# Are infinite reflections between two mirrors really infinite?

As written in the title it's a somewhat rhetoric question, so let me be more clear.

Let's say I have two mirrors facing each other. They are perfectly aligned and perfectly reflective. I'm looking through one of them (using the one-sided mirror effect) and see what seems to be infinite reflections.

If I look closely at the vanishing point, I obviously can't see very well what goes on there, because the reflections are too small. My question is, are there other limitations?

a) If I look close enough to the vanishing point, is there a point at which the light would have to travel too much distance to reach me that no photon would actually make it?

b) If I start looking immediately after turning on the light, it should take a tiny amount of time until all the reflections are created, i.e., it takes time for light to do all the back and forth trips it takes to create the very small images. How good a microscope (telescope?) would I need to be able to see the image being created? That is, to look at a spot and see nothing, and then some epsilon of time later see an image appear.

• Are the mirrors perfectly reflective, and do they perfectly reflect all light directly back toward whatever direction it came from? And what is this "infinitely small amount of time" that you speak of? (Something tells me that means something other than what you have in mind. Very small is not infinitely small.) – user Jan 15 '17 at 21:17
• @MichaelKjörling perfectly reflective, very small. Thanks – Dotan Jan 15 '17 at 21:31
• If you're looking through one of the mirrors, how does the perfectly reflecting mirror let any light through to you? – DJohnM Jan 15 '17 at 21:36
• You are definitely limited by the fact the no material is a truly continuous medium, I.e. There are atoms – ClassicStyle Jan 15 '17 at 22:37
• There are from memory at least 3 near duplicates or closely related questions to this post, physics.stackexchange.com/q/55254. And links from that. Related physics.stackexchange.com/q/13500. – user140606 Jan 15 '17 at 22:58

You will only ever see a finite number of images, for practical reasons.

1. No mirror is perfectly reflecting. Some small fraction of light is always absorbed each time light is reflected. As you can see from your photo, the images get darker, and by the 8th image they are too dark to distinguish. This is caused by light losing energy due to the bounces between the mirrors. Light takes about 3ns to travel 1m, so if the mirrors are 2m apart it take approx 0.1 micro-seconds (one ten-millionth of a second) for the light from the 8th image to reach you. The delay between the first and last image is too small for you to notice.

2. Even with mirrors which are perfectly reflecting, the images get smaller. At some stage they are too small for your eyes to resolve clearly. But even allowing for a telescope which can resolve images perfectly, at some stage they will be not much bigger than either the wavelength of light or the atoms in the mirror. At this scale the reflections are too fuzzy to distinguish, and they merge into each other.

• Is the fact the image gets smaller due to the shape of mirrors? Is it impossible to create shapes that prevent this ever increasing size reduction? – Winston Sep 17 '20 at 18:05
• @Exocytosis The images get smaller because they are further away from the observer. Smallness here refers to their angular size. ... An object which is close to the focal point of a concave mirror has a magnified image. This image acts as the object for the 2nd mirror but it might be further away from the focal point of this mirror and therefore smaller. You could experiment with some mirror arrangements to see if you can stop the images getting smaller. – sammy gerbil Sep 17 '20 at 23:47
• Thank you, after checking your recent activity I thought you did not come to SE so I asked a question about it. I get your point, which I believe is a better answer to this question than the ones I received so far. Can I quote you or would you contribute an answer to this link? In any case, thank you for your interesting explanation. – Winston Sep 18 '20 at 5:58

You are forgetting one important very trivial third factor.

When you introduce a probe, you are blocking the infinite perspective point just in the position of the probe, hence you will never be able to capture the perspective point, even if no dissipation exists at all.

Note that in the shown image you look how the perspective curve goes out from the visual image, losting the perpective point.

This is equivalent to say that any photon reaching the perspective point must be perfectlty perpendicular to both mirrors, and also, perpendicular to your probe. That is clearly not possible, thus, you will never see the "bottom of the infinity".

At most, you will see a curvature produced by the multiple reflections, which at some point, will escape from one of the mirrors, effect which will seem like seeing a "rounded wall", lile walking at the side of a giant circle surrounded by infinite copies of your mirrored testing room, just a little twisted by the very small curvature....

As pointed out in the comments, if the probe used for tracing the light rays are is a wire loop, there is actually NO impediments to have infinite reflections, rather than the decay of the environment.

• I will put a graphical proof sketch later, it can be seen the only way to see the perspective point is putting a transparent probe, which is impossible. Any other probes will catch angles rays of light falling outside the perspective line. – Brethlosze Jul 20 '17 at 14:00
• The probe doesn't have to block the vanishing point, because it can be a wire loop that measures the curl of the electric field inside of it. – Display Name Jan 9 '19 at 15:08
• @DisplayName That is true. I have updated the question to expose that fact. – Brethlosze Jan 10 '19 at 20:46

I love these answers, but there's also a more fundamental problem with this thought experiment. Because the speed of light is finite, at any given point in time, you will only be able to see a finite number of reflections. Because, at any instance, there are only a finite number of reflections, then at any instance you are seeing something finite; growing rapidly, yes, but still finite.

Imagine, for instance, that we were somehow able to see fast enough to perceive each image as it was formed. I would be able to tell you exactly how many images had been formed at any instance; since there would always exist a larger number than that of the current number of images, the number of images would never be able to "reach" infinity.

Mathematically speaking YES ... there are infinite reflections as it gets closer to the center... (By Mathematically, I mean we ignore all physical limitations).

It's like asking how many decimal numbers between exists two whole numbers. As long as you know its there... [example: $$0 < 1\cdot 10^{-1}, 1 \cdot 10^{-2}$$, ... , $$1\cdot10^{-\infty} < 1$$], and you cannot really write out all the examples, so you just use "infinite" to define such situation. So in our case, the infinite reflections are there, you just cannot see it.

But Physically Speaking, however, NO... Human eye has limitation; Light source has limitation; Mirror has its limitation; Your Telescope/Microscope will have limitation... there will not be infinite reflections, because at the center, it gets too dark to see.

And Philosophically... THE QUESTION IS TOO VAGUE this question cannot be answered, because infinite itself cannot be measured/prove outside of the field of Mathematics. Therefore, even if you have infinite reflections, no body can prove that it exist other than having faith... maybe you should just take the leap of faith?!

If you place two mirrors with a 1-meter distance and assuming the light is attached on top of one mirror: 3ns/meter => 333 million bounce/second => 166 million images per mirror per second.

So, if you will wait 1 minute => 9.9 billion images.

To get an infinite number of images, you need to wait for an infinite time (assuming no loss during bounce).

I'm looking through one of them (using the one-sided mirror effect) and see what seems to be infinite reflections.

If you look at the reflected reflections of infinite reflected reflections through a one-sided mirror this means that the laser light will die out exponentially (except when the laser is continuously lasering.
The beam of coherent photons will after a period of time (in a vacuum) diverge as the photons wavefunctions will spread in space.

In your mirror case (let's also put these in a vacuum) the same happens. On top of that, there is a chance (depending on the laser's frequency) propagating a pair of virtual particles hiding in the dark to a real pair. After an infinite time, you can wave goodbye to your beam.