Trying to rotate a flashlight over a finite distance in infinite time means the flashlight would never move at all. This is essentially the same argument Zeno made.
I agree with Chronos’, summation.
So you are saying, essentially, that this problem is simply a modern version of Zeno’s paradoxes of motion.
Let the light take infinite time to move over the angle of 1 rad. True, it does not move on any finite scale. But it spans an arc on a circle with infinite diameter. The infinities cancel each other out, so the linear velocity on that circle still can be 1.
Wait… I thought Zeno’s paradox was about halfing a distance an infinite number of times. The thing he missed is that the summation is bounded. Sum(1/2)^n for n = 1…infinity = 1.
In the case of turning a flashlight infinitely slow, well, I don’t see a summation going to infinity.
Okay, angular velocity is 0. (1 rad/infinite time). So far so good.
Circle of infinite diamter = line. How an unmoving light spans this is where I think I start to get confused.
Linear velocity? Thought this was a circle. But a circle with infinite diamater is a line, so maybe. I really don’t know.
I think your argument is that velocity = distance/time. Since the distance = infinity and the time = infinity, they cancel. I don’t think I agree with that. infinity/infinity can be anything from 0 to infinity. (L’hopital’s rule please!)
I can’t see how you can spin a flashlight infinitely slow and ever expect it to move. But I’m not the brightest person, so that really isn’t saying much.
But perhaps you brought out the answer… a line IS a circle with infinite diameter. Therefore, the flashlight is shining on a circle, and as such, you can never NOT shine it on that circle. Hmm… that actually makes sense to me. (But see last sentence of previous paragraph.)
Chronos -
Why would not the last photon be headed for the point at infinity?
Also, if the claim is that the photons are emitted discreetly and not continuously, then even when you shine the flashlight 90 degrees to the wall, there are still very brief moments when photon are NOT colliding with the wall and therefore turning the light becomes irrelevant. My assumption is that the emission of photons was not a discreet event. It was an idealization on my part.
Infinity is not a number. You can’t divide by it, if you are expecting to have an answer that has mathematical meaning. As the value approaches infinity, a limit may be approached, but there is no solution for x divided by infinity. Infinity can’t even be divided by infinity. It isn’t a number.
Infinity means: "I don’t know, but I think it’s really really big.
If space curves, then parallel lines need a new definition, since the current one includes having a constant distance for perpendicular distance between them. Under the new definition of parallel, they can meet in Pittsburgh, if that is what you want in your new definition.
Light is an entirely different kettle of photons, as well. Spherical propagation of single photons, Probability, Heisenburg, don’t even go there.
Tris
“Here Kitty, Kitty, Kitty.” ~ Erwin Schrodinger ~
Theoretically, even after it takes an infinitely long amount of time for the last photon to strike the wall, there is still an infinitely long amount of time left.
It seems that a lot of people are confused by the concept of infinity. It just doesn’t act like any finite items.
Velocity on the line (or the wall), as opposed to angular velocity.
You can see a circle in every direction, not so a line. A circle has a middle point, a line has not. Until you treat it as an infinite circle, then the middle point goes infinite. But then you have again a point surrounded by infinitely distant points, so both are the same. Or not.
Call it a straight line and we need tan(90º)/[symbol]¥[/symbol] to compute a length, call it a circle and we only need [symbol]¥/¥[/symbol]. 
The point is both have no meaning until you define one.
Equalizing ‘round’ and ‘flat’ for infinity or talking about lengths is just a matter of definition.
Sorry, but that would mean [symbol]¥[/symbol] + [symbol]¥[/symbol] = 2[symbol]¥[/symbol] ?
Oh, and L’Hospital tells us where the fraction will go with both parts approaching infinity. The confusion arises from our assumption that they already got there. Nothing stops us from saying circumference=2[symbol]p[/symbol]*radius still is valid.
Wow. Sounds like I opened up a can of worms. I also posed this question to a co-worker. He replied, “The question cannot be known since man cannot undestand infinity. It can’t ever be plugged into a mathmatical equation as it is not a number.”
This led me to think, how do we know the characteristics of an infinitely long wall? I mean, how can we apply any of the laws of math or science to an equation which includes an “infinite”. Since we don’t know what infinity is, how can we assume it obeys any of our laws?
I’ll go ahead and add another monkey wrench. Forget math for a second and let’s talk physics. This wall is infinitely long, and we can assume it’s made of some type of matter. Therefore, it’s infinitely massive. Wouldn’t this tend to bend the light toward the wall, no matter what angle you turn the beam to?
Seems to me we are making A LOT of idealizations in this problem. For one, that an infinite wall can even exist. Another is that the light is not dispersive. If you want to put real world limitations on it the answer is really, really simple: who cares? None of it can ever be created in the first place.
to ummm… yeah:
What is the background of your co-worker? I can think of LOTS of times infinity is used in math and science. To calculate potential we very often start with an integral from infinity to wherever. 1/infinity is defined and is 0. As was stated earlier, the SUM(1/2)^n for n = 0 to infinity = 2. In all these cases infinity is used and understood. There are times when it causes problems, but I don’t think so in this case.
Again, there is a one-to-one mapping of every degree between -89.999r to 89.999r in the circle of rotation of the flashlight to a spot on the wall. (i.e, using Pathagarus we can easily calculate where on the wall the light will shine if we know the angle of the flashlight to the wall. For every angle between -90 and 90 exclusive, that will correspond to a single point on the wall.) Because the flashlight is continuously rotated to 90 degrees, it MUST pass through the angle corressponding to the infinite point on the wall. The light will take an infinite amount of time to reach that point. The light will always shine on the wall.
Not if we make it infinitely thin. HA! 
And I won’t even think about where the center of gravity of an infinite massive wall is. Or if an infinite static universe would collapse. Triskadecamus is right. Don’t even go there.
You have never seen them talk about 0.999rep=1, do you? 
(That reminds me, I have a question. Will someone take the blame?)
Our entire math is based on axioms that can’t be proven. We start with “there is a something” and call it “one”. We call “different” what distinguishes another something we call “two” from the first. We don’t know what “another” means, but if we take two “ones”, do something to them and get out what we say is not different from two, then we can call that “addition”. Quite circular reasoning, but now we can add one and two and call it “three”. Thus we can build our natural numbers. We better deduce a “zero” and “multiplication”, and “negative” numbers. From there we get rational numbers, they can be completed with irrational numbers, complex numbers, etc. etc.
Infinity is just another concept deduced from our first “somethings” and obeys any law we had to make up to define what a number is in the first place. Not very intuitive laws, but pure logic. Some time ago the number Zero was thought of as sent by the devil, until all the laws were figured out to incorporate it in our math. We can define infinite infinities greater than infinity, and there are infinitesimals smaller than any division. And there are omega-numbers greater and smaller than that, if I got it right.
Like there are many levels of numbers, the riddle can be solved on many levels of infinity:
[list=1][li]There is no infinite wall. Period.[/li][li]Assume there is. The light will never reach an end in any time.[/li][li]Say you have infinite time. Oh yes it will![/li][li]Take infinite time to turn the light. No it won’t![/list=1]And I’m sure it can get even more complex.[/li]
I got a headache.