Chapter 27 — Clutch Move
Rufus describes a conundrum raised by the algebraic concept of dimension.
Followed by Chapter 27 —— Clutch Move, in which Mica sees an analogy between Saskia’s switching her flow through time and changing gears in a car.
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Hello Friends,
Two dimensional space gets a mention in today’s chapter, and I wanted to give a little aside on the idea of adding dimensions algebraically. Mathematician’s intuition is designed around geometry, and in that case going from one dimension to two dimensions is the idea of switching from tracing backwards and forwards along a thread, to permitting movement up and down as well; adding warp to the weft, if you will. Combine the two possibilities and you can reach any point on an expanse of cloth by starting anywhere else and first traveling up the warp and then across the weft. Similarly, you could imagine going up to three dimensions by running threads up and down through a layered stack of fabric. In that case, to get from one random point to another, first you’d trace one of the up-down threads to rise up, or sink down, to the second point’s layer, after which you revert to traveling along warp and weft.
The algebraic concept of dimensions is similar. It simply adds up scalar multiples of different fundamental “directions”. A point in 29-dimensional space might represent a farmers’ market vegetable stand’s inventory. The first dimension counts how many apples they have, the second how many apricots, etc, etc. After I’ve stocked up for the week, the farmer will reduce their inventory by the size of my haul along each dimension, yielding a new point in their 29-dimensional inventory tally space.
In our 2-dimensional cloth example you might describe how many threads of warp you pass along the weft, and then, conversely, how many threads of weft you pass along the warp.
Sometimes, however, the algebraic concept of dimensions can give rise to conundrums when we try reconciling our picture with our geometric intuition. Today, I’d like to take a quick detour and blow some of your minds with such an example.
Back in chapter 23 I mentioned quadratic equations. In that discussion I kind of used, and yet elided over the concept, that, for instance the square root of 3——that is the solution of the equation x^2 = 3——cannot be expressed as a fraction (or rational number).
Thinking about this the other day, I realized there are some really fun subtleties buried here, so let’s take a moment and double-click on what’s happening.
A quadratic, that is a polynomial with x—squareds in it, either has solutions somewhere in the field of scalars we’re considering, or, if it doesn’t, then adding scalar multiples of a solution literally doubles the total field of numbers we now have (the term `field’ might feel like a slightly purple description of our collection of scalars, but it’s a genuine mathematical term). Of course if the field of scalars we started with were infinite, as the rational (aka fractional) numbers are, then doubling is kind of non-sensical. Twice infinity is still infinity!
This is where thinking spatially can help (or induce a headache for the uninitiated, but bear with me).
In the case of x^2 = 3. There are no rational solutions (since the square root of 3 is cannot be expressed as a fraction), but if we add all rational multiples of the square root of 3 (the new solution) to the rationals that we started with, we end up with numbers of the form:
x + y * square root of three
And thought of this way, the geometric intuition makes sense: we can think of numbers in our new solution space as x rationals (across the weft), and y rational multiples of the square root of three (up the warp). Considered this way, the solution space is clearly 2-dimensional.
So far, so good.
The problem is we already have a sense of how “big” the square root of three is. Somewhere between 1.7 = 17/10 and 1.8 = 18/10. So, in our mind, maybe we can “place the square root of three on the line, somewhere between 1.7 and 1.8”. Indeed, any of the multiples
y * square root of three
can be slotted in an “empty spot” on our rational number line; that’s what it means for the square root of three not to be rational.
Thought about this way, we’ve sort of geometrically lost our second dimension! Weirdly, our second dimension sort of sits in the holes left along the line of the first dimension.
Of course if we started with the field of all the real numbers then adding in square root of 3 wouldn’t change how many numbers we had, as we would already have had it (since the square root of 3 is a real number; it represents a distance, if you will). That said, as some of you might know, we could, of course, add to the reals the solution to x^2 = -1, typically denoted i. And, as those of you who have encountered the imaginary number i before probably know, in that case the visual representation of numbers of the form x + i y is of a plane, in which we think of the x an y as representing the traditional Cartesian coordinates.
As I said above, first encounters with new mathematical concepts, like dimensions in algebra, can indeed feel strange. Unfortunately, today’s musings are not the right place to really unpack what is going on here. That, I’ll leave to a course in Galois theory, named after the French mathematician, Evariste Galois, who unfortunately died in a duel at the tender young age of 21. But now we’re veering into the realm of history.
Perhaps we’d better get back to The Curve of Time.
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 twenty seven, in which Mica sees an analogy between Saskia’s switching her flow through time and changing gears in a car.
— 27 —
Clutch Move
“I’ve been thinking about it all since Wassily explained the problem,” Mica continued, “and I think I might have a solution: what if there’s a clutch move.”
“A clutch move?”
“Somehow, whenever you’re about to turn time around, you depress a clutch——like in a car——and you temporarily disengage from space-time. As long as you’re out of your own way when you let the clutch out, you’re fine. It’s like in a car, only instead of changing gears, you’re changing directions in time.”
“What about in bed the other morning?” Saskia asked.
“You let the clutch out before you got out of your own way.” It all made sense to Mica. It was a classic example of a situation in which having no technical background opened the doors to thinking really laterally. Why not invoke cars?
“I was lying really still when it happened,” Saskia mused, running with Mica’s theory.
Again, Mica nodded. “But it wouldn’t have mattered, even if you had moved a bit. All your wobbly parallel lines could still run into each other. You can’t release the clutch while you’re still between switching gears. You’d just get a violent crunch.”
“So what is my clutch move?”
“I don’t know what it is exactly, but I bet it buys you some time. It must be instinctual. When you first did it——like a bird flying for the first time. Thankfully you had the instinct for it. Mind you, those without the instinct——they would be pretty quickly eliminated from the genetic pool.”
Thinking about it, the two woman reasoned that it obviously didn’t last forever. Indeed, Saskia had seen what happened if you didn’t move out of your own way quickly enough; it created a kind of stalling event: Saskia’s experience in the bed the other day.
However, although Saskia could try experimenting with it, to see how much time she had, she wasn’t particularly inclined to do so, given how much it hurt. “I’m going to assume the clutch move, whatever it is, buys me a few seconds. I doubt it’s a minute, just long enough that I can move out of my own way. Like a little hop, and it’s probably good to remember where I was.”
“Were you always moving when you slipped back in time in the past?”
Saskia paused to consider. “The Lottery, yes. I was walking from the park. When I met you, yes; back from my bike. The racetrack; back to the betting window.”
“Any others?”
Saskia rubbed her forehead. “My first time, I was heading back to see what happened the first time I slipped in time. And the other night, I was walking around your kitchen table.”
As Mica listened, she was looking, again, at the almost parallel curves Wassily had drawn, and where they intersected. Suddenly, she realized they didn’t just intersect near the tops and bottoms, the places where Saskia was changing the direction that time travelled. Mica pointed to an intersection point near where Saskia simply changed physical direction, not the direction time was flowing. “What about this point? Your bum and tits are criss-crossing, but you’re not changing the direction that time is flowing.”
Saskia focused on the intersection point Mica had indicated. It was a problem with her theory, to be sure. But then she saw it. “That’s just a problem with our two dimensional representation. It’s an over-simplified picture.” Saskia explained to Mica how the drawing misrepresented reality: “I can happily turn about in three dimensions, but if you projected where I am onto two dimensions, and only look at it there, it might look like I’m self-intersecting.” To show Mica what she meant, Saskia directed Mica to watch her as she turned about.
Mica did as she was told.
Saskia spun about and gave her arms a flourish at the end. “See, no problem!”
“Great, so you can do a pirouette.” Mica hadn’t understood what Saskia had just shown her.
Saskia crossed back to Mica, and lifted her hands. She extended the pointer finger on each hand and straightened Mica’s arms. Then she moved back across the room and stood in profile. “Track the back of my head with one finger, and my nose with your other.”
Mica complied, still unsure of the significance of what she was doing. Then, slowly, Saskia turned from profile to face Mica. As she did so, Mica’s fingers collided.
“See?” Saskia encouraged Mica. “My head and nose didn’t collide, just your fingers did. There’s plenty of room for me to turn about in three dimensions without the need for a clutch.”
As the penny dropped for Mica, Saskia picked up Wassily’s drawing again and indicated the x-axis. “Wassily’s picture only gives one dimension to position, this backwards–forwards tightrope, there isn’t even the second dimension for height. Far less a third for depth.”
“Huh.” Mica was struck by the elegant simplicity of Saskia’s argument. Together, they sat with everything they’d considered.
“It’s a good thing we’ve got three dimensions to live in,” Saskia chirped happily.
Mica nodded, and then something occurred to her: “And a good thing we’re alive now, and not in twenty years, because battery electric vehicles don’t have clutches to help crutch our thinking.” Mica winked at Saskia.
∞
Later, as they walked along the beach, each eating a burrito, Mica turned back from the setting sun. “You know, one other thing occurred to me when I was talking with Wassily, that UCLA mathematician.”
Saskia turned to Mica, curious. “What?”
“Well, thinking about his tightrope and looking out and seeing yourself multiple times again——it made me think that perhaps the you I saw outside the bathroom at Cleo’s——you don’t remember it, but maybe that’s because you’re yet to go back.”
Saskia looked up the beach. She’d dug herself a hole on this one. Was there an answer up ahead? . . . Nothing that jumped out at her. She turned back to Mica. “Maybe. I don’t have a plan to, though.”
Mica shrugged. “The past is still in the future.”
Saskia smiled back at her, but her mind filled with other worries. Was dodging her former self enough to avoid disaster? What about everything else? Was she destined to bump into things every time she turned the flow of time about? Even clear air wasn’t empty, after all.
