Contained 1-D space would not be perceived, unless she were to make a point-hole in her 2-D plane in order to construct a Möbius-strip.

We would have to do the same—make a 2- D hole in our planar surface in order to construct a Klein-bottle. But a Hyperspherelander could see it all from here perspective, like we can see it all in Flatlander’s world from our own Spherelander perspective. Actually, a Klein-bottle can be constructed from two Möbius-bands, on right-handed and the other left-handed, like this…

In this manner, just as the ‘twist’ in the 2-D Möbius-band can be created only within 3-D space, so also the ‘hole’ in the Klein-bottle can be created only within 4-D space.