I Seem To Be a Klein Bottle

A Unicorn could exist in our world…but it doesn’t. A Klein Bottle (KB), on the other hand, cannot exist in our familiar 3-dimensional space, and yet…I seem to be a Klein Bottle. 

To exist in our 3-D world, KB would have to pass through itself, which is topologically impossible. A Klein Bottle is an example of a ‘non-orientable’ object. That’s a 3-D object with a 2-D Möbius strip embedded in it.

And in English? It’s a bottle that won’t hold water. (Note to self: DO NOT USE when decanting wine, especially if the wine is red, and you’re standing on a priceless oriental.) 

KB can’t hold water, or wine, because it has no inside. That would be a problem for any ‘bottle’, wouldn’t it? A Möbius strip is a ‘strip’ that has no obverse/reverse sides. i.e., no front or back. If it were a coin, it would always come up ‘heads’ (because it doesn’t have a ‘tail’.) 

So, non-orientable objects, be they strips of paper or glass bottles, have no ‘sides’ - no inside, no outside, no upside, no downside. But what does that have to do with me? Clearly, I do have an inside and an outside.

Clearly? Well, it’s hot outside; I’m anxious inside. While the ‘heat’ may be a product of global warming, I experience that heat internally via my subdural nervous system. My skin is a membrane that separates the internal from the external for me, right?

Well, sort of. Skin is a membrane, but it is also part of a dissipative system: it does not so much wall off the inside from the outside as it regulates the traffic – the way Customs is supposed to operate on a national border.

The defining characteristic of a Klein Bottle is that it cannot hold water…or wine. Certainly, that’s me! Pour a 1982 Lafitte down my gullet…please, and then watch: soon it will emerge from my other end…with no notes of tobacco or cassis. 

But this still leaves one apparently insurmountable problem: a Klein Bottle can’t exist in 3-D space. But do we really live in three dimensions? Didn’t Einstein show that we actually live in 4-D spacetime? And didn’t Aquinas (and Hawking?) argue that ‘time’ is another name for ‘motion’? And what is motion other than a measure of the discontinuity between some figure (e.g., me) and some ground (the world)?

100 years ago, a biologist by the name of D’Arcy Thompson (On Growth and Form) suggested that an organism could be regarded as a Klein Bottle if its perpetual motions were taken into consideration. For him, motion is a 3-D analog of time. 

Jean-Paul Sartre’s signature contribution to Western philosophy was his recognition that ‘everything (etre) is outside (hors)’, always; literally, ‘nothing (neant) is inside (dedans)’, ever! The sun is shining; I feel the sun’s heat on my skin, and I feel my internal body temperature rising. I am becoming feverish. I am beginning to hallucinate.

Does it help to know that all this is external to ‘me’? My body, its temperature, my fever, even my hallucinations…all ‘outside’. Or is this a time when I’d trade the wisdom of the ages for a cup of water? (Sorry, Job!) 

So if the sun, its heat, my skin, nerves, and brain are all ‘outside’, what’s ‘inside’? If they’re not me, what is? Now, if you’ve been paying attention, you already know the answer: ‘Nothing’ (neant)! Nothing’s inside. In the words of Odysseus, I am Nemo, ‘nothing’. 

In fact, I am the active negation of whatever is (etre). To paraphrase Robert Oppenheimer, “I am Death, the destroyer (and creator) of worlds.” My role as Neant is to destroy worlds, but only so that new worlds may emerge. I am (or I am a reflection or manifestation of) the primal unrest at the core of Being. 

I am aware of X, which means that I am aware of X not being me. Everything I am aware of, I am not…by definition, or I wouldn’t be aware of it. Awareness implies separation. So does self-awareness (aka consciousness). I am self-conscious to the extent that I make myself an ‘other’, i.e., to the extent that I view myself from the outside-in.

When I view myself from outside, I effectively turn the universe inside out. What was in is now out and what was out is now in. That’s the essence of the Christian doctrine of incarnation: incarnation turns the universe inside out. The whole becomes its own quantum element.

To borrow an idea from Jacques Derrida, consciousness inserts differance between the knower and the known, even if the known is the self. So I can be both the knower and the known, separated by ‘differance’, which Derrida defines as a quantum of ‘difference’.

Derrida bravely fights a war on two fronts: on one side, Zeno, arguing that all motion is impossible; on the other side, Newton and Leibniz, arguing that spacetime is continuous. Zeno throws ‘pussy’ (arithmetic) down a proverbial well, while Newton assumes the role of ‘Little Tommy Stout who pulled pussy out’ (with calculus). 

For Zeno, spacetime is granular. He understood Planck long before Planck was born. For Newton, spacetime is continuous; for him, a quantum of difference is infinitesimal (infinitely small). Derrida marries the two: ‘Differance’ is minimally small but not infinitely small.

Here, Derrida recalls Sartre. Being is existence without essence. ‘To be’ is pure; it is unadulterated by any qualifiers. Differance is what comes from abstracting existence from essence.

Remember the Real Number Line? Take any two points on the line, call them A and B; between A and B there will always be an intermediate term, C, right? Not according to Zeno! And not according to Derrida. For Derrida, B > A (or A > B) but there is ‘nothing’ separating them: A ≠ B but A – B = 0.   

In the Odyssey, Polyphemus asks Odysseus his name. Our hero replies, “Outis” (Nemo in Latin), meaning ‘nobody’. Cyclops is a nominalist; he mistakes the name ‘No One’ for the reality of ‘no one being there’. Like many of us today, he fails to draw a straight line: “if someone is saying ‘no one’ to me, then someone must be here, saying it.” Does this remind you of the physicists who clearly hear Fiat Lux but deny there is ‘anyone’ saying it?

But this is not the end of our story. Outis has a secondary meaning: ‘everyone’. Being ‘no one’ means that I am also ‘everyone’. (Math analogy: the null set ø is a subset of every other set.) If I were ‘someone’, then I could not be ‘anyone’. I could not be ‘anyone’ because I’d already be ‘someone’, and I could not even be myself since I am ‘no one’. Confusing enough for you? 

Because I am ‘no one’, nothing, I can be ‘anyone’, anything (consistent with the physical limitations imposed by the universe): “I know who I am, and I know that I can be whoever I want to be.” (Mae Jemison, Astronaut) 

If I can be anyone, then I am everyone. Odysseus is everyone, as is Leopold Bloom in Joyce’s reprise. Because I am not actually anything, I am potentially everything. I am a bottle that is ‘no bottle’. I am in the form of a bottle, but I lack the functionality of a bottle. I am made in the image of ‘bottle’ but not in its likeness. I ‘look’ like a bottle, but I don’t ‘act’ like a bottle.

Is this separation of form from function what’s called ‘crisis’ in psychology, spirituality, and philosophy? 

When you are a baby and trying to figure out the world around you, nothing is more baffling than a mirror. There are people in the mirror who look like Mommy and Daddy but aren’t; there’s a baby in the mirror who might be you but isn’t. 

If you’re on track for Oxbridge, you might notice that the mirror is like a ‘window on the world’ but a window that looks inward rather than outward. If you’re really clever, you might even think to call it an ‘indow’. So, next time your little niece points to the mirror and shouts, “indow, indow,” don’t correct her. She knows the difference between an indow and a window; she’s just smarter than you. Deal with it!

While a mirror is not itself a non-orientable object, you and the mirror together form a system that functions like one. Right and left are ‘reversed’ in a mirror. In your ‘orientable’ world, objects display 360° symmetry: what goes around comes around. In my ‘non-orientable’ world, what goes around still comes around…only in reverse.

But don’t give up! What comes around needs to go once more around to get us back to ‘the way we were’, i.e., how things were at the get-go. Non-orientable systems display 720° symmetry (vs. 360°symmetry). The image of my image is myself.

So I am a Klein Bottle; like a mirror, I reflect everything but contain nothing. I am a topological twist...a knot in the fabric of being. Through me, the Universe gazes back upon itself; through me, Universe is conscious of itself. I am what the Universe sees when it looks in the mirror.

David Cowles is the founder and editor-in-chief of Aletheia Today Magazine. He lives with his family in Massachusetts where he studies and writes about philosophy, science, theology, and scripture. He can be reached at david@aletheiatoday.com.

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