I’ve had this question for years, and was just made to think about it again in reading the wiki on Hamiltonian paths.
Okay. You’ve got a box, and a laser, and a bunch of small mirrors. The mirrors reflect on both sides - they aren’t “one-way” mirrors.
You put a hole in the box,* and point the laser into the hole. The goal is to trap the beam in the box. You must arrange the mirrors inside the box to reflect the laser into some kind of infinite loop. If the laser touches the inside of the box, the game is over. If the laser escapes back out the hole, the game is over. Disregard all physical limitations of the materials - they’re all made out of unobtanium.
Is this game winnable? Is there some arrangement of mirrors that allows you to loop the laser? My gut tells me that the game is not winnable**, but is there a mathematical proof to back this up?
This problem was probably solved in the 17th century, but for the life of me, I don’t know how to search for it.
*No dick-in-a-box jokes, please.
**For many reasons, but for one, if it were winnable, then at a given point in the looped laser’s path, there would be essentially an infinite number of recursions (limited only by c), and some impossible amount of potential energy.
I can’t do it mathematically, but I think I can do it logically.
If the laser is going to be looped, it must pass through the “origin” again at some point. However, this origin is going to have to now be a mirror. Since the mirror can’t be there when the beam is on, since it would block the beam, it would be impossible to get the mirror into place fast enough to “catch” the loop.
Depends on how big the loop is. It could take seconds for the loop to get back to the point of origination depending on how complex the mirror angling is. Heck, with todays technology you could probably close the mirror hole in several thousands of a second so the laser loop wouldn’t have to be quite so complicated.
Why can’t you just put a mirror at the back of the box at some angle, then place another mirror angled perpendicular to the beam coming off the first mirror.
It should reflect back to the first mirror, and continue to bounce back and forth between the two, no?
This assumption gets at the heart of the problem, I think. Is it provably true? Must every loop include the origin? Can a loop be created that doesn’t? (I’m asking in earnest.)
Trunk - no. The beam will hit the first mirror, go to the second, go back to the first, and then exit the box back out through the hole. Game over.
It will reflect back to the first mirror, but by reflective symmetry the beam will then go out the entry hole, and you lose the game.
Here’s a solution: Situate four mirrors in a closed rectangle inside the box, and ensure the laser enters thru one corner of this rectangle at a 45-degree angle. If the dimensions of the rectangle are 1 x sqrt(2) units, the beam will never strike another corner (because sqrt(2) is irrational). Of course, this assumes you can make precise-length mirrors and the hole is infinitesimally small, but this is exactly what unobtainum is for…
You want there to be a closed loop. But the laser path has to enter this loop. So at some point in this loop, there is one “outgoing” path and two “incoming” paths. I assume a mirror maps each “incoming” path onto one and only one “outgoing” path. I don’t see how this is possible.
Or to look at it another way, reverse the beam. It starts in a closed loop. Then after a kajillion iterations it suddenly takes a different path, which after a bunch of bounces, ends up in a tiny hole in the box. That sounds kinda crazy to me.
My first thought is to construct a curved, spiraling mirror, such that the beam also follows a spiral path; approaching the center assymptotically. Tracing the beam from the point where it entered the box, it would always be getting closer to the center but never reach it. It doesn’t go into a closed loop, but takes a path of infinite length.
Couldn’t be done in the real world; the thickness of the mirror would block the path eventually. But you said to disregard the physical limitations of the materials.
Here’s one way you might solve it. Of course, it assumes that you can set up mirrors with infinite precision. Set the system up so that the laster beam first bounced back-and-forth between mirrors A1 and B1 5 times, then between mirrors A2 and B2 25 times, then between A3 and B3 125 times, etc., always bouncing 5 more times between each pair of mirrors. Clearly, you have a length of path which rapidly diverges. With a fairly small number of pairs of mirrors, you can get very long paths. By my calculations, if the mirrors are 1 metre apart, with 23 pairs you get a path about 1.6 light years long. With 38 pairs, the light will need to travel longer than the estimated age of the universe to get to the end, so in practical terms it will be trapped forever in a box just 1 metre by 40 metres.
Let’s say you have this box but there is no hole in it. Completely sealed and mirrored on the inside. Possibly in the shape of an octagon looking from the top down. One of the sides IS a two way mirror so you CAN shoot a laser through it and created a looped path. You use some smoke in the box so you can properly see the laser to align it. Once aligned you pump all the smoke out of the box, any and all particles out of the box, creating a vacuum.
Now once you turn off the laser beam source, even though not visible, is that beam of light still bouncing around in there infinately?
Or will the impurities of the mirrors eventually and quickly refract and scatter the beam till it disapates?
By that definition, I doubt it’s possible. What you need is some way to take two incoming beams and focus them to a single beam (which would then loop around to be one of the two sources) and I don’t think that’s possible with just mirrors.
I think the OP vague on this point, since it says “The goal is to trap the beam in the box. You must arrange the mirrors inside the box to reflect the laser into some kind of infinite loop.”
However, it is easy to see that trapping the beam so that it ends up following the same path in a repeated loop is impossible. For any given point on a mirror and any given incident path, there is only one possible reflective path. And since mirror reflection is symmetric, the incident and reflective path can be swapped. Therefore, there is a 1:1 correspondence between all incident and reflective paths at any mirror point: No two, different incident paths at any single point of the mirror can be reflected in the exact same direction (we’re talking about ideal rays here).
Thus, there is no way for the laser ray to get into a loop. A ray already in the loop of mirrors is bouncing off the same mirror points and following the same reflective paths repeatedly. It’s impossible to enter this loop because no two, different incident paths (i.e. the “entry” path and the “looping” path) at any single mirror point can be reflected in the exact same direction (i.e. the path of the loop reflected after the first mirror).
On preview, Robot arm has the idea as well; I suspect a similar argument can be made for any combination of lenses, prisms, what have you (weird, QM-optic devices notwithstanding ).
I’m not really sure what the OP is asking for. It’s certainly popssible to create a mathematical path where beams continue to bounce around and, if the beam is of zero width, not exit the box. But the OP wants a “loop”. Any way you go through a place where you’ve passed, unless you can close the opening, the light will get back out. If you CAN close the opath, just send the light into a big box and have it follow a symmetric path, but close the window that let the light in. Now you’ve got a loop with no light getting out. Simple.
as a practical matter, you can’t do this with lasers and mirrors.
1.) If you used curved surfaces, the light beam is going to expand (maybe it will focus first, then expand, but expand it will). The expanding beam won’t follow your neat little path, and eventually leaks out of the hjoles.
2.) even if you use all flat mirrors, the beam will expand. Contrary to popular belief, lasers don’t stay collimated forever, and once you get beyond the Rayleigh length you;'re subject to that inverse square law again – the beam expands.
3.) There’s a neat little solution to the expanding beam – it’s called a Herriott Cell. You refocus the beam from a curved mirror to counteract the tendency of the beam to expand. The stability conditions for a Herriott Cell are the same as those of a laser resonator. They really are used in the real world to send beams through large distances inside a closed package, or as optical delay lines. BUT, the beam is finite. You can make the beam walk a long path inside a Herriott cell, but part of it will eventually get out. Unless you close that entry port with a moving mirror.
4.) as for a laser beam bouncing around inside a box – yes, it will eventually be absorbed by the mirrors and go into the walls as heat. The typical number of bounces is (rho)/(1-rho), where rho is the reflectivity of the mirror. So if your mirror reflects 0.999999 of the light, it will make about a million bounces before it’s absorbed. For a box 0.5 meters on a side, this takes about 3 milliseconds.
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