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

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

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

FIGURE 23

FIGURE 23

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

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

FIGURE 5MöBIUS LINE, MöBIUS BAND

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

FIGURE 5MöBIUS LINE, MöBIUS BAND

Contained1-D space Containing

2-D spaceUncontained

3-D space

We would have to do the same—make a We would have to do the same—make a 2-D hole in our planar surface in order 2-D hole in our planar surface in order to construct a Klein-bottle. But a to construct a Klein-bottle. But a Hyperspherelander could see it all from Hyperspherelander could see it all from here perspective, like we can see it all here perspective, like we can see it all in Flatlander’s world from our own in Flatlander’s world from our own Spherelander perspective.Spherelander perspective.

Actually, a Klein-bottle can be Actually, a Klein-bottle can be constructed from two Möbius-bands, on constructed from two Möbius-bands, on right-handed and the other left-handed, right-handed and the other left-handed, like this…like this…

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