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No. 2251: Moebius Strip Strip

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Moebius Strip Strip

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Strip by: Alien@System

{Garfield paces around on a table, with John watching}
Garfield: No matter what he rolls
Garfield: If I roll a 4, I would still win when he rolls a 1, but I'd lose on 6 and 8
Garfield: Or course, even if I roll a 2, he will lose on a 1 and win with 6 and 8
Garfield: So, it would be a safe win, or a two thirds chance to lose
Garfield: Multiply that by two, add three...
Garfield: Red beats Blue!
John: You're wearing a groove on the table
Garfield: Or maybe the green one
Garfield: Rolling a 7 would be a win, unless he has a 9
Garfield: Even if it is only a 5, it would beat his 4 and 2 but not when he rolls a 9
Garfield: Heck, even if it's a 3, the 2 would still be a loss for him
Garfield: In total, twice two thirds to win, once one third
Garfield: Four... Add five... Carry the zero...
Garfield: Green is better than Red!
John: You've been at this for two hours now
Garfield: Then there is Blue...
Garfield: The 8 would be a sure win, with his best of 7
Garfield: Sure, the 6 would loose against that, but win over both his 3 and 5
Garfield: Only problem to consider is if I roll a 1, which will always lose
Garfield: One sure win, one sure loss, but in between two thirds win
Garfield: Adding three and two... Divide by nine...
Garfield: Blue will beat Green!
John: You told me how you plan to trick me
Garfield: I could take the red die
Garfield: If I roll a 9, I will win

The author writes:

Back for #1682, I said I wanted to do a strip using Moebius symmetry, and here it is. You might be familiar with the idea of Moebius storytelling from xkcd #381, although this isn't quite as sophisticated. Mostly, it allows Garfield to swap places with John between every panel, in a circular strip with 21 panels.

For those that don't know what a Moebius strip is, here is a short explanation: A Moebius strip is the most easily understood example of a non-orientable two-dimensional manifold in three-dimensional space. It can be described via a parametric — you know what, let's forget math and start simpler: Take a long strip of paper, twist one of the ends by 180°, then glue them together (or use a paper clip if you don't want to be wasteful). What you hold in your hands now is not quite a Moebius strip, but a pretty good model to understand how it works. The object now has only one edge, and one side. You can test this by taking a pen, and painting along the edge, or on the side, until you reach the start again. You will see that you have painted the entire edge of the object, in contrast to a strip of paper glued together normally, which would have only one of its two sides or edges coloured if you do that.

Why is the object not a true Moebius strip? Because the paper you are using has a thickness. At each point, there is basically a "front" and "back", and although it's possible to get from one to the other without going through the paper, they are different. For a true Moebius strip, those two points would be equal, so to understand it properly, you'd have to take transparent foil instead of paper, and ignore which side you're writing on.

Since you have access to transparent foil and glue, let's do an experiment to illustrate the non-orientability better. Take a small piece of foil, and draw two perpendicular arrows on it, with a common point of origin. Put a small glob of glue on that point, it stands in for a third arrow. Let the glue dry, then place this piece against a surface in such a way that it lies flat against it. The glob of glue now points "up". Now, move the piece around as you like, as long as it remains flat against the surface. If you do it on a normal surface, like, say, a sphere, or the non-twisted strip you made earlier, you will notice that if your piece of foil rests at the same location, the "up" direction will always be the same, no matter how much you moved it around in between. Now, try it on the Moebius strip. You will quickly see that by moving it around its length once will allow you to place the two arrows on the same position as before (ignoring the thickness of the foil), but with the "up" direction reversed. This is what a mathematician means when they talk about non-orientability: The inability to define a continuous "up" direction over the surface of the object in question.

As you played around with the Moebius strip, you might have noticed that you can't twist it in a way to realize the way it is folded in the above picture. This is because the strip above actually twists by 540° over its length. This doesn't change the basic attributes of the strip, however, just makes it easier to read.

In the field of topology, which only cares about homeomorphology, the Moebius Strip is not only AN example of non-orientability, it is THE example of non-orientability. How so? As you might, or might not know, in topology, edgeless orientable surfaces can be put into equivalence classes, where two objects in it can be deformed (as if they were made of infinitely stretchable rubber) from one into the other without cutting or glueing. They can be identified by their number of holes; the row starts with the sphere, then comes the coffee cup (or doughnut), then goggles, then a pretzel, and so on. The same is possible for non-orientable edgeless surfaces, and here's the fun fact: Archetypes for each of those classes can be constructed by taking an orientable object with the same amount of holes, cutting a circle out of it somewhere, and then glueing a Moebius strip in there. Along its edge. It only has one, after all. If your brain just broke trying to visualize that, don't feel bad: to unfold a Moebius strip so that its edge is a circle requires the surface to intersect itself.

If you still feel fine after trying to visualize that, here comes a real fun fact, which will also answer this clever question you just thought of: It doesn't matter how many Moebius strips you glue in there this way. The object might look weirder like this, but it won't be different in terms of homeomorphy. A Klein Bottle, which is the basic equivalence class of non-orientable edgeless objects (having no hole), can be cut in such a way that you get two Moebius strips, but it can also be cut in a way that gets you only one.

Of course, all of this doesn't tell you a thing about what Garfield is blathering about in the strip above. The answer is: He is trying to choose between three nontransitive dice. Given what your brain went through above, it shouldn't be hard to accept now that, despite what common sense might tell you, it is not possible to order dice based on which is more likely to roll higher. Specifically, there exist sets of dice so that the ordering of which is more likely to "win" will result in a circle, that is, the red die will win over the blue one, the blue over the green, and green over red.

If you have such dice (Garfield's ramblings should be enough to tell you which numbers are on the faces), it is possible to play a mean game with someone: Each of you rolls a die, and who rolls lower has to pay the other person a dollar (or loses a point, if you're not into gambling). You can switch die as often as you like, but your opponent is allowed to switch as well afterwards. Make your mark choose dice first, and there will always be one you can choose that will beat theirs.

Have more than one friend? There also exists a set with seven dice, where for each pair of dice your two opponents choose, there exists one that beats them both.

Note: The author of this text takes no responsibility for any grudges, fights, arguments or insanity-inducing brain damage that may result from trying to apply the contents of this text to reality. Neither does he take responsibility for any eldritch abominations summoned as a result of the above experiments, regardless of their rugosity. Play with Moebius Strips at your own risk.

Original strip: 2008-12-21.