Oct 24, 2023
“…Congratulations…you just created your first universe. May I call you God? Or would you prefer Lord of ‘the Ring’?”
Riddle: What does a cash register receipt from CVS have in common with an electron, human consciousness, and political revolution? Give up? Answer: They all exhibit 720° symmetry.
What? 720° symmetry? That’s ridiculous! You can have 90° symmetry, 180° symmetry, 360° symmetry, but never 720°. That’s just not a thing. The idea itself is ridiculous.
I agree with you. It is ridiculous…and yet, it is a thing. May I demonstrate?
Next time you visit a CVS, please don’t throw away the mile-long cash register receipt they give you at check-out. Instead, turn it into a loop—but not just any loop—a special kind of loop known as a Möbius strip. Here’s how (not recommended for readers under the age of 3):
Hold the two ends of the receipt, one in each hand.
Twist one end 180 degrees.
Scotch tape the two ends together.
Not too challenging, but congratulations anyhow: you just created your first universe. May I call you God? Or would you prefer Lord of ‘the Ring’?
But now what can you do with your Precious? Run your finger along its surface. Keep going. Still going? It’s endless, isn’t it? You’ve taken an everyday rectangle, albeit elongated, with well-defined sides and edges and turned it into a one-sided loop with no boundary. You’ve created what topologists call a ‘non-orientable’ surface.
Now pick a spot on the ‘strip’ and imagine an arrow on that spot, pointing up; slide that arrow to the right or left until you come back to where you started (360°). Hmm, something’s different, isn’t it? Your arrow is pointing down now, as if it were a reflection of the original arrow. Of course, it is the original arrow, only now it’s ‘disoriented’, a bit like me after a long night at my local.
But no problem! Just keep going in the same direction. Another 360° et voilà, your arrow’s pointing up again. Your universe is symmetrical after all, but that symmetry requires 720° of revolution, not the meager 360°, as in my boring universe.
At first, it seems amazing that we can create 720°symmetry in 360° space, but the phenomenon is not as rare as you might suppose. Ever heard of an electron?
Electrons have something called spin – the subatomic analog of your finger running along the Möbius strip. We’re familiar with spheroids. Obviously, they exhibit 360° symmetry, right? Not necessarily. Turns out that the electron, and the proton, all massive subatomic particles, in fact, exhibit 720° symmetry. They behave like figures on a Möbius strip, not like baseballs.
BTW, just in case you were wondering, massless particles like the photon generally exhibit boring old 360°symmetry, but the graviton is a bit of a twist: it exhibits 180° symmetry.
720° symmetry is important in another context: consciousness. I am aware of a table. For the sake of argument, let’s assume it’s a real table existing in a material world. Thanks to the properties of my central nervous system, I am aware of the look and feel of this table, but I am also aware of myself, and I am aware of myself being aware of the table. That’s consciousness!
But what’s that? Is it some sort of transcendental substance, a ‘soul’ perhaps? Or is it just another manifestation of a material ‘neural net’? Neither. It’s a topological feature of Universe per se. Like mass and energy, particles and waves, it’s akin to the property of reflection: the universe reflects itself.
Apparently, self-reflection is an irreducible property of Being. Perhaps it’s what Being is. That’s not to say that self-reflection is always, or even usually, conscious. Consciousness rises to the level of a phenomenon only in the context of certain well-defined, but currently unspecified, physical structures.
Once upon a time, people broke Universe down into ‘mind’ and ‘matter." Gilbert Ryle (The Concept of Mind) put an end to that. He demonstrated that that sort of dualism makes no sense. Mind and matter, like mind and body, are just different aspects of one phenomenon.
Today our models are more nuanced. But using the old terminology, it turns out that both mind (consciousness) and matter (massive particles) exhibit 720° symmetry. This appears to be the real, deep structure of Being, not the anemic ‘special case’ abstraction we know as 360° symmetry.
As usual, we have things upside down! Universe is characterized by 720° symmetry; 360° symmetry is a relatively rare, ‘degenerate’ exception. Yet we’ve mistaken the exception to the rule…once again! We’ve imprisoned ourselves in a ‘toy universe’. We ‘swim’ in Maya.
“You say you want a revolution” (The Beatles)? Well, you’re at home here! And what is a revolution other than the inversion of the socio-economic arrow? Marx said it well: a “Dictatorship of the Proletariat”. The Psalmists, Isaiah and Mark, did even better: “The stone rejected by the builders has become the cornerstone.” But the prize for Best Said goes to The Who: “Meet the new boss, same as the old boss.”
The Theory of Revolution requires that the fabric of society (the political logos) be non-orientable. How else can we turn an up-arrow down?
But, of course that could never happen in the real world, could it? We’ve never seen a revolutionary cadre turn into a new aristocracy, have we? Oh and BTW, I have a bridge for sale in Brooklyn that has your name written all over it. Let’s chat.