Two visual structures that I use a lot are the Möbius strip and the Klein bottle because they embody a paradox. Specifically, they have only one side although it seems to be two sides. That concept is very important for what’s to come, which is why I am introducing it early on and without other content. So without further adieu, let me introduce to you, first, the Möbius strip.

**The Möbius Strip**** **

The Möbius strip, named after August Ferdinand Möbius, is a two-dimensional surface that requires three dimensions for its existence.

But the interesting thing about it is that it has only one side. Locally, it seems to have two sides. If you took a snapshot of a piece of the Möbius strip, you could point out what seem to be a front and a back; however, when you consider the entire surface globally, there is only one side. For example, if you draw a line down the middle, you will never lift your pencil up yet the line will be on both sides. To make a model of a Möbius strip, take a piece of paper that is longer than it is wide. Join the narrow ends together, like you are making a loop, EXCEPT give one end a half twist just before you join it to the other end.

As a result of the half twist, the Möbius strip has only one side and one edge!

Test it for yourself by drawing a line down its center until you return to your starting point. Did you ever cross an edge? Or, hold the edge of a Möbius strip against the tip of a felt-tipped pen. Color the edge of the Möbius strip by holding the highlighter still and rotating the Mobius Strip around. You were able to color “both” edges without lifting the pen, right? For something completely different, cut the Mobius Strip along the center line that you drew. Then draw a line down the center of the resulting band, and cut along it. What happened?

**The Klein Bottle**

Whereas the Möbius strip is a 2-dimensional surface that requires three dimensions (for the twist), the Klein bottle is a 3-dimensional surface that requires four dimensions. It’s not as easy to imagine a Klein bottle because it is represented in only two dimensions in the figure. The Klein bottle was invented (or imagined) by Felix Klein (1849-1925), another German mathematician.

The Klein bottle, proper, does not self-intersect. Nonetheless, there is a way to visualize the Klein bottle as being contained in four dimensions. By adding a fourth dimension to the three dimensional space, the self-intersection can be eliminated.

I like to use the Klein bottle as a metaphor because by its very nature it requires a higher dimension that isn’t part of our everyday reality. It’s an entity, like us, that exists in three dimensions but requires four dimensions to do so. It points to the mystery of our existence, to the existence of the unknown, the n+1 dimension.

Similar to the Möbius strip, the Klein bottle embodies a continuum that encompasses a seeming duality, e.g., where inside and outside are not distinct but one continuous unity that flows from one to the other. It shows how the labels of a duality or polarity are only labels of aspects of a whole that are not, in fact, separate.

I use Klein bottles to help us think of concepts as “linguistic containers.” If we conceive of words as being containers for meaning, which enable the speaker to convey her ideas to others, then adding content words about interconnectedness is like pouring new wine into old bottles. I’m suggesting instead that we develop new types of bottles. How will these new “containers” look? How will they function?

Wine bottle | Klein bottle | |

Inside vs. outside | Inside and outside are a continuum; one merges into the other | |

Holds and contains something | Embodies the notions of contained and uncontained | |

Either/or paradigm | Both/and paradigm | |

Static unity | Dynamic, interpenetrating unity | |

Words used to convey separateness and distinctions | Words used to convey interconnectedness, process, paradox, and unity |

It exemplifies the concept of a merging continuum or union of opposites. The Klein bottle embodies the type of paradox that could be incorporated into language to be able to speak into being a world that works for everyone—us *and* them, old *and* young, rich *and* poor, conservative *and* liberal, black, white, yellow, *and* brown—at the same time. For the world to work for all, I propose a linguistic structure based in the notion of both/and.

How would *you* devise such concepts? Leave a comment.

Hello Lisa

I came to know about your book which I look forward to read. Perhaps you would be interested in my work on the Klein bottle and the surmountal of the Cartesian cut, much available in the web or you may write to me for the articles

best wishes

Diego

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Yes, I’d love to work this blog into an e-book at some point. (There is still much more to write…still in draft form.) And I wouldn’t mind guest authoring. What is your blog? Perhaps my readers would be interested in it.

This is an insightful blog. An odd omission from the English language is the word outsight. The notion of duality establishes an inside and an outside. A kleinbottle possesses no inside and outside and has no boundary to make any such distinction. The notion of one-sideness is therefore imported from the secondary notion of two-sided duality. The notion that people can claim to possess intellectual insight, but not outsight, reveals the fact that intelligent language is well-founded on the one-sided (universal) logic of the kleinbottle structure.

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This is an insightful blog. An odd omsosiin from the English language is the word outsight. The notion of duality establishes an inside and an outside. A kleinbottle possesses no inside and outside and has no boundary to make any such distinction. The notion of one-sideness is therefore imported from the secondary notion of two-sided duality. The notion that people can claim to possess intellectual insight, but not outsight, reveals the fact that intelligent language is well-founded on the one-sided (universal) logic of the kleinbottle structure.