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)

8 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

Leave a Reply

Your email address will not be published. Required fields are marked *