I have trouble with three-dimensional space. Yes, I do live in it, and I get by without hurting myself too badly too often, but honestly, I miss a lot of what’s going on. There’s no reason for you to care about my problem, except that it touches on the issue of “What does it take to be a mathematician?”, and the details of my limitations might serve as a useful antidote to the idea that math ability is always linked to spatial intuition. But I’ve got an ulterior motive for these confessions: I need some help, and I’m hoping one of you can provide it.
BERKELEY, CALIFORNIA, 1983
I remember my zeroth day of grad school, the day before the start of classes. I was attending a workshop on how to be a good teacher — something the university required all incoming Ph.D. students to do, since most of us would serve as Teaching Assistants for at least one semester. The seasoned TA leading our orientation wanted us to experience the sort of Aha! moment that good teachers instigate, so she led us through the famous “handcuffs puzzle” in which two people tied together by cords must extricate themselves from one another. Since some of you haven’t seen this trick, I won’t spoil it (though you can spoil it for yourself by going to this link; it also appears in many books on recreational mathematics). At the end of the exercise, when the workshop leader asked us to raise our hand if we’d had the prescribed Aha! moment, I was one person who didn’t raise his hand. She asked me to explain why I hadn’t, and I said that while I could do the trick, I didn’t get it.
The workshop leader thought that this blunt confession of ignorance indicated insecurity on my part, so to console me, she said “That’s okay; the point of the exercise isn’t getting this specific Aha! moment, but getting the idea of how to lead people to have these kinds of moments when you teach. Anyway, I’m guessing you’re not a math or science student; you came here to study music or literature or something like that, right?”
“No, I said; “I’m here to study math.”
The workshop leader’s dismay (which she tried to hide, but I’m guessing she wasn’t an acting student) told me that she thought I’d made a big mistake in coming to Berkeley, exceeded only by the mistake the Berkeley mathematics department had made in accepting me.
In a way, she was right to fear that I’d have trouble doing graduate work in math; someone with my spatial ineptitude would probably have a lot of trouble in certain branches of math, such as topology (Endnote #1 notwithstanding). But research mathematics has a very long, winding frontier that hosts a lot of very different mathematical ecosystems, and there was a lot for me to learn at Berkeley in game theory, combinatorics, and dynamical systems that didn’t cause me to trip over my limitations as a three-dimensional thinker. So I did just fine in the math Ph.D. program.
Something the workshop leader may not have realized is that a person who says “I don’t get it” is sometimes an exacting thinker rather than someone who is dense. In my own case, I don’t feel I’ve truly understood something unless I feel that I could apply my new knowledge in a different domain where the same principle is at work. Truly understanding the handcuffs puzzle would mean that if someone gave me (say) a metal wire disentanglement puzzle whose solution involved the same trick, I would be able to solve it by transferring what I knew about the handcuffs puzzle. I was pretty sure I hadn’t absorbed the lesson of the handcuffs puzzle at this deep a level.
It’s possible that I understood the handcuffs puzzle as well as other people in the room but simply didn’t experience the satisfaction that they did with that level of understanding. But I suspect that I understood it less well, and am both an exacting thinker and dense, because I have independent evidence that my spatial abilities are lacking. Athletics and dance have been difficult for me all my life, because there are so many things I have to think about all at once as I move my body through space. Also, when I tie knots, it’s pretty much hit-or-miss. This is especially true when I’m tying up a parcel and my pressed-down finger is obscuring the half of the knot-in-the-making I’ve already prepared; once the crossing is out of sight, I no longer have a mental picture of which strand went over the other, so I have only a 50 percent chance of making the other half of the knot in the appropriate way. This was a problem just a few days ago, when I was tying up broken down cardboard boxes for recycling.
But there’s a different problem in three-dimensional geometry that’s vexing me even more than knot-tying these days.
BELMONT, MASSACHUSETTS, 2017
A very tangible symbol of my inability to transfer knowledge from one three-dimensional domain to another is the not-quite-fully-collapsed heat-box that (excepting those times when my wife and I have company over) lives in a corner of our living room, silently taunting me with my inability to fully collapse it. It’s a box that kills the bedbugs and bedbug eggs that may be hiding in our luggage when we come back from a hotel. “Cimecticide” isn’t in any dictionary yet, but in another decade I bet it will be a common word for the practice of killing members of the genus Cimex, better known as bedbugs. These literal bedfellows of the human race are infiltrating more and more of our homes, and while they don’t currently carry any serious pathogens (“Growth mindset!”) yet, I suspect it’s only a matter of time before — oh, but you came to me to learn about math, not bugs.
Anyway, my wife and I are fanatically cimecticidal, and the box in the living room was to have been our biggest and best bedbug-zapping box yet, but alas, after unpacking it we discovered that it requires two separate heaters, and our home’s antiquated electrical wiring can’t handle the load. So we decided to pack the thing up again, to return or resell it. And this is where we faced a problem, because the manufacturers of the ZappBug™ Oven have deliberately made it hard to pack the unit up.
(That last bit is not a surmise on my part: when in desperation I called customer service, they said they’d intentionally tried to make it impossible to return the oven to its original compact state. I guess they’re worried that people who give other people their bedbug zappers might, in the process, also give them their bedbugs.)
In packing up the ZappBug™ Oven and shipping it to people like me, the folks at ZappBug make use of the same three-dimensional trick that makers of pop-up tents use: if you have a loop of bendable metal, say of length L, there’s a way to fold it so that it become a triply-wrapped loop of length L/3. Easy to say, right? But not so easy to do. Check out Evelyn Lamb’s article “Mathematicians Solve Problem of Folding a Pop-up Tent” or the article “Overcurvature describes the buckling and folding of rings from curved origami to foldable tents” by Pierre-Olivier Mouthuy, Michael Coulombier, Thomas Pardoen, Jean-Pierre Raskin and Alain M. Jonas.
One incarnation of the triple-wrapped loop problem is the problem of folding a band saw blade. Watch Paul Sellers’ video on this topic. The difference between a band saw blade and a pop-up tent is, when you start folding a band saw blade against itself, there’s a lot of empty space that you can stick your hands through. With the pop-up tent (or the bedbug box), there’s a flexible but impassable membrane threaded on the loop that gets in the way of where you might want to put your hands, so you really need to have a clearer sense of what you’re trying to do. Though I get the feeling that if you see the membrane as being an obstacle, you’re looking at the situation the wrong way. (Which I realize sounds like one of those annoying inspirational mantras about how the way you get through the wall is to not see the wall.)
I’ve watched the video about folding a band saw blade and then tried to fold up the bedbug box. And then I’ve watched the video again and tackled the box again. Each time, I’m unable to transfer the geometric trick from the one domain to the other.
A flattened but not fully collapsed Play-Hut.
A fully collapsed (and bagged) Play-Hut.
What’s especially galling about the situation is that I am able to fully collapse a pop-up tent! — specifically, the Play-Hut I got for my son when he was four. Like the ZappBug™ Oven, the Play-Hut is just four circular metal hoops joined by fabric. In both cases, it’s easy to flatten the structure, but the final step of taking that flattened structure and looping it around itself three times is harder. For the pop-up tent, I can make that final step; for the ZappBug™ oven, I can’t. I’m sure it would help me collapse the oven if I could psych myself into believing I was actually collapsing the tent instead, or if I could just focus on the topology and nothing but the topology, so that the cosmetic differences between the tent and the oven would recede into oblivion, but I can’t seem to do it.
Likewise, I’ve tried to apply the prescription from Evelyn Lamb’s article: “To fold a ring into three loops, place your hands on opposite sides of the ring. As you lift up, bring your hands together and grab the opposite sides in one hand. Use your free hand to coax the two opposite sides down and toward each other to form a saddle shape. At both the top and the bottom, push one side over the other and collapse the loops together.” Even with another person to help me, I’m defeated.
One thing that consoles me is that I’m not the only one who finds this hard. After all, there wouldn’t be videos about folding band saw blades and articles about the problem of folding pop-up tents if other people didn’t find it problematic! So it’s good to know that I’m not alone in finding these aspects of our three-dimensional world confusing. In fact, in a sense we humans (unlike birds and fish) live in a “two-and-a-half-dimensional” world; most of our movement occurs in two directions, and our kinesthetic intuitions are accordingly limited.
ODD AND EVEN
As a way to try to get a better feeling for triply-wrapped bands, I’ve gotten in the habit of photographing rubber bands that have fallen into a triply-wrapped or quintuply-wrapped state not because of deliberate action by their human taskmasters, but because they’re just doing what rubber bands do naturally. It’s an odd sort of domestic nature-photography, I’ve included the most interesting-looking of them below.
You’ll notice that there are no doubly-wrapped or quadruply-wrapped bands; if you count the number of loops properly, you always get an odd number. (Most of my photos show triply-wrapped bands; in this grouping of eight photos you’ll see one quintuply-wrapped band and another one that’s so tangled that I’m not sure whether I should call it quintuply-wrapped or septuply-wrapped.)
If you want an explanation of why the wrapping factor is always an odd number, you’ll have to ask someone with more topological intuition and technique than I. (And if you are such a person, and you have a good intuitive way to explain it, please share it in the comments!) This phenomenon is related to the fact that, without letting go of a lit candle, you can give it two full turns, or four, or any even number, but not one full turn, or three, or any odd number. See Thane Plambeck’s Balinese candle-dance video. The candle trick is also known as the plate trick; it was used in the musical Waitress, and was featured in a segment of the 2016 Tony Awards. Evelyn Lamb calls it the pie trick in her essay, which is a nice pun, since the waitresses’ plates are holding pie and the maneuver’s mathematical name is the 
James Tanton, in his book “Solve This”, has a short discussion of rubber bands. In section 12.2 (“Rubber Bands and Pencils”), Tanton writes:
“There is only one way to produce loops in a band of paper that is sitting on a table top while maintaining its vertical ‘walls’. Pinch a fold of paper into the center of the loop. Pick up the end of this fold and flip it back over to the outer edge of the band to form two loops (with upright walls) within the band. Any loop you produce in the band must be balanced by another loop counteracting the effects of the production of the first. For this reason, two loops appear in the procedure illustrated; and, in arbitrary manipulations, only even numbers of loops can appear. Adjusting the paper (or the rubber band) so as to ‘stack’ these loops along with the original circuit of paper thus produces an odd number of loops to wrap around the pencil. And conversely, given any specific odd number, one can clearly wrap a band around a pencil that many times.”
In lieu of the diagram from Tanton’s book, I’ve included a photo of what he describes.
Tanton’s words get me most of the way toward really getting it, but I still feel there’s something I’m missing. I can see that manipulations of the kind he describes always change the number of loops by two, so an odd number stays odd. But how can I be sure that every way of putting the rubber band around the pencil can be obtained by a succession of manipulations of that kind, starting from the singly-wrapped rubber band? Or, conversely (and equivalently), how can I be sure that every way of putting the rubber band around the pencil can, be means of a succession of manipulations of the kind Tanton allows, be reduced to the condition of a singly-wrapped rubber band?
Here you see some of the comical wariness of a seasoned mathematician, who may distrust steps that to the less battle-scarred seem safe or even obvious. Viewed in isolation, my qualm seems fussy and unreasonable, but what you don’t necessarily see is my tragic back-story: all the times in my education when I’ve been burned making intuitive leaps of exactly this sort that turned out to be fallacious. Too many grassy fields I’ve trod turned out to harbor hidden landmines. So, when it comes to committing to a proposition, I, like many other mathematicians, can be a bit of a commitment-phobe.
In the case of the odd-number-of-loops theorem, I feel that there’s a different, more rigorous way to see why it’s true; can any of you help me find it?
(And do any of you have similar stories to tell, about mathematical facts that you sort of get but sort of don’t?)
But my real problem isn’t understanding rubber bands and pencils; it’s getting rid of the box in my living room. Here’s what I’ve learned (or think I’ve learned) from the rubber bands so far. First, the number of times a rubber band wraps around an imaginary spine depends on where you think the spine is. (See the dark green rubber band, which requires that you imagine a very bent spine.) So maybe the trick to collapsing a pop-up tent or a bedbug box in practice is not just to ignore the membrane, but to actively imagine a spine passing through it, at the very start of the process; the imaginary spine starts out bent, and you imagine you’re straightening it. (I can say it, but I can’t see it.) Second, when you’re doing the triple-wrapping trick, you’ve got to break the symmetry of the original situation twice. The first time you break the symmetry is when you pick up the rubber band, grabbing it by two particular points; the second time is right after you’ve brought the two “wings” together, as illustrated by the light green rubber band. So that’s the time you really have to do the right thing.
If any of you are Boston-area residents and want to take a shot at collapsing the box, let me know! I’ll pay $20 to whoever can collapse it for me. That would enable me to put it in the mail to another family that wants to buy it from me but doesn’t live in the Boston area. In addition to earning $20 (and my gratitude), you’ll also earn kudos in the Endnotes — especially if, in addition to solving the problem for me, you teach me the right way to think about it so that in the future I can solve this kind of problem on my own!
Next time: More about .999…
Thanks to Sandi Gubin, Scott Kim, Evelyn Lamb, Jenny Moseley, Shecky Riemann, Rich Schroeppel, Rich Schwartz, James Tanton, and Allan Wechsler. This essay is dedicated to the memory of my sixth grade teacher Jesse Stern.
ENDNOTES
#1: I implied that people with limited spatial intuition shouldn’t go into fields like topology, but in an important sense that’s not true at all (even if it is true in my own particular case). One reason is that in higher-dimensional space, we don’t have much kinesthetic intuition for how things behave, and the intuitions that we do have are sometimes wrong, so being devoid of intuition can impart a useful freedom from prejudice. But more importantly, there’s value in going into a field that you find difficult to grasp, as long as you’re willing to be really persistent, because if you find a different way to think about things, something that works even for someone like you, chances are that other people will find it useful too. One might call this the Schmendrick effect, after the character in Peter Beagle’s The Last Unicorn who is told by one of the great magicians of the age: “My son, your ineptitude is so vast, your incompetence so profound, that I am certain you are inhabited by greater power than I have ever known. Unfortunately, it seems to be working backward at the moment, and even I can find no way to set it right.” The late mathematician Raoul Bott, who among other things figured out why (n+8)-dimensional spheres are a lot more like n-dimensional spheres than they are like spheres of intermediate dimensionality (“Bott periodicity“), credited his success to the fact that he couldn’t understand the way other people did topology, so that he had to find his own ways to think about it. As he put it, “There are two ways to do great mathematics. The first is to be smarter than everybody else. The second way is to be stupider than everybody else—but persistent.” (Bott periodicity is about higher homotopy groups of spheres; an accessible introduction to this topic is Evelyn Lamb’s essay.)
#2: My interest in pushing against my own limitations as a topological thinker goes back a long way. When I was at MIT in the ’90s, I used to run an activity during IAP (Independent Activities Period, held in January every year) called “Choreographic Topology”, partly inspired by work that Scott Kim, Karl Schaeffer and Erik Stern have done on bodily math (go to MathDance.org and SternWorks.org, or watch their TED talk). In a way I had no business running such an activity, on account of the fact that when it comes to topology I am not merely an amateur but an untalented one. Would you want a screechy violinist to conduct a chamber ensemble? Still, I put together a suite of activities that would give students a chance to collectively form knots and links of various kinds and study their deeds and sufferings performatively, moving their bodies in space in a coordinated fashion. It was a lot of fun for me and the students, though I can’t say that my kinesthetic appreciation of three-dimensional space gained much from the process; for that, I think a daily practice, rather than a yearly one, would be required. And even then, being just one part of a knot doesn’t necessarily impart an understanding of what the knot as a whole is doing. I would need to be magically transformed into some snake-like creature to get a true kinesthetic appreciation of what’s possible in three-dimensional space.
REFERENCES
Pierre-Olivier Mouthuy, Michael Coulombier, Thomas Pardoen, Jean-Pierre Raskin and Alain M. Jonas, Overcurvature describes the buckling and folding of rings from curved origami to foldable tents. Nature Communications, Vol. 3, No. 1290, December 18, 2012.
James Tanton, Solve This: Math Activities for Students and Clubs, MAA, 2001.
mathenchant.wordpress.com 
When it comes to rubber bands, it sounds like you’d be interested in the concept of writhe. This concept shows up in various guises, but the simplest might be the writhe of an oriented knot drawn in the plane. The orientation (a little field of arrows tangent to the knot) and the distinction between an overcrossing and undercrossing allow you to distinguish between “right-handed” and “left-handed” crossings. To compute the write, add up numbers for each crossing: count each right-handed crossing as 1 and each left-handed crossing as -1. A figure-8 curve drawn in the plane has writhe ±1, while a plain old circle has writhe 0. This is probably why none of your photos of rubber bands lying more or less flat on the plane show figure-8’s: unless I’m confused, they all show curves of writhe 0. If they had nonzero writhe, this would need to be compensated for twisting in the material of the rubber band, which would increase the rubber band’s energy.
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Nice! So one way to see why an even number of windings around the pencil is impossible would be to show that the number of crossings would have to be odd (implying that the writhe is non-zero), and to show that writhe is conserved. I guess you could prove both by using Reidemeister moves; is there a more direct way to see it?
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I don’t know if there’s a more direct way. When doing knot theory, it’s probably good to bite the bullet and prove that the Reidemeister moves work as advertised, and then enjoy the easy proofs of many other facts. Note that the first Reidemeister move does not preserve the writhe!
If you look at the picture I just linked to, you’ll see it being used to change the writhe by 1, by creating an extra twist in a length of rope. Here’s a fun puzzle: suppose we have a length of rope with two extra twists like this, but with opposite handedness, so that their total writhe is zero. Use the second and third Reidemeister moves to make both twists go away!
This is fairly tricky.
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Not sure how I missed this post in March. Having erected and put away a pop-up tent on the beach for my nephew a few times this week, the problem is near the top of my mind. As happens increasingly often, my collection of odd maths facts has the answer.
John had the right idea – you look at the linking number of the loop.
The paper “Topology Explains Why Automobile Sunshades Fold Oddly” by Curtis Feist and Ramin Naimi proves the odd-number-of-folds theorem by looking at the linking number and using a bit of braid theory. It’s a very short paper, most of which is taken up by definitions from knot theory.
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While I can’t help with the box problem, I did want to leave two comments. First, thanks for another intriguing and well-written post about mathematics! I’m very interested to pursue this topic further through some of your referenced materials.
Second, I really appreciate your sharing your experience of “not getting it” the way you did in this article; that really resonated with me. I often have the experience of learning some math in a group or class and feeling that I “don’t get it” when the people around me seem to be getting it (or at least not expressing the same level of discomfort as I am). I’ve always been at least a little perplexed by the frequency with which this happens since I am a math teacher, and did very well at university (which suggests at least a reasonable propensity for the subject). Over the years I’ve wondered if it is because I am just a little slow on the uptake (dense in your words), or if it is that I have a higher threshold for when I consider myself to have “gotten it.”
Just last week I was in a room of teachers being shown an algorithm to find the square root of an arbitrarily large number by hand. While I could easily perform the process, there was a step that was mysterious to me; I couldn’t make sense of it. The teacher did their best to explain it to me with the diagram, but I couldn’t make the connection. I could understand what they were saying, but it didn’t rise to the level where I really had a “sense” for it, so even though everyone else seemed to understand it, I kept saying I didn’t. I’ve thought about it a lot since, and I’m close to understanding it, but still not quite there yet : )
I may never know whether I’m “dense,” cautious, or a combination of both, but I’ve come to see this as an asset. It forces me to think longer and harder about things, and be honest with myself about my level of understanding. In the end, I think it makes me a better teacher, and student.
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After reading your article I took a second look at my IKEA laundry hamper. I can’t post a picture here but I’m including a link that shows it. I think it is the perfect tool to visualize the action that James Tanton describes of making a saddle and pinching the middle together. Then twisting and meeting. I was having trouble replicating the described action with a rubber band in a way that I could slide it onto a pencil.
The laundry bag is mesh so you can see through the sides and therefore not lose any visual info. The bottom is a wire oval which does not collapse or twist. The sides are made up of one continuous wire which is conveniently in a necessary saddle shape already! https://www.amazon.com/Ikea-Laundry-Basket-Collapsible-Black/dp/B00BKOQ464
I have to admit that I have collapsed it several times before with no problem. But after reading this article and experimenting with rubber bands I saw the light in this bag but suddenly could not perform the necessary collapsing action. After some trial and error I can do it consistently and doing it with the mesh bag somehow makes it more clear. I hope this can be of some use to you
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Whilst trawling through the internet I found this blog. After I finished reading this I entertained about some of the values mentioned here and it definitely gave me many views on the different things referred to within this blog.
https://toolsadventure.com/best-band-saw-reviews-and-guides/
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