Here’s a debate I heard in college that was never resolved. Keep in mind that I was a “Communications” major so I have average intelligence at best.
There is a wall that runs infinitely from east to west. There is also a beam of light that runs infinitely. If you start the beam at a 90 degree angle to the wall and move it out east or west, will there ever be a time when that beam leaves the wall?
Sorry if this has been asked before. Thanks in advance for your imput…
Sorry, let me try to clarify. Again, this was a questoin someone else posed so this is my interpretation of their question. I am assuming that by “moving the beam of light” means rotating it around 360 degrees, like a lighthouse. As far as the wall part. I don’t know. It could be like earth, but it could also be a straight line in space. What difference would it make in each scenario? Hope that clarifies… Thanks.
THE WALL
<>
/\




THE LIGHT
Now, assuming this light travels in a straight line and the beam does not widen, once it is rotated so it is paralell to the wall, the beam does not touch it.
Again, it all depends on the interpretation of the question. I’ll go through several.
If the beam is pointed at the wall and then the source of the beam is moved along the wall so that the beam is still at a 90 degree angle. The beam will always be on the wall. This holds true for both if the wall is along the Earth’s surface or simply a straight line in space. The path of the beam emitter follows the same “East to West” path as the wall and thus continues to point at it throughout it’s infinite travel.
This question could be a mangled riddle. The beam could be at a 90 degree angle and yet face away from the wall. Thus it would never hit the wall to begin with. Corrolary 1.): if it never hits the wall it could never have “left”. Corrolary 2.) If it hits the wall, the light beam stops, is no longer infinite, and the “infinite beam” does not touch the wall.
If the beam is said to rotate like a light house, it only slightly matters if “EastWest” is on the Earth or a line in space. If it is a line through space, with constant Z axis values for both the wall and beam, the beam will diverge when it reaches exactly 0/180 degrees or parrallel to the wall  but not a 1/infinigoogle of a degree before because the wall itself is infinite. If the wall is along the surface of the Earth, the beam will “leave” it much sooner in it’s lighthouse like rotation as the curvation of the Earth will cause the wall, but not the beam, to change Z axis values (because the wall is a ring). If it’s a line through space  departs when it’s parralel; If on Earth, it will go “over” the wall before it reaches parralel (assuming the wall isn’t also infinitely high  if so, parralel)
Final corrolary: If the Universe itself is finite, yet curved, the beam could bend around the edges of the rim of the universe, or bounce, or whatever, and the infinite beam would never leave the wall because an infinitely sized object in a finite universe would be everywhere at once. Obviously, two “everywheres” would have to share the same space and could never leave each other. Also, the beam cannot be blocked by another object because, well, it wouldn’t be infinite then.
I suspect that what they really meant is that the beam is parrallel to the wall always, and they are asking does it ever reach past the end of the wall. Tantamount to asking: “Is infinity infinite?” The answer to that should be obvious.
Your diagram is correct, as I understand it. The arguement I heard against this is that if a beam of light starts on a wall and the wall is infinite then it can never leave the wall. There was some technical reason for this but I don’t remember. Does that make sense? (Does is make dollars?)
The effect on a wall in space will be different from a wall on the earth, as well.
On earth, assuming a noninfinite wall height, the beam of light will not hit the wall at the same height off the ground as the beam is rotated to the east or west (seeing these directions leans me toward a terrestrial wall). As the spot at which the light beam hits the wall follows the wall face along the curvature of the earth, that spot will “rise” in relation to the ground. Depending on the distance away from the wall from which the beam originates and the height of the wall, it is possible the beam will shine “over” the wall before becoming parallel to the wall.
For a straight wall in space, or a wall of infinite height on earth, the beam would leave the wall when it became parallel to the face of the wall.
Upon Preview, I see other replies. This MAY be that old “arrow in flight never reaching target” question of halving distance (or angles) to reach the target (or parallelism).
What you are basically asking is, do two parallel lines meet?
(at least that that is what I understand by the comment that the beam of light is infinite in both directions. If it has a known starting point, like a flashlight, point it away from the wall and go home.)
So, we have this laser which is generating two tight beams in inverse directions. Making a lot of hypothetical assumptions, we treat this as a line in a plane.
The wall goes on infinitely, so we make a lot of hypothetical assumptions and call this a line in the same plane.
Obviously, if they are not parallel, you will see a dot where the laser hits the wall.
If they are parallel, Euclid says they do not intersect. But Euclid wasn’t that comfortable doing math with infinity in it. My advanced kinematics class says that a line intersects every line it is parallel to “at infinity”. In planar geometry, infinity is a line, each point on the line has a unique set of parallel lines which intersect at it.
This probably confused you. The short answer is that mathematically they do intersect, but you have trouble visualizing what happens at infinity. This is natural, since nothing in reality is infinite. The wall will bend with the curve of the earth, the tightest laser beam will still form a conical beam and it will take time for the beam to get that far.
I think UncleBill has it. The light emitter is rotating, and the spot of light is sweeping away from it down the length of the wall. At some point, the light emitter will be aimed exactly parallel to the wall, and no spot of light will be on the wall any longer. Whoever originally posed the question couldn’t get his/her head around the idea of the instant when the spot of light stops sweeping down this infinitely long wall and swings out parallel.
Yabbut, even when the flashlight has rotated past the point where it’s parallel to the wall, there will still be photons on their way to hitting the wall very far away, so you can’t really say the the light beam has left the wall, can you?
Let’s say you have an infinite wall and light beam which does not disperse. The light is emiited from a flashlight and is at a 90 degree angle to the wall:
 < wall (infinite)
^

 < light beam

–
 
/


Is it possible to turn the flashlight such that the beam will no longer shine on the wall?
Obviously, you say, because all you need to do is turn the flashlight greater than 90 degrees.
But wait. As you turn the flashlight the beam will strike the wall at greater and greater distances from the original point of intersection (call it the ‘spot’). Since the wall is infinite, and the beam will travel the entire distance, the ‘spot’ will travel an infinite distance in a finite amount of time. How is that possible?
My guess at an answer:
It will always be tracking along the wall somewhere. Even when you have the flashlight parallel to the wall, the moment before you had it parallel that photon will be heading to some point on the wall, and it will take an infinite amount of time to get there. However, at some point there is a ‘spot’ on the wall from photons that have completed the journey. Therefore, there will always be a ‘spot’.
[Rodney Dangerfield]No respect, I tell ya. No respect. Here I cover several angles including that one and the credit goes to my Uncle Bill, who didn’t even mention the beam swinging past parrallel. I tell you, no respect. But what should I expect? Everyone hates me. I told my psychiatrist that and he said “don’t be ridiculous, not everyone has met you yet.” Ohhh! I tell ya, no respect [/Rodney Dangerfield]
But now that I think about it, CurtC is right. We all forgot the time component. As the beam rotates through the last 1/infinity of a degree before it reaches parrallel, the photons will have a distance approaching infinity to get to the wall  and thus a travel time approaching infinity. An infinite travel time before the beam “leaves” the wall would be “forever”.
And wouldn’t speed make a difference? CurtC is saying that light photons will always be travelling towards hitting the wall, even when the light is parallel. But if you stopped rotating the light for even a second wouldn’t the light travelling to the wall have timt to catch up? As you can tell I have no experience in this field. Ya’lls responses have been great but this is the question that has popped up as I am reading. BTW, CurtC’s explanation was the theory I heard back in college but could not recall. Good job!
If the wall were of infinite length, then as the laser spot swept along the face of the wall as the source turned, it would take longer and longer for the light photons to reach the wall. At the last 1/infinity degree before becoming parallel, the light has a hypothetically infinite distance to travel to reach that last spot on the wall, and in theory would not reach that spot. The spot would never leave the wall.
The light is a continuous beam of photons, so as the light “caught up”, there are more bundles of stuff right behind that!
Wait, who said that we had a continuous stream of photons? The flashlight emits only a finite number of photons, one of which muct be the last photon emitted before parallel. That photon would have a finite distance to travel to the wall, and would do so in finite time.
Assigning a solution to [symbol]¥[/symbol]/0? Looks more like a definitional crutch to avoid dividing by zero, or something. But correct me if I missed a theory.
Chronos, Do we exactly know the time of emmision and the direction of the photons? No one will know where the wall gets hit last, until that point is measured. “uncertainty principle going infinite”. Yikes. Did we discover the infinite improbability drive? I’m sure there’s a bowl of petunias at the end of the wall.