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R U Orientable?

David Cowles

Apr 18, 2024

This is why I was never a track star in high school; I was the only kid running on a non-orientable race course.”

Do you believe the Earth is round? 

“Round? Why? Does it look round to you?” 

Even before Socrates, philosophers and scientists debated the shape of our planet. There are many theoretical reasons to think that the Earth might be roundish. Against this, however, is the evidence of naive sense perception: it looks flat! I can drive from Boston to Seattle without ever having to consider the overall curvature of the federal highway system. Even if it isn’t flat, it might as well be.

Yet, you probably answered my opening question with a full-throated “Yes!” Most of us are convinced that the theoretical arguments for the roundness of our planet override the immediate testimony of our eyes. 

For the most part, the debate has moved from the shape of the planet to the shape of the Universe; but on this later question, the jury is ‘still out’. I won’t go into all the open issues; I’m only interested in one: Is the Universe orientable…or not? 

When I first started contemplating this problem, the textbooks were unanimous: the Universe is orientable, period. This is a big claim since the presence of a single embedded non-orientable topological feature (e.g. a Mobius Strip) is enough to make the entire Universe non-orientable. So it doesn’t matter how the Universe looks from (N – 1) perspectives; we need to know how it looks from the Nth perspective.

There are precedents. 400 years ago, the Universe looked Newtonian; we accepted the ‘fact’ (sic) that Newton had captured the ultimate order of things. 300 years later, the work-a-day Universe still looked Newtonian, but by then we had learned to look further and deeper.

In the process, we discovered that the Universe was not ultimately Newtonian after all. It still looked Newtonian at most scales, but now we knew it wasn't—it was Relativistic. Newton’s model turns out to be just a very useful approximation of Einstein’s. 

But we’re not done. Now we are struggling to fit Einstein’s Relativity into an even broader conceptual framework that includes Quantum Mechanics and Non-Locality. It’s a work in progress. Again, on most scales, the Universe looks relativistic, but now we know that ultimately, it can’t be. 

Which brings us to the matter of topology. Imagine an extra-long cash register receipt like the one the cashier at Target handed you the day after you got last year’s tax refund. Yeah, like that! Stretch it out. It’s bounded on 4 edges, and it’s two-sided. Your receipt is like an episode of Stranger Things. If you live on the obverse side, you have no knowledge of the reverse, and vice versa. 

You can write a short story on one side and draw a landscape on the other. You can even join the two ends together to form a corral or ring. Obverse/reverse becomes inside/outside; still, the twain can’t meet. 

Now instead, twist your paper (receipt) 180° and then join the ends together. What a difference a twist makes! Now there’s no inside/outside. In fact, there are no sides at all; your 4-bounded register tape just became 2-bounded. God sure had some fun creating this universe!

But is this your world? Well, your world is twisted (in more ways than one), and it can seem a bit one sided, at least according to some contemporary news channels, but we need to dig much deeper. 

Place a triangle pointing up (˄) on the Mobius strip and slide it 360° along the tape until you get back where you started. Easy-peasy! Except it’s not. You have not restored its initial position. Something’s happened to your triangle; now it’s pointing down (˅). What goes around still comes around, but it comes around upside down. 

Objection: If you continue to slide your triangle along the Mobius strip, it will eventually come back right side up. 

Answer: Exactly! Every 360° spin around the ‘strip’ reverses your orientation; the point is that you no longer live in a world with 360° symmetry. This non-orientable space exhibits 720° symmetry.  

So, is ‘orientable cosmology’ the contemporary analog of Newtonian physics? Are there reasons for thinking that the Universe might not be orientable after all?

First, massive elementary particles (e.g. quarks, electrons) have non-integer spins – which means that they exhibit symmetry other than 360°, e.g. 720°. As with our Mobius Strip, it takes two laps around the track for these buggers to go from the start to the finish line upright. This is why I was never a track star in high school; I was the only kid running on a non-orientable race course.

Also, in one- and two-dimensional systems, it has been shown that the properties of some particles simply express the topology of their space. For example, electrons can have fractional charges (e.g., 1/3 or 1/5 rather than 1), and these quantities are derived directly from topology. While particles in orientable space can sometimes be teased into ‘exotic states’, those states exist naturally in non-orientable spaces. 

Riddle: How is Particle Physics like the Dick Tracy movie? Answer: They both call their villains things like ‘Full Face’ (360° Photon), ‘Two Face’ (180° Graviton), and ‘Half Face’ (720° Fermion). 

Riddle: How is the Standard Model of Particle Physics like Roman Mythology? Answer: They both have a two faced God (Graviton, Janus) in their Parthenon.

The idea that the stuff of Universe is a function of its underlying topology is not new. Plato advanced the idea in his Timaeus as did R. Buckminster Fuller in Synergetics. There’s a lot more to explore here but as a bartender once said to me, “Sir, are  you orientable?” 


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