Chapter 50 — Around the World and Back Again
Rufus gives a shout out to a Primed Minds Exploration.
Followed by Chapter 50 —— Around the World and Back Again, in which Saskia sees Wassily for the first time in a decade.
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Hello Friends,
A few years ago, I ran a website called Primed Minds that celebrated mathematics by taking visitors on curated journeys into the mathematical wilds. I called these journeys Explorations, because they were designed to be a bit of a guided expedition into a joyous part of mathematics that most people unfortunately missed in school. I envisioned these Explorations as the the equivalent in your math class of watching a film in your English class on the last day of term. That is, they were designed to inspire. Anyway, given that we’re in the holiday season it seemed like a fun time to highlight one.
Last week’s discussion of infinity represented to bones of an Exploration that I never finished putting together, but happily this week’s chapter mentions Möbius strips, reminds me of perhaps my most popular of mathematical adventures. So, I’m going to give the Möbius Cuts Exploration a quick shout out today. I’ve shared this content with both mathematicians and self-professed math-phobics, and everyone who has engaged, both enjoyed it and was surprised.
To tempt you to take a look yourself, let me whet your appetite by walking you through the opening gambit. The Exploration starts by encouraging the visitor to make a paper cylinder and draw a line around the center. I then ask what would happen if you cut along the line. When presenting this in person, I generally prefer that everyone has their own paper, pen, sticky tape and scissors, but the advantage I’ve got here and now is that you won’t be tempted to simply make the cut without thinking. OK, so this opening gambit is maybe too straightforward. Yes, you do end up with two shorter squatter cylinders. It was the warm up! It wasn’t supposed to challenge you.
Next I want you to imagine making a Möbius strip with a piece of paper. That is, take a strip of paper, put a half-twist in it and tape the ends together. Now imagine drawing a line down the center. Can you visualize that? Perhaps you see a problem? Maybe you don’t. In any event, once you’ve drawn your line down the center, predict what will happen when you cut along that line.
Have you got a guess in mind? Good. I suspect that some portion of you already have a pretty good idea what happens, but, if that’s not you, then you should try it for yourself. What I will say here is that, no, you don’t end up with two shorter squatter Möbius strips. I also suspect that most of you——even those who’ve seen everything so far before——would be less sure what happens if you make two lines, each a third of the way into your Möbius strips and cut along those instead.
Obviously I’m biased, but, if you’re even remotely intrigued, I’d definitely encourage you to have a look at the rest of the Exploration, maybe with your niece, or your grandmother, your dad or your kid brother. I can assure you that whoever you take with you on your journey will thank you.
In any event, visualizing this sort of thing is much closer to doing real mathematics than anything you likely saw in your high school math class. It’s almost certainly a lot more fun too. And though it’s not exactly the context in which Möbius strips crop up in today’s chapter, it is in spirit very related; both instances belong to a branch of mathematics called topology. So, now that I’ve primed your minds, let’s see where Saskia and Wassily take us.
Until next week, be kind to someone and keep an eye out for the ripples of joy you’ve seeded.
Cheerio
Rufus
PS. If you think of someone who might enjoy joining us on this experiment, please forward them this email. And if you are one of those someone’s and you’d like to read more
And now, without further ado, here’s chapter fifty, in which Saskia sees Wassily for the first time in a decade.
— 50 —
Around the World and Back Again
Wassily’s true reason for showing up was principally the outside hope he harbored that there might be a chance for him with Mica. He hadn’t really believed there was much chance of prevailing, but hope pressed up against impossibilities, like a seed sending tendrils through a crack in the sidewalk. Saskia was the boot that crushed that dream, albeit as inadvertently as a passer-by walking down the street.
“Wassily, this is——”
“Hello Saskia,” Wassily short-circuited Mica’s introduction. His immediate hope was dashed, yet a new opportunity sprung up with it. If the universe had any sort of plan, then it had to be significant that Saskia was sitting on Mica’s kitchen stool in front of him.
∞
For her part, Saskia immediately recognized Wassily’s hope with Mica. Of more immediate consequence, though, was Mica’s attention on her.
“You two know each other?” Mica asked.
“We had some classes together at school,” Saskia dodged the core of Mica’s question, then turned to Wassily. “Linear algebra?”
“Yes,” Wassily agreed, struggling to follow Saskia’s lead. “Topology too.”
Mica studied Saskia. “You didn’t think...?”
Saskia could see that Mica was annoyed. But she wasn’t sure if it was just the truths she’d elided, or that Wassily’s attention had now shifted to her. It belied the insignificance of their having merely known each other. “I didn’t realize that your Wassily was mine,” she equivocated.
“Is that you?” Wassily interrupted the moment, indicating the open photo album on the counter, his eyes flicking up to Mica.
“And that was her college crush,” Saskia indicated Wangari’s puckered lips on the opposite page.
Mica’s eyes darted to Saskia.
Wassily examined the photograph. “Makes you wonder about the path not traveled. Right? Branching timelines.”
“Branching timelines never made sense to me,” Saskia declared, eager that the conversation not stop. Still addressing Wassily, she asked: “You think the world could not be a smooth manifold?”
Wassily cocked his head in consideration, and then his eyes fell back on Mica who looked baffled. “What the hell is a smooth manifold?” he asked rhetorically.
Mica squinted her eyes and silently mouthed “What?”
With the uneasy mood, Mica’s question was enough to motivate his launch into an explanation of the mathematical concept. He explained that a manifold was a geometric object that locally looked like Euclidean space. “Like a line, or a plane, or a solid. So a curve is a 1-dimensional manifold. Locally it can only go backwards of forwards.” Wassily traced a flowing squiggle in the air in front of him.
Despite herself, Wassily’s diversion piqued Mica’s curiosity. “What do you mean locally?”
Wassily was away: “Well, taking the curve. It might wobble about on a page, or like the arc I just traced in three-space. But at any one point on the curve you can either go forwards or backwards——if you zoomed in close enough it would look exactly like a little line segment. You wouldn’t even see a bend. Mathematically, you could map it smoothly, point for point, onto a line segment.”
“But that doesn’t work for a branch point,” Saskia clarified, though when Mica’s eyes turned back to her she could feel Mica’s focus shifting again.
“And it doesn’t have to hold globally, either,” Wassily added. “You might not even be able to straighten it out. To untangle it.”
Mica’s eyes pinged back to Wassily.
“Take a rubber band,” he elaborated. “You can’t stretch it out, or bend it, to look like a line segment. Not unless you commit some violence to it. Break it.”
“It’s a circle,” Saskia added. “Although you could zoom in locally at any point, and it’d look like a line segment, globally it’s not just a deformed line; the two ends are connected.”
“Knots can get even more complicated!” Wassily enthused.
“By a knot, Wassily just means a piece of string with the two ends fused together,” Saskia made Wassily’s comment no more clear. “Otherwise you could simply unwrap whatever tangle you’d made.”
“And a rubber band is just an unknot,” Wassily hit Saskia’s lob back. “It’s a piece of string without any ‘knot’ in it. Just looped back, with the two ends fused together.”
Saskia could see that she and Wassily were chipping away at Mica with their infectious enthusiasm.
“In two-space,” Wassily sailed on, “the unknot is the only knot; every other knot would have to intersect with itself, since there isn’t enough room for it to go over or under itself.”
“If it intersected itself it wouldn’t even be a smooth manifold anymore,” Saskia clarified.
“But in three dimensions, the string can go under or over itself. The way you and I think about knots in the real world.”
“And in four dimensions——” Saskia started.
“Hey don’t step on my punchline!” Wassily enthused back. “In four-space every knot is the unknot.”
Mica looked at them both, utterly bewildered.
“OK, imagine you wanted to untangle a knot by lifting one strand from going under another to going over the top. In three dimensions you can’t necessarily do that——”
“It might even be impossible,” Saskia chimed in.
“But in four dimensions: you can slide the bottom piece forward in time, and then lift it up at a time where the top piece isn’t in the way. Then move it back in time and wallah! Keep doing that and you can untangle any knot.”
Mica looked a bit deflated. “That sounds less interesting.”
“No,” Wassily protested, “it just means the space that your manifold is embedded in is pretty important. My specialty is four dimensions.” Wassily was grinning again.
“I thought you just said everything was the same in four dimensions?”
“No.” Wassily gave Mica a gentle finger waggle. “I only pointed out that all one-dimensional manifolds are boring when embedded in four-space. The raw manifolds get more complex as the dimensions go up, irrespective of their embeddings.” Wassily held both of his hands splayed out in front of him and began gradually swooshing them through the air, as if delineating a surface. It was a visual echo of the one-dimensional squiggle he’d traced out earlier with a single fingertip. “There are two-dimensional manifolds that can’t even be embedded in three dimensions.” And, as if to underscore his point, he collided the pinkie finger of his right hand across the back of his left hand in a sort of karate-chop. “You need four dimensions just so they have enough room to not self-intersect.”
“It’s exactly the same problem that the knots had in two-space,” Saskia made Wassily’s analogy more explicit.
“And, just as we fused the ends of a string together to make knots, we can identify the edges of a sheet of paper to describe surfaces, that is two-dimensional manifolds.” Wassily then cast his eyes about Mica’s kitchen.
It was Saskia who obliged him, passing him the pen and notebook that Mica had given her the other day.
“Alright,” Wassily enthused. “Let’s start simple. Imagine identifying the single arrow edges of this square, in the direction they point. Then do the same for the double arrowed edges.” Wassily the sketched the following picture:
“What would your surface look like?”
“Imagine you cut the square out,” Saskia clarified, “and then stitched the left side to the bottom side, and the top side to the right side.”
Mica imagined, as she’d been instructed.
Wassily continued adding instructions: “You should imagine the whole square is stretchy, so the only points that are going to touch are the sides.”
“Would it make a sort of spinach filo pastry triangle?”
Wassily cocked his head sideways and let his eyes run to the ceiling. “Yes! But, topologically, we’d call that a sphere——you could stretch it without tearing it or gluing any more points together and you’d end up with a ball.”
“What about the corners?” Mica objected.
“Even the corners you could stretch out flat. Imagine it was a balloon and you inflated it right up to the point before it burst.”
“OK.” Mica didn’t seem completely convinced but she let it slide.
“Point is,” Saskia added, “it’s not just a two dimensional plane anymore. In the same way that the earth isn’t actually flat. Even though you can see that it looks flat at any point——imagine the edges are just like those on a computer game where you go off the left side, but come back in the bottom; so even on the edges you can imagine that you can go left or right or up or down at any point. Locally, the sphere——just the surface, not the inside——is two dimensional; it looks like the plane, even if globally there’s no way it could be situated in the plane.”
“Alright, now do the same with this picture.” Wassily scribbled a new square, but this time indicated for different edges to be identified:
Mica blinked, and blinked again.
“Do it in stages,” Saskia encouraged her.
Mica looked over at Saskia, then back at Wassily’s second square. “Well, if I stitch the top and bottom together I get a cylinder ... ”
“Right,” Wassily encouraged her. “Now stretch it out and bend the ends back together.” He mimed his instruction, starting with his hands palms up, and curling them so that the tips of his thumbs touched the tips of his respective pointer fingers. Then, he lifted his hands with a twist that brought the two thumb-pointer-finger end-circles together.
“A donut?” Mica asked tentatively.
“Exactly! Though I’d call it a torus. That’s the word a mathematician would use. The same way I’d call a ball a sphere.” He grinned. “OK, how about this one?” This time Wassily’s square aligned the top and bottom, but his ends matched with opposing orientations.
Again, Mica stared at Wassily’s picture and blinked. Eventually she looked up at Saskia and Wassily, unsure.
“You can see something funky is going to happen if you join the left and right ends first,” Wassily cautioned. “You’ve now got a Möbius strip that you’re going to somehow stitch the edge together on. The thing is, imagine you’re an ant standing in the center of the square. If you now walk off the righthand side of the paper and back onto the lefthand side, that would switch you to the other side of the paper.”
“So whatever the surface is,” Saskia picked up where Wassily left off, “there is no ‘inside’ or ‘outside’, not the way a donut or a sphere has an inside and an outside.” The banter took Saskia back to the Topology class she had shared with Wassily a decade ago. She smiled at both the mathematics and the memory. Then, suddenly, Saskia wondered if the non-orientable looping back of the Möbius strip was somehow connected to her experience with the backpack and pen, back in the hotel room, when she first arrived in Dallas.
∞
Mica was torn watching Saskia and Wassily keep one-upping each other. Their connection seemed like more than just mathematics. She hadn’t had any interest in Wassily, and still didn’t, but his easy repartee with Saskia felt like an inadvertent slight.
She wondered what it must be like for a mathematician to find themselves in the company of another mathematician. Was it two non-native English speakers happening upon one another in a crowded room? Their shared tongue creating an island in the conversation around them, opening them up.
Her mind rewound to where the conversation had started. “Wait, what does all of this have to do with branching timelines?”
Wassily turned back to Mica. “Well, if the universe branches, then it’s not a smooth manifold.”
“But if it does branch——that’d just be weird; that’d mean facsimiles of the entire universe suddenly springing out of themselves.” Saskia rubbed her forehead. “It’s not like mitosis. You’re demanding perfect copies appear spontaneously.”
“And if the world branched like that, then it’d have to sit in a really high dimensional embedding space,” Wassily completed Saskia’s thought. “Just to avoid self-intersections.”
Mica looked directly at Saskia. “Maybe the other you I saw at Cleo’s wasn’t your future you.”
Abruptly, Wassily’s eyes flicked between the two women, before they settled on Saskia. “You’re the time traveler?”
The needle on the proverbial track of the universe screeched to a halt. Even Mica wasn’t sure if her comment had been an inadvertent slip, or if it reflected an irritated jealousy with Saskia. Whatever the case, she’d stopped believing that the other Saskia she’d seen at Cleo’s was the Saskia in front of her having later gone back in time.
“So your objection to branching points wasn’t just in the abstract?” Wassily looked from Saskia to Mica and back again. “Have you felt it? Have you felt yourself splitting in two?”
