Chapter 23 — Life on a Rope
Rufus takes a bit of a risk and describes what algebraic geometry is with the example that first delighted him.
Followed by Chapter 23 —— Life on a Rope, in which Mica seeks outside expertise to make sense of time travel.
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Hello Friends,
Today I’m going to take a risk and share a little mathematics!
I’ve mentioned algebraic geometers before, and today’s chapter is going to introduce a topologist but really Wassily is a bit of a placeholder for many of the mathematicians I’ve encountered over the years. I want to be upfront though, Wassily really is an amalgam of many of the mathematicians I’ve known, not all. If you’re an algebraic geometer, or a topologist, or any other variety of mathematician, please don’t feel misrepresented if you don’t recognize yourself. Mathematicians, like everyone else, come in many shapes and sizes.
That said, I realize most of you out there are not mathematicians. So, for those whose eyes have glazed over when I’ve mentioned algebraic geometry in the past, take a deep breath and buckle up, because I’d like to give you a glimpse of the beauty and power of mixing symbolic variables with pictures (which is kind of what algebraic geometry is). Specifically, I’m going to describe for you my own first brush with algebraic geometry.
It wasn’t until grad school that my eyes were really opened to algebraic geometry, but the thing is that foundational moment could have been experienced by much earlier, so today I want to take a minute to recall it. It was the introductory lecture, after all.
The lecturer was one Brian Conrad whose self-described reputation for pedantry made him intimidating, and I confess to being lost before the end of the course, but this opening salvo lives with me to this day, over a quarter of a century later.
So, let’s get started!
I’m hoping that most of you will vaguely recall Pythagorus’ theorem, which itself is an interplay between numbers and pictures. As a refresher, it says that the area of a square sitting against the longest side of a right triangle——a triangle with one corner the same angle as that in the corner of any square——is the same as the sum of the areas of the two squares sitting against the other two sides (those either side of the right angle that defines the right triangle). Pictorially,
You see the connection between the numbers (the sum of pink areas, and the black area) and the geometry of the picture, right?
And maybe you recall that the triangle with sides 3, 4, and 5 is one such triangle; a quick examinations verifies that the algebra at least checks out:
3x3 + 4x4 = 5x5
The wonder of Brian’s opening lecture was to describe a way of enumerating all such triangles with whole number sides x, y, and z. That is, all whole number solutions to:
x^2 + y^2 = z^2
To see this we’re going to employ a few nifty tricks, the first of which is a piece of algebra: dividing through by z^2. Which yields:
(x/z)^2 + (y/z)^2 = 1
But if we think of X = x/z and Y = y/z as just two rational numbers, which is fancy math speak for fractions, then really we’re looking for rational——that is fraction number——solutions to:
X^2 + Y^2 = 1
which some of you might recall as the equation for a circle with radius 1, centered at the origin. That is, we’ve reduced our task to finding points on the unit circle with rational (aka fraction number) solutions.
Now comes the geometry. We’re going to draw a line that intersects this circle and note that finding the intersection points amounts to solving a quadratic equation (since, as you may recall, a line is something of the form:
y = m x + b
so we’re really just replacing the Y in the circle equation with “m X + b”). For those of you listening along, this is easier to see on the page, but bear with me.
Now, the brilliant insight here is that if the m and b are rational numbers, then the X’s that we’re looking for will have the form:
some rational number plus or minus the square root of some other rational number
which means the only possible non-rational thing going on is the square root (remember, non-rational just means “not a fraction”, and not “crazy”). And this means that if we knew ONE of the roots is rational, then we’d know the other one must be too.
This is exactly the kind of high-level thinking that goes into the creative part of mathematics. Notice that we haven’t made any actual calculations yet; instead we’re just thinking about ideas and implications. Moreover, the idea that solutions of quadratic equations come in related pairs is something very natural to any mathematician——and again, not surprising, since a generic line will intersect a circle in two points (or one double point, or no points which really just means imaginary points, but I’m getting sidetracked).
Anyway, we certainly know one point on that circle, namely (-1,0). Though just to be clear there is nothing special about this point (except that it is rational); we could just as easily have chosen (3/5, 4/5). Here’s a picture of what I’m talking about:
So the thing is any other rational point on this circle will define a line between the point we’re starting with and (-1,0), and moreover a little careful consideration will show that such a line will have rational coefficients m and b. The magic thing, is that this correspondence goes both ways. We could also parametrize (which is a fancy way of saying: “list off”) our lines through our starting point (-1,0), using the slope m alone (the b’s will come for free, if you will; the point (-1,0) kind of makes that happen).
Thus, every Pythagorean triple is parametrized——or counted, or enumerated, or listed——using our starting point (-1,0) and a rational slope m. Amazing, huh!
OK, I grant that if this is the first time you’ve seen or heard this your eyes are almost certainly swimming right now. But, I also claim that working through the details is really just a matter of dotting i’s and crossing t’s. The amazing thing is that this simple picture and the algebraic insight about roots coming in pairs have given us an algorithm for listing every Pythagorean triple that exists! With some luck, in your heart you now have some hope and expectation as to why this is all true. To be sure, there are details to square away, but you have in your hot hands a mathematical pirate map. Maybe it’s been stained by tea, and sullied by dirt and grime. And maybe it’s even got a burnt hole in the center where someone brought it too close to the flame when trying to make sense of it all, but it’s a map and you could try following it if the whim struck you.
Alright, thanks for indulging me, and let me know if this was helpful or interesting. I doubt that such excursions will become a regular part of these commentaries but I wanted to give you some insight and given that today’s chapter gives another glimpse into mathematical thought it seemed like a fun one to pair this recollection with. Of course if you all love this, I can always add more of these sorts of elements in the commentary section.
Lastly, if you’re listening to this and wish you had a picture to see, know that you can always sign up for the weekly newsletter version of this podcast on my website www.writtenbyrufus.com or go back and have a look at the text version of any past episode under the Episodes tab. The newsletter version is also an easy thing to forward to friends who might find this podcast interesting, just remember to remind them that it’ll probably make more sense if they start at the beginning.
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
PPS. Just for fun I put today’s episode image into a new AI text to video generator and asked it to animate it. The output is pretty cool, especially given how little was required to make it. Anyway, thought I’d share:
And now, without further ado, here’s chapter twenty three, in which Mica seeks outside expertise to make sense of time travel.
— 23 —
Life on a Rope
After Saskia left, Mica texted a man she hadn’t spoken with in six months. Wassily texted right back. He had to give a lecture that afternoon, but was otherwise free, and happy for her to drop by. Mica packed her bag for the day, pulled the battery from its charging station and hoped on her bike.
Believing now that Saskia could slip in time, Mica wanted to consult an expert on the subject. She didn’t know any physicists, but she’d interviewed a mathematician a few months back for a piece she’d written about climate change tipping points. Though his explanations had left her with the distinct feeling that she was swimming in much deeper waters than she’d at first thought, he’d seemed eager to help if she ever had another question that she thought might be down his alley. His department bio described him as a four dimensional topologist. Mica reasoned that time was the fourth dimension. Worst case, he would no doubt be able to direct her to someone more on point.
Forty minutes later, and a little more sweaty than she’d have liked, she arrived at UCLA. Still, she hadn’t regretted her decision to ride. It was a nightmare to find parking anywhere near campus unless you were prepared to fork over an arm and a leg to the parking structures, and risk a dent from the narrow spaces. Besides, riding her bike was better for her and better for the world, even if it was electric.
Wassily laughed when she first mentioned time travel, but moments later he was talking to her about life being like a curve in four dimensional space-time. “Instead of thinking of us as occupying a point in three space, add a fourth dimension and follow the path that point——you or me——cuts out in the new space. The point becomes a curve, and that curve is your life line. Like a rope in four-space.” Wassily stopped for a moment to marvel at his construction. But as he beamed at Mica, he realized, by her deer-in-the-headlights stare, that he’d moved too fast. “No worries.” He waved his hands reassuringly, as if to tamp down the air in front of him. “Let me back up. Sit.”
Mica settled herself on a well worn couch that rested against one wall in Wassily’s office. She was about to place her hand on the armrest, but noticing cushion stuffing spilling out of a hole that was generated more by wear than tear, she opted for her lap instead.
“If you were a point in space,”——Wassily started a new attack on his explanation——“which obviously you’re not, I mean you take up a bunch of space——”
“Excuse you.”
Wassily laughed nervously. “No, I’m not calling you fat or overweight. You’re just not an idealized point.”
“Not fat, just not ideal?” Mica checked, no less affronted.
“You’re wonderful, just not a single point in space. You wouldn’t want to be a single point in space——”
“So why keep harping on it?”
Wassily sighed. He was uncomfortable around attractive women, even if the woman in question was one he knew he had no shot with. In any case, his point was more than nomenclature. To explain it he reduced the dimensions: “Say you live on a line——you could think of this as the rope we were talking about earlier. Actually, don’t think of this as the rope, think of the line as a tightrope. You can walk backwards and forwards along it, but that’s it. Actually, you’re an ant——on a pogo stick, on the tightrope.”
“And it’s important that I’m ant and that I’m on a pogo stick?” Mica was mystified as to why Wassily would introduce these details.
But Wassily persisted. He had chosen an ant because that was small enough that it was essentially just a point on the tightrope. But then he had realized that an ant had six legs, so it actually took multiple points on the tightrope. That was why he had put the ant on a pogo stick.
“And this is how you normally think about the world?” Mica asked him.
“It’s supposed to help give a sense of what is going on,” Wassily stated, without irony. “But, now that I think about it, a pogo stick would keep bouncing off the tightrope.” He paused. “Maybe a unicycle is a better model.”
“I’m an ant on a unicycle on an imaginary tightrope?”
Wassily nodded enthusiastically. “And we’re really just interested in the point where your wheel touches the tightrope.”
Mica got the sense that this rabbit hole would keep descending if she asked for more clarification, so she too nodded.
Wassily then sat beside her and laid a sheet of paper on the coffee table in front of them. A blank sheet, upon which he drew two axes. Indicating the line that crossed the page, he said, “This is how far along the tightrope you are.”
Mica nodded again. If she squinted at it, she could pretend it was the tightrope.
Wassily then pointed to the vertical line, bisecting the page. “Anywhere on this line means you’re at the center of your tightrope.”
Mica cocked her head, askance. It looked to her that all but one point——the intersection point at the center——were off the tightrope altogether.
“Left to right on the page, is where you are on the tightrope,” Wassily clarified, “Up down is forward or backward in time.”
“Got it.” Mica nodded, understanding his meaning. “Center of the page equals center of the tightrope——”
“At the start of time.”
“Higher on the center line is later in time, but still in the middle of the tightrope. Lower is earlier.” So far, so good.
“Great!” Wassily clapped his hands. “Now here’s your life curve.” And with that, he drew the following delightfully flowing curve on his axes:
Wassily then placed his finger on the left end of the curve. “Our ant starts on the left end of the tightrope. And if you, the ant, look out at that point in time, you’ll see yourself at two other points along the tightrope. All at the same moment in time.” He indicated quickly, the other two places that his curve crossed the horizontal axis. “You are simultaneously at three points on the tightrope at the beginning of time. Now, as time creeps forward you pogo smoothly along the tightrope.”
“I’m not sure smoothly pogoing is a thing,” Mica objected. “And didn’t we switch to a unicycle anyway?”
Wassily paused. “Right, imagine your pogo stick has a little wheel on the bottom——a really little wheel——that’s your unicycle. The point where that wheel touches the tightrope is where you ‘are’.”
For a moment Mica wondered to herself if she had had a teacher like Wassily when she was in school ...would she have enjoyed mathematics more or less? And what was it about Wassily and Saskia, did all real mathematicians love drawing pictures? It wasn’t the way she conceived of mathematics, but maybe she’d had it pegged wrong.
Concerned that Mica’s attention was wandering, Wassily checked with his pupil. “Are you with me?”
“Sure,” she assured him. “Sure.”
“So time is creeping forward, and you are rolling out along the tightrope. At the top of my little curve, you are almost as far along the tightrope as the second you that you saw was, when you started. But the fact that my curve now starts to gradually dip down means you’ve started to go back in time. You’re still rolling forward along the tightrope for a while, at least until the curve starts to go left again, but you’re doing so as time goes backwards. And when you start rolling backwards along the tightrope, you’re still moving backwards in time too. Because my curve is now going left and down.”
Mica seemed to be taking it all in, so Wassily decided to test her, to make sure: “If I were me, standing in the center of the tightrope watching you——and remember, for me, time just moves forward, or in our pictures’ terms up the page——would you be coming towards me or moving away from me at that point?”
“I’m riding backwards along the tightrope, right?” Mica asked. At this point she no longer trusted her intuition.
“Yes, but time is also running backwards for you; you’re coming down the curve to the second place it crosses the x-axis.”
“Is this a trick question? Backwards-backwards means, to you, it would look like forwards-forwards?”
“Exactly! To me, it looks like you’re riding towards me as I just move up the y-axis. Though, you’d be facing backwards, from my perspective.”
“Wait, where did the y-axis come into play?”
“I’m on the y-axis. I’m just standing there, in the middle of the tightrope the whole time. Watching you.” This last sentence, as soon as he’d uttered it, made Wassily feel a little uncomfortable.
But Mica blazed past his self-conscious boyish blush. She’d understood what he was saying, and the thrill of that understanding was intoxicating. Wassily had told her before that this was the reason mathematicians did mathematics. It was like a drug. Still, she couldn’t help but check that she’d followed along correctly. She pointed near the bottom of her curve, where it crossed the y-axis. “Do we crash right there?”
“Well you do cycle right by,” Wassily conceded. “Maybe imagine I’ve dropped underneath the rope and am just holding on. And you roll on by.”
Mica reached out to Wassily’s hand and stroked it, apologizing for her imagined blithe crushing of his digits.
“Don’t worry.” Wassily smiled——as much at his gallantly waving off her concern, as enjoying the attention he was receiving. “You’re just an ant, remember. I’m sure it wouldn’t have hurt.”
Mica kissed his fingers better all the same. She knew and understood the power of her tender touch.
“One last question.” Wassily’s awkwardness forced him to break the moment. “From my perspective, from which direction did you roll over my fingers?”
Recognizing that there was more to his question than the obvious, Mica paused. And then she saw it: “From the right. Since time moves forward, or upwards on our graph, so I start on the right hand side of you. That’s super weird though”——Mica was now marveling to herself——“I’m really moving backwards through time to the right, which, for you, is the same as forwards through time from the right.”
“That’s relativity,” Wassily enthused. “Your relative motion.”
But Mica’s mind was elsewhere. She wondered if Saskia had any inkling of how all of this might actually feel.
And then she realized that Saskia’s double, the woman who had turned her back at Cleo’s, must have been an older Saskia, come back from the future. She wondered, though: exactly how far in the future had that Saskia come back from? If Saskia this morning didn’t recall having done so, then she was yet to go back!
What Mica didn’t realize, and what Wassily’s model didn’t account for, was the possibility of life-paths splitting and multiplying, or existing in parallel. But how one life path could split into two, and what that would mean for the person living the split——it wasn’t surprising that that possibility had been glossed over; it would be difficult to imagine that as an option, that it would even be possible to survive such a split at all.
