How many images in a mirror?

This is certainly a trivial question but has bugged me since I was a little kid. Assume you are in a room 10x10 across and there is a 10x10 foot mirror on two facing walls, how many images would appear in each mirror? I used to think it was infinite but now realize that at some point the molecules must lose their ability to reflect light. So how many images would appear and how small would the last image be? And would images become fuzzier has they got smaller or would the images just stop?

It depends on several variables, such as your ability to see in low-light conditions, the reflectivity of the mirror and the ambient lighting strength. It’s safe to assume the reflectivity of a typical wall mirror is on the order of 85-90%. Once you’ve determined the amount of incident lighting and the cutoff below which your eyes cannot detect light, the rest is a fairly straightforward calculation. You’ll never get an exact answer, of course, because there are too many other minor variables, such as the spectrum of the incident light, the condition and cleanliness of the mirror surface, the color, reflectivity and texture of the objects being observed and others. Human vision is a complex topic.

It wouldn’t be infinite, because no material is 100% reflective. Hold two mirrors opposite each other and you’ll notice each recessive image gets dimmer and dimmer. The point at which this happens probably depends on the quality of the material. Not all mirrors are made equal.

I read somewhere (I think it was Carl Sagan’s Cosmos) that when you look at 2 mirrors like that, you are not looking into infinity because the mirrors aren’t perfectly flat and they aren’t perfectly parallel. If they were…I don’t know.

It depends on the quality of the mirror. There is some light loss with each reflection, so the more distant images are dimmer, and at some point it becomes too dim to see. I’m not sure what the reflectivity of household mirrors is, but a telescope mirror with a fresh aluminum coating (but no “enhanced” coating) is about 90% reflective. If we assume 85% reflectivity for your wall mirrors, that means after 4 reflections, a light beam is reduced to half the original brightness. After 28 reflections it’s down to 1% of original brightness.

The surface quality is also an issue. Distortion adds up after many reflections, so yes, the images get fuzzier and fuzzier.

Well, this kind of depends on what you define as an image in real-life terms. Silver does not reflect all the light that falls on it, so the reflections in the series would grow darker and darker, the more mirrors they were reflected in. At some point, both through distance and darkness, they would be extremely hard to make out, but it would be hard to define any one point where the images stop because the brightness of the images would be a decreasing geometric series, converging on zero without ever getting there. The molecules do not ‘lose’ their ability to reflect light any more than a wall loses its ability to have balls bounced off it - though unlike the wall a mirror occasionally absorbs photons instead of bouncing them.

As an aside, there is a form of reflection that is practically lossless, called total internal reflection IIRC. This has to do with bouncing a beam of light at a very acute angle against the inner surface of a piece of glass, say… because the glass has a higher refractive index than the air around it, it will bounce away from the boundary with no light absorbed by the boundary itself, (though a very tiny amount of light is absorbed by the glass.) You can’t get total internal reflection in with light bouncing at a 90 degree angle to the reflecting surface, though, like if you were standing in front of a silver mirror – it only works at much lower angles of incidence.

I remember learning that the US millitary had managed to rig up some complicated refractive structures to get total-internal-reflection periscopes, which were noticeably brighter and clearer to look through in poor light conditions than traditional silver-mirror periscopes.

Hope that this rambling helped.
Hi to everyone else who posted while I was writing this.

That’s why fiber optics is so efficient - light reflects off the boundary between the core of the optical fiber and the outer cladding. Prisms in binoculars work that way too. But if you just want one reflection, a total internal reflection prism is not particularly efficient. Light has to enter the block of glass first before it reaches the reflecting surface, so there’s light loss at the entrance surface.

The way to improve a mirror is through multilayer coating, as explained here. Reflectivity of 99.999% has been achieved for lasers (i.e. for a specific wavelength).

Here goes nothin’…

Some limiting facts:

  • light is attenuated by the air
  • light is absorbed by the mirror (<100% reflection)
  • line-of-sight is not perfect (your head gets in the way, so you have to look at a slight angle)
  • the eye has a diffraction-limited angular resolution
  • the eye has a limit intensity sensitivity
  • the mirror’s reflection become non-specular at small scales

You suggested a 10 ft x 10 ft x 10 ft (?) room, which I’ll take to be 3 m x 3 m x 3 m.

Attenuation by the air
Perfectly clean air has a Rayleigh-scattering-limited visibility of about 300 km. (Visibility here means the distance over which the light intensity falls by 1/e=0.368.) While pollutants can take this number down to tens of km, let’s assume your test room has clean air (using a HEPA filter, etc.) Why not? So, we can ignore this mechanism unless we find we are getting something much more than 10[sup]5[/sup] room crossings. Something else will surely dominate before that. Continuing, then…

Absorption by the mirror
95% reflection is not too hard to find. 99% gets expensive, and beyond that is specialty stuff. We’ll take 95%, since that’s a good quality bathroom mirror. (You spent all your money on the HEPA filter.) Here’s where the sensitivity of the eye will start to matter, but maybe something else will dominate still so we can ignore this one, too. The intensity goes down by 1/e after 20 room crossings.

Line-of-sight
A rough measure of my face tells me that if I stick myself in the corner of the room, one side of my head will be 15 cm from the wall, and my eye will be about 4 cm in from that. Thus, I must look at an angle of tan[sup]-1[/sup](4 cm / 2 / 300 cm)=3.770 degrees in order to focus on the the mirror behind me. The actually angle doesn’t matter, though, because the number of interest in the 4 cm – each room crossing must eat up at least 4 cm of wall. I’ve used up 15 cm with my head, so that leaves 71 room crossings, max, before I am just viewing the wall on the other side of the room. YheadMV.

Diffraction limit for the eye
The eye is most sensitive at around 570 nm, so let’s take that as our wavelength. In a dark room, your pupil can approach 1 cm, which leads to a diffraction limit of 0.003 degrees. (The eye actually has an angular resolution that is a factor of 10 or so worse than the diffraction limit.) The line-of-sight problem definitely dominates this one.

Mirror becomes non-specular
This will occur when the apparent size of the object is about the size of the wavelength of the light. That is, on the last reflection, the object would have an angular size of (570 nm / 3 m) = 10[sup]-5[/sup] degrees. The above two items kick in well before this matters.

Sensitivity
We’re talking about seeing an actual object – not just poorly resolved photons – so we need an object. One possible “object” would be two yellow LEDs separated by a small distance. We already know we’re limited to about 70 room crossings, so using a 1 cm pupil, we’ll see (1 cm)[sup]2[/sup]/(16)/(70300 cm)[sup]2[/sup]=1.4x10[sup]-10[/sup] of the emitted light. We’ll need about 1000 photons per second to stay above the brain’s noise threshold, so that means the LEDs would need a power output of (1000 photons/sec)(3.5x10[sup]-19[/sup] J/photon)/(1.4x10[sup]-10[/sup])=2.6 microwatts. Flipping this calculation around: an LED has a power output of order 100 mW, so we can take another hit of sqrt(100 mW / 2.6 [symbol]m[/symbol]W) in the distance. Thus, 70 room crossings could go to >10000 room crossings before we were completely blind to the light.

So…
I have to say I’m a little disappointed to find that the limiting factor is that your head gets in the way, keeping the distance under two football fields.

That’s only a limit on the number of images you can see. The images are still there, though.

Well, that depends on what an ‘image’ is. Is it really an image if you can’t see it?

I am always amazed at how detailed and scientific the answers in these posts are.

I have to admit, though, that I never thought about the “real world” factors that could influence the problem. I am sure even the quality of the mirror manufacturing process and the thickness of the glass could also play a role. It is like Zeno’s Paradox - mathematically, the arrow never reaches the target and factors such as the type of wood, the length of the arrow, the quality of the feathers are not considered.

So I guess this is a similar puzzle in that if you look for a purely mathematical explanation you get a non-sensical answer that is technically correct. The closest I guess we could get to “an ideal real world” solution would be to ask how many images you would get if you flashed more and more powerful beams of laser lights at the mirror. At some point a burst of light of twice the power might only add a small percentage more images and this would be our accepted maximum number of images.

The figure that’s used in calculating the number of bounces from integrating spheres and the like is p/(1 - p), where p (actually the Greek letter “rho”, but I’m too lazy right now to look up the code for it) is the reflectivity of the surface (expressed as a fraction). That number is about the number of effective “bounces” you’ll get.

as an example, put is 99.99% as 0.9999, and you get 9999, or, to round it out, 10,000 bounces. That’s about how many images you’ll see. But is your reflectivity is only 99%, you get 99 bounces.

The OP mentions a factor that I don’t see discussed here. The discussion has focused on light loss. Let’s assume no light loss, for the sake of argument. As the image bounces again, the virtual image gets farther and farther away, and therefore smaller. At some point the image is so small as to be reflected by, say, a single molecule of the mirror silvering. So how many bounces until it gets so small that it can’t be reflected at all?

There is only one image in a mirror. That the image shows mirrors within mirrors doesn’t matter, the mirror still has only one image.

There’s a maximum resolution your eye can see at, given by the size of the retinal cells and diameter of pupil, amongst other things. So I guess that would limit how many images you’d see in perfect mirrors made of high-grade unobtainium.

Similar effects can be acheived with positive feedback in a video circuit, which can be demonstrated when a camera is pointed at its own monitor. This was a regular special effect for many a 1970s pop video.

Or you can do it with cats looking at pictures of cats on computers, as in the Infinite Cat Project. After a while your cat will become smaller than a pixel, so it’s not as infinte as it claims.

Molecule size has nothing to do with it. Light is a wave, and as long as the molecules are much smaller than the wavelength of light (which they are), the wave is reflected without degrading its quality.

But there is another limitation - the size of the mirror. Light is a wave, so if it passes through a finite aperture, it bends arround the edge of the aperture (diffraction). This is why larger telescopes have better resolution: if you force the light to pass through a small telescope lens, then no matter how good the lens is, there is image degredation due to diffraction. This isn’t a small factor either - it’s the limiting factor on any high-quality telescope, including amateur telescopes. (That’s what they mean by “diffraction-limited optics.”)

The 10x10 ft mirror in the OP would have the same problem - every time light passes through it, the light is diffracted and the image quality degrades. I’m afraid I’m not sure how to calculate the diffraction effect of multiple reflections, but I think this would set the theoretical limit if mirror reflectivity and surface quality are perfect.

(That’s assuming you use a telescope to overcome the human eye’s limit, which would otherwise be the limiting factor as Fridgemagnet correctly stated.)

One of Chas Addams’s classic cartoons pictured a barber and his customer endlessly repeated in opposing mirrors. About ten images deep, the customer is shown as a werewolf.