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”
“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.