Is there ever a case in which our current best physics would predict, of a situation involving only rational measurements in its initial state, that after some rational interval, there will be a pair of objects the distance between which is exactly pi? (Any measurement units should do here I think, as long as they’re used consistently through the whole description of the scenario.)
At first I thought this would have to be answered “no” since pi is transcendental and everything starts out with rational values. (I didn’t have a strict argument for this “no” answer, it was just a hunch.) Then I thought that for all I know, there are physical laws that can introduce transcendental values like pi into mechanical calculations, in which case, for all I know, you might start out with all rational values and end up with transcendental ones.
Anyway, hopefully the question makes sense. If not, please ask clarifying questions.
(Also, the straight dope is awesome for being a place I think it plausible to ask questions like this. I’m curious to know of other places where laymen can ask halfway informed and possibly dumb physics and math questions like this.)
It depends on what your units are. You don’t say if it is yards, or miles, or parsecs or whatever. So, pick any two objects and just define the distance between them as pi [imaginary units].
I doubt it. Pi is a ratio so there is no reason to expect it to come up as the distance between two things in our arbitrarily-chosen systems of measurement.
I’m not sure the question is well-founded. In particular, I would think that by redefining our units of measurement, we could easily make something pi units away.
But even before that, consider this. Let’s say you and I start out 1 meter apart and then increase our distance until we’re 4 meters apart. At some point, we must have been pi meters apart.
ETA: Amazing. The OP sits for 20 minutes with no answers, and then suddenly there’s three answers! I’ll leave my post regardless, although it’s now pretty well redundant.
I thought about this (hence the parenthetical in the OP about how I think any unit will do) but I didn’t make it clear how I intended to head off this concern.
Here’s what I mean to ask. Can you have an initial state where all measurements are rational, where length is measured in some unit we’ll call “u-units”, and where a prediction of physics is that after an amount of time with a rational measurement, some pair of objects will have a distance that is equal to pi u-units?
If I’m reading this right, you’re asking if for, say, a universe in which space is quantized, with a minimum distance of a planck length, could a process put two objects a distance pi apart.
If that is the correct interpretation, then the answer is no. Pi assumes a theoretical infinite division. It cannot be reached by rational intervals in a finite time.
But you know that since it was in your OP. I’m trying to wiggle that around to get at your question, but I think all formulas that put pi into them assume mathematical infinities.
Let’s say you and I start out 1 meter apart and then increase our distance until we’re 4 meters apart. At some point, we must have been pi meters apart.
A key part of the OP’s criteria was “After a rational amount of time”. Your example would not work if you and I were moving apart at a rational speed, since, in that case, the time it will take for us to be π meters apart will be irrational.
To the OP:
What do you mean to say with “all measurements are rational”? For example, it’s not possible to have a square all of whose corners are a rational distance (in whatever units) from each other (since the distance between opposite corners will be sqrt(2) times as large as the distance between adjacent corners); are we therefore prohibited from considering situations involving squares?
More to the point, of course, any Euclidean circle induces two lengths which are in a ratio of 2π, by definition: its circumference and its radius. Surely you aren’t going to prohibit situations which involve circles? If not, naturally, even though the circumference’s length is not directly the distance between two points, it’s easy to use this to construct a distance ratio of 2π. If you do prohibit circular objects, circular motion, and all the rest on grounds such as “Well, sure, in a Newtonian world, but with quantum uncertainty blah blah…”, the difficulty becomes, if objects do not have a definite position, there aren’t definite distances between them either, so it no longer makes sense to ask if the distance between them is exactly this or that.
If we’re talking about sufficient (that is to say, infinite) precision to tell the difference between rational and irrational numbers, then we run into the problem that there’s no such thing as a precisely-defined position, and hence no such thing as a precisely-defined distance.
I don’t think this satisfies the OP’s question, since for all anyone knows the exact moment that you were pi meters apart is a trancendental measument of time.
Could you come up with a position (or velocity or acceleration) equation which is itself polynomial or rational and which at time T where (T IS Rational), position is Pi?
If you simply defined a constant velocity of 1 unit/second from units 0 to units 10, then Pi seconds later (T=Pi), distance will be Pi, but the OP is not looking for that since he doesn’t want a transcendental T and Pi seconds is trancendental.
You guys are looking hard to NOT find something that satifies the OP’s question or something in a similiar vein.
Hell, in the real world I might as well say you can’t have a circle that had a circumference equal to pi times D because of quantum mechanics or curved space time or something, therefore pi isnt an interesting number because it doesnt really exist.
Imagine a 2-stroke piston engine where the piston’s motion in and out is 1 length unit. The crankshaft has a diameter of 1 length unit.
One stroke takes 1 time unit.
The crankshaft drives a wheeled vehicle whose drive wheels have a diameter of 1 length unit.
Run the engine. For each engine stroke (1 time unit), the vehicle moves 2*pi length units, therefore over 1/2 time units the vehicle moves pi length units.
Good point. I’m tempted to re-ask by saying the initial state can be described using only rational values. (I.e., “A and B are 1 unit apart, B and C are 1 unit apart, and angle ABC is 90 degrees”) but I’m afraid that probably just ends up sneaking in irrational values if you fill in the details of what it takes to describe the state as fully as necessary. (Esp. since a “degree” has alot to do with a circle, which has a lot to do with a certain transcendental value…)
So maybe there’s just no way to fill in the antecedent here… Maybe irrational values must necessary inhabit the description of any initial state. Of course, really what I was after was an initial state where “all measurements are non-transcendental,” (I asked in terms of rationality because I figured it’d be easier to parse and it’d be a stronger question, yielding an answer to my actual question) but I suspect (because of the “degrees” thing for example) that transcendental values “infect” every description of an initial state as well.
Tempted to ask about descriptions of implausible scenarios where all objects lie exactly on some particular straight line, but this gets away from my purpose in asking the original. (That purpose was to help myself think about the question whether physics presupposes that things can actually have transcendental (or even irrational) measures in the real world. (I mean to distinguish between measures and measurements–a measurement being something we do, a measure being the actual value we’re attempting to arrive at when we make a measurement.)
I was trying to bypass the issue of precision in measurement. I was just trying to ask whether, if you just give the physics (perhaps artificially) precise rational measurements for the initial state, does the physics ever predict transcendental results after a rational amount of time.
I’m not sure if this answers your question or not (in fact, I think it might sort of side step the question) but if two objects started out 4 units apart and then moved to 3 units the intermediate value theorem says they HAVE to be at Pi units apart at some point.
But since Pi is irrational I think the best we could do is give a really good approximation of when that would be.
And if you want a single stroke to move the vehicle pi length units (such as if the angular velocity of the crankshaft is not constant while the engine is running), just add a gearbox.
Not quite. I meant to be asking about a non-quantized space, actually, since I figured if space is quantized you’ll never get actual measurements of pi. (To my knowledge, which may well be wrong, the quantization of space is controversial. Is that right, though?)
Probably subsequent posts have made this clearer (and Indistinguishable’s post has possibly highlighted a fact about physics that makes it impossible to satisfy the antecedent conditions of my hypothetical…) but the idea is to wonder whether you can ever end up with transcendental values predicted given an initial state that included only rational values. (Actually, for my purposes, really all I need in the initial state is that all values be non-transcendental, though I didn’t say that in the OP because I figured rational values are easier to talk about.)
That may be it (although to my shame I’m not quite sure how crankshafts, pistons etc relate to each other mechanically) but I’d also want to know about force measurements and so on.
Hmm, are you saying that the force applied in the cylinder due to the ignition of the fuel might inherently be a transcendental equation? I’m not certain - does anyone know?
We could say that a human being is pushing and pulling the piston open and they have enough dexterity to never apply a transcendental force, and we could use the gearbox or redefine our crankshaft and piston motion length to be 1/2 length units to make the distance moved in one stroke to be pi length units rather than a multiple or fraction of it.