The Deep Symbolism of the Mobius Strip

The Mobius Strip

If ever there was something which merited the name “God” in my eyes, it would be the Mobius Strip. But I don’t believe in a personal, let-alone sentient, god. I’d be far more inclined to call it “Tao” instead. Buddhists might call it “Om” (or “Aum”). Mathematicians should call it “i” (the square root of negative one), but there are even more examples in Mathematics (the involution, the half-rotation, inconsistency, contradiction, “not” or the symbol ¬). Electronics circuits represent it as the inverter whose ouput feeds back into its input. Philosophers might call it “contradiction” or more formally the “paradox of self-reference” epitomized in the Liar Paradox:

“This statement is False.”

The Mobius Strip is the physical manifestation of this paradox. Observe:

Take a longish strip of paper (I cut a 3cm strip off the long side of an A4 sheet of paper) and simply glue (or tape it or whatever you choose) both ends together. This is a (admittedly very short, squat) cylinder. Now draw a line all the way around the loop (inside or outside, it matters little). When you un-glue the ends of your loop, you’ll find the line you drew is only on one side of the strip.

Now do the same again, but this time give the strip of paper a half-twist before gluing it. This is the Mobius strip, or ‘twisted cylinder‘. Now if you draw a line going around that loop, you’ll go on and on and eventually come back to where you started, just like before… But the tricky thing here is that when you separate the ends again, and lay it flat, you’ll find that the line you drew is now on both sides as opposed to just one!

And how peculiar it is that what is clearly a 2D object in a 3D environment has only one ‘side’ (where one might expect an object having only one side to be 1D).

If we were to call one side of a fresh strip “IS”, and the other “ISN’T”, then construct a Mobius Strip, another interesting thing happens: As we go around the loop, we find ourselves ‘alternating’ between “IS” and “ISN’T”. This harkens to another situation that we’ve just seen – The Liar Paradox:

If it’s True, then when it says it’s False, then it’s False. But if it’s False, then it certainly wasn’t True. So if it isn’t True, then when it says it’s False, then it’s True… and so on, flipping back and forth from True to False and back again.

What if we were to call one side “i=1” and the other side “i=-1”? Going round the loop we flip between -1 and 1. There is an equation that does this is: “i = -1 / i” where if “i” had the value “1”, then “-1/1” is “-1” – so “i = -1.” But if “i = -1”, then “-1/-1” is “1”, and so “i = 1″… see what’s happening here? Now, solving for i, we would multiply both sides of the equation by “i”, giving us “i*i = -1”. Starting to look familiar? Take the square root of both sides and we have “i=squareroot(-1)” – the ‘imaginary number’ “i”.

But there’s more! On your Mobius Strip, draw a line going across the narrowest part (perpendicular to your long line going around the loop). Label this shorter line “D”, and then label the longer line “C”. The Mobius Strip represents our two ‘modes’ of thinking: “D”iscrete and “C”ontinuous (or Digital and Analog, if you prefer). Going back to the IS/ISN’T dichotomy: by going the “Discrete” route – that is, the action of flipping the strip of paper over to get to the ‘other side’ – we see the clear and reliable alternation of “IS/ISN’T”. But going the “Continuous” route we also see the clear and reliable alternation of “IS/ISN’T”! So despite the ‘paradoxical’ nature of these things, they still remain consistent. Yet both views seem perpendicular to each other.

If we instead labelled both sides “IS” and were to instead label the edge of the strip “NOT”, then it would seem more ‘coherent’: Going around the strip (taking the “C” path) everything is “IS” – i.e. “it’s all the SAME”. But flipping over the edge of the strip, so that if you were to speak it out loud, you would have “IS” (where you start) “NOT” (flip) “IS” (where you end), or “ISN’T IS” or, if you were to say that the other side is ‘transformed by’ the “NOT” then you could speak it as “IS IS NOT” and thus “IS ISN’T”.

Finally, to tie it back to my earlier inclination of referring to it as “Tao”, I want you to picture the Taijitu symbol, with yin and yang going around each other in a coherent whole – where further still each is infused with a spot of the other. It needs little imagination to see what labelling one side “yin” and the other “yang”. Now, to parallel the other version, where the edge is the inversion, label both sides “TAO” and I will leave you with the opening stanza of the Tao Te Ching:

“The Tao that can be named is not the Eternal Tao”

or, more simply:

“The Tao is not the Tao”

and finally:

“TAO NOT TAO”

The Mobius Strip is such a deeply symbolic representation of so many recurring ideas that I see in the mind and in nature – representative of constant change (isn’t that an oxymoron!), both Discrete and Continuous and how both can reside together in one coherent whole reality… That such a rich set of ideas can come from such a simple construction is one of the things that makes mathematics beautiful.

Thank you for reading.

Thomas (Taomath)

15 thoughts on “The Deep Symbolism of the Mobius Strip

    1. … or the complex number “i” being isomorphic to the liar paradox? Or the reflection to the half-rotation? Or the self-reference to the negation? All of the same family! Pretty wild isn’t it?

    1. Hi Todd,

      That’s very honest and forthcoming of you, thank you for approaching me.

      I don’t really need credit – certainly if you quote what I wrote then yes, but for the idea itself, nah. I mean the object is there, you just have to make one to see how strange it is, and begin to ponder some of the mysteries of life and begin to make parallels.

      The parallel between the Liar’s Paradox and the imaginary number “i” is not my idea! That’s actually in the pages of George Spencer-Brown’s “Laws of Form” (I don’t have my copy on hand so I can’t point you to the exact pages, sorry). And there’s Louis Kaufmann and his paper “Virtual Logic” (pdf here: http://homepages.math.uic.edu/~kauffman/VirtualLogic.pdf). The idea that the imaginary number “i” represents rotation (and thereby alternation) goes all the way back to Euler himself with his beautiful equation e^(i*pi)+1=0. While it might be possible that the parallel of the Möbius Strip and the Liar’s Paradox may be original to me (only because I haven’t exhaustively researched to see if someone did it before me), I don’t think it’s very likely. And finally, the parallel between the Imaginary number and the Möbius strip is more probably originally mine (just coz it’s such a kooky notion I don’t think any ‘serious’ mathematician would consider it!).

      What is important to me is that people begin to think about these things so if you can orient your readership to my blog or the authors I mentioned above, or provoke in them curiosity about what, exactly, existence is ‘made of’ – simply put as long as the idea is preserved in your story then I’m happy.

      (Oh and purely selfishly I would love to see a copy of the story when you’re ready!)

      Thanks and best regards,

      Thomas

  1. I find all of this beyond fascinating. I’m 43, didn’t take advantage of school when I was young. I’m finding myself now, reading whatever I can get my hands on. Symbolism, philosophy and understanding the concept that everything in life can be mathematically explained. Yet, this all does indeed seem to be a complete paradox of how I am understanding the majority of what I’ve since learned. Is it possible to get the pdf version? Or any other recommend readings? I’m absolutely enthralled.

    1. Hi Carrie! I’m thrilled to find a sibling soul and thank you for having taken the time to comment. I too didn’t take advantage of school and everything I’m putting up here is stuff I’ve been learning on my own. I’m not sure /everything/ can be explained mathematically, but mathematics is a pretty good language in which to try, certainly. Sometimes the mathematics isn’t adequate enough to truly reflect reality. You ask for a pdf version… of what? This little blog entry? Something I refer to in it? PDFs are widely available on a huge variety of subjects and I’ll gladly point you to a few (I have already provided links to a few in other entries on this blog) if that’s what you meant to ask. You can also check out my ‘inspiration’ page where I list a few books that have helped to inspire me and point me in the direction I’ve gone with all of this thinking. One of my regular haunts is the “Stanford Encyclopedia of Philosophy” (https://plato.stanford.edu/) which another commenter has also pointed-out. I had also joined a philosophy forum (which has since closed, sadly) which helped me sort-out my ideas and exposed me to a vast list of authors and similar thinkers.
      Do stick around and feel free to share your experience and questions as I welcome all feedback and am always hungry for challenges to this line of thinking (it’s the only way to improve it!).
      Thanks again,

      Thomas

  2. Any book recommendations (for us laymen) would be greatly appreciated. Feel free to contact me via email.

    So many questions, so little life.

    1. Hi Carrie! Thank you for having taken the time to leave a comment. I’m afraid I don’t have a reading list in particular but I can recommend Alan Watts’s book “The Way of Zen” for starters. I would also gladly discuss these topics with you via email. I’ll send you one soon 🙂

  3. Hello Thomas. I’m really glad to have found your blog … it makes me feel a little less alone! I think of the Mobius as a MODEL of the truth and, for me, it links to the Christian trinity (two opposites joined by a spirit) and to the Schrodinger’s Cat thought experiment, where the cat in the box is Both alive and dead. I’ve posted loads of stuff on this subject in my own cack-handed way on G+. I believe that the Truth is both orthodox AND heterodox, i.e. Paradox.

    1. Hi Tommy!
      Thanks for leaving a comment – indeed makes me feel a little less alone 😉
      Oh the holy trinity! I hadn’t thought of that – I can see how you tie that in.
      I’ll have a look for your notes on G+ otherwise I’ll write to you if I can’t find ya!

  4. Hi Thomas,

    I agree with your conclusion “both Discrete and Continuous and how both can reside together in one coherent whole reality…”

    It speaks to me because of the work I have done on attention over last 8 years (providing I understood correctly what you were trying to say 🙂

    I believe it is correct to say that the reality consists of separated objects which are simultaneously connected. It depends on us, and more precisely on our attention style how we perceive reality. (see this short post which explain it https://www.openfocusattentiontraining.com/2015/01/19/true-freedom-3/)

    Perceiving reality as a whole (continuous) is a very old tradition and I believe this a main difference between western and eastern way of thinking (see this post https://www.openfocusattentiontraining.com/2015/01/22/flexible-attention-where-the-west-meets-the-east/).

    I found interesting you called your blog Thao Math. I believe Thao belongs to continuos (nondual) reality and Math is used to describe discrete (dual, filled with separate objects) reality. Those two approaches are smilingly completely different from each other but truly they can be included together in one elegant formula called the four attention styles theory and attention flexibility (see this picture https://www.openfocusattentiontraining.com/2015/01/29/one-picture/)
    In other words objects a function of continuity.

    I would like to share with you a theory which I am researching now and I think you might find interesting. It shows that math belonging to objective (dual, discrete) reality struggles when applied to continuous reality. It other words scientific based approach cannot describe reality as it is (see this post for start https://www.openfocusattentiontraining.com/2014/12/13/science-nonduality-and-attentional-flexibility/). Math is helpful but cannot answer all questions.

    The simples way to present how math does not help in describing continuous reality is to say that in continuous reality two equals one (2=1).

    I believe Mobius String is a good example of this. It has simultaneously one and two sides.

    Another example is directly related to the way we perceive the world.

    You might be aware that all stimuli like sounds, light, touch and smell once registered by receptors, (inside your ears, eyes, skin and nose) become small electric currents travelling along nerves towards our brain. The brain does not hear, see or touch. It operates using electric currents which oscillate along brain cells.

    It means that when you observe something you are detecting play of electricity in your brain. Say, you see two chairs and space between them. The chairs and space are coded in your brain as electric currents. Chairs swimmingly differ from each other and from space but truly there are a function of the same medium. There are two and one at the same time.

    More importantly the impression that you yourself are separated from the world around you is produced by your brain too. It means whatever is you and not you are different and the same at the same time.
    To make is short the reality outside and inside you is one.
    The strong impression that you are separated from the rest is an illusion produced by your brain.
    There is no other way.

    I have a feeling you may add a lot to it.
    I wonder what you think.

    I am sorry for links to my blog.
    Please, do not approve this post if you find it not appropriate.

    1. Hi tomasz!
      First I must thank you very much for taking the time to write. So many people read this article or many and never even say hi , so thank you. I will try to dedicate some time for a proper reply soon but right now I have been unable to free enough time. Please keep an eye open here and I will write more because you have given me much to respond to!
      Speak to you soon!
      Thomas (P.S. do you spell it Thao in Polish? In English it is either Tao or Dao. I chose to put math in the title because I wanted to let people understand this was serious work and to keep out the crazies… only to find out I gradually became crazy myself haha!)

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