67. The Con Badge Problem

(With thanks to TT and IL for prompting me to think about this. Happy Pi Day.)

Picture this: you're at a convention. You've got multiple badges to juggle - an attendee badge, two or three beautiful art badges to display, and maybe even an access or volunteer badge - so you've put them all on the same lanyard. In the morning, you carefully arrange them all facing the right way, put the lanyard on, and go out to the con. By noon, at least one badge will already have flipped backwards. By evening, another two have flipped backwards, and you could swear that you briefly saw one of them upside down. Over the course of the con, you fight a losing battle: you notice a backwards badge and flip it and it flips back on its own; you take off your lanyard altogether and carefully reorient all the badges correctly, and within an hour, half the badges are backwards and chaos reigns once more.

It's long been a frustration of many who wear badges - the rectangular plastic kind that might also be used for card readers or have art on them, I mean, not the medallion-y kind - that they seem to very frequently end up flipping backwards no matter what you do. Worse yet, if you manage to wear multiple badges on the same lanyard, it seems like no matter what you try, like half of them wind up flipping backwards no matter how often you try to flip them right-side-out again. But how can that possibly be possible? I mean, surely if you arrange a lanyard or flip a badge to make the badge face the correct direction, that should be the end of it, right? ...Right?

Wrong, actually! And as usual, the answer involves math: more specifically, topology and differential geometry. This is the Con Badge Problem, and it has a beautiful topological explanation that won't help you solve it but will at least let you understand why you're doomed. As ever, we begin with some definitions.

In math terms, we define a "ribbon" to be a simple closed curve, along with a vector normal to the curve at every point on that curve, which we can think of as a choice of normal vector to the "plane" of the ribbon. We'll keep things simple and only consider orientable ribbons - that is, not Mobius strips. Let's further sketch out definitions of two more related concepts that might initially look unrelated to each other: "twist" and "writhe". Twist is straightforward: it's just the number of full counterclockwise rotations that the normal vector makes while going around the ribbon once. We can think of this as the number of full counterclockwise turns we'd make with respect to the center of the ribbon if we walked along the whole thing. This number is an integer; if we wanted to think about nonorientable ribbons, we'd need to think about half-integer twists, too. Writhe is a little subtler, measuring how the curve coils around itself in 3D space. Pick a direction around the curve to count as "forwards". Looking from any viewpoint, start at 0 and check all of the crossings: +1 if the top strand crosses left-to-right relative to the bottom strand, -1 if right-to-left. Sum them up - that's your writhe value; it measures, roughly, how often the strands of the circle twist around each other clockwise or counterclockwise.

Now here's where it gets interesting: if you think about it, a badge lanyard is exactly this kind of mathematical object - a simple closed curve with a bit of surface extending from it. Accordingly, we can say what twist and writhe are in this setting - twist is how many times the lanyard twists around its center as it lies roughly flat on your body, and writhe is how many times the lanyard as a whole curls arond itself. Second, there is a remarkable theorem of Călugăreanu that for our purposes says that the writhe value is always exactly the negative of the twist value, and thus that you can convert twist to writhe and back again.

This last point gives us both a major piece of the puzzle for why badges on lanyards behave so frustratingly and hints at how to fix it - at least, temporarily. Consider that a badge is attached to a lanyard via a rivet or clip which might be able to spin freely. This is by design - a badge should hang naturally downward regardless of the lanyard's orientation, always settling at the point of lowest gravitational potential energy. But this has topological ramifications: namely, the rivet acts as a sink for twist. Whenever there's half a turn or more of twist in the lanyard near the badge, the badge turns upside-down and turns that twist into writhe, and then the freely-spinning rivet lets the badge settle back down in the reversed orientation. That is: the badge flips over and your friends might point that out to you if they weren't all too familiar with the same happening to them, taking it in stride at this point. The key here is that a badge hanging from a lanyard has precisely two stable equilibrium states: "correct", with front facing out, and "backwards", with back facing out; in both cases, the badge hangs straight down. But through the mechanism I've described, any small perturbation - leaning over, someone bumping you, the badge catching on something, or especially taking the lanyard off and putting it back on - can flip the badge from one state to the other, making use of the fact that now, a full turn's worth of twist or writhe isn't necessary to make the transition. Removing and re-donning the lanyard and even particularly vigorous walking around can locally invert the orientation and add enough twist to mess up your badge array.

It gets even worse. Consider the naive solution: what if you just... flip the badge back over? Not only doesn't this work, but it also tends to make your problems worse. Even if we model a badge's attachment to the lanyard as infinitely strong, it's still the case that flipping the badge naively adds more twist to the lanyard, which is now locally constrained by the rivet itself. The rivet absorbed twist from the lanyard to flip the badge in the first place; manually flipping it back not only won't remove that twist from the lanyard, but it splits that twist between both sides of the rivet. So now you've got local twist accumulating near the badge, which will migrate along the lanyard and flip yet more badges. Flipping the badge manually doesn't solve the problem and it doesn't just move the problem - it makes it actively worse! The more badges you have, the more rivets you have, the more opportunities for twist to get absorbed and generated and migrate in complicated ways and turn to writhe and back again and create more chaos. And every time you manually flip a badge back, or remove it altogether and put it back the right way, the more pockets of twist you add to the system, making the problem all the worse.

Alright, so... what are you supposed to do to fix the problem, if not just naively flip the badge over? Here's the bad news: you can't actually permanently solve the con badge problem, at least, not without radically changing something about the system. As long a you have a closed loop with freely-spinning rivets, you're going to have badges flipping. Here's a few options for what you can do, though:

• You can just give up. Embrace the chaos. Your badges will flip, and that's fine, and no one will judge you for it. Everyone's been there. The iron dictates of topology weigh heavily upon us all.
• You can carefully minimize twist introduction. Any time that the lanyard goes on or off your neck is another chance for more twist to be introduced. Try to lift it straight up and off of your neck rather than remove it haphazardly and introduce new pockets of twist, and try to check for such introduced pockets of twist while the lanyard is lying flat before you put it back on.
• You could change something about the lanyard itself. If your lanyard comes with a buckle or clasp, you can undo it and remove the twist all at once. Of course, that risks introducing inherently unremovable twist if you're not careful about how you fasten it back up. You could also use fixed rather than freely rotation rivets, but then you've got badges that stick straight up at an annoying angle rather than hang straight down. In either case, you've just traded one problem for another arguably worse one.
• Best of all, you can take the topologically sophisticated option. Start by picking a side of the lanyard to be "forward"; the other will be "backwards". When you attach your badges, make sure at the start that all of them face forwards. Minimize twist introduction when you put on or take off your lanyard, and any time you see a badge flip backwards, don't just flip it over, because that won't work. Instead, start by checking for local twist you can turn back into a badge orientation change. If there's none in evidence, rotate it upside-down first and then flip it, and be prepared to carefully propagate the change past all the other rivets. The important thing is that all the badges face the same way - problems can arise when you remove and reattach a badge.

It's worth noting that throughout this whole rigamarole, the shape of the lanyard stays fixed - at least, up to reasonable lanyard-able transformations. (More formally we might call these diffeomorphisms.) Accordingly, the value of (twist - writhe) also stays constant, and since a lanyard being worn has zero writhe, in effect global twist stays constant at zero as well. So if that's the case, why do we care so much about introducing local twist? The key is that even though global twist is conserved, local twist need not be, and it's local twist that we care about when thinking about what happens to badges and rivets at any given moment. What's more, when you wear a lanyard, whatever local twist situations arise are usually frozen in place, because the lanyard is pressed up against your body and also subject to gravity, though that latter part matters much less.

Another key observation is that from the perspective of differential topology, several different-looking badge states are all equivalent. A badge in the correct orientation with rivet facing inward is equivalent to a badge upside-down and backwards with rivet facing outward - which, because the rivet spins freely, is equivalent to a badge backwards but right-side-up with rivet facing outward. The first equivalence follows from considering what happens if you naively try to turn a badge upside down, and the second follows from the fact that the rivets spin freely.

 

Notably, this state of affairs can be fixed with a little thought and intentionality, in the manner of the last bullet point above; however, most people don't think to check or track which direction the rivet is facing or don't consider it to matter, and if they do, they get as far as realizing that you have to flip the rivet and the entire local section of the lanyard over, but when the badge is in the correct facing but upside-down, they forget that the rivet can spin freely, believe themselves to be off-track, and give up. Worse yet, they might completely remove and reattach the badge in the wrong facing: while a badge in the correct orientation with rivet facing inward is equivalent a badge in the wrong orientation but correctly hanging down with rivet facing outward, both are inequivalent to a badge in the correct orientation with rivet facing outward.

 

So what's the takeaway? Sometimes, a problem has no good solution because the constraints that create the problem are so fundamental to the system that no solution can exist. You can't have freely-hanging, correctly-oriented badges on a closed-loop lanyard without accepting that they will sometimes flip. The topology doesn't allow it.

This is actually pretty comforting to me. It's not that you're doing something wrong. It's not that there's some trick you're missing. Mathematics and the universe just don't permit the thing that you want. Your badges are backwards because linking number is conserved, free rotation creates twist sinks, and twist, writhe, and rivet rotation all trade off, and no amount of wishful thinking could ever change the differential geometry of curves in 3-space. But at least knowledge of mathematics - especially the kind that at first blush seems like it could never have any practical application past yet more abstruse science - tells you why you can't get exactly what you want, and it even tells you how best to put a temporary patch on the problem. Greater knowledge and a more detailed, more attentive understanding of the system and its constraints give you both the impossibility of a permanent solution and the right way to make a temporary fix, and it falls to you to accept them both with grace.