Even very simple optics can reveal very interesting and surprising phenomena, if one looks carefully enough! I was recently looking into the optics of a so-called “infinity mirror”, which in its simplest incarnation is simply two parallel mirrors on opposite sides of a room or elevator. The result is a multiplication of images, seemingly stretching out to infinity (source):
I started mulling over the nature of the images — assuming one could see all of the images in an infinity mirror, all the way to infinity, would their total apparent area be finite or infinite? It is probably clear from the photo that they’re finite, but there is nevertheless a surprising twist, which I will reveal below, after some math!
First, a few words about the “infinity effect”. As the picture above implies, to see an infinite number of images of an object one has to look at it in the mirror off-center: the cameraman in the photograph is standing behind and to the left of the woman. A single person standing in front of the mirror will only see a single image; the infinity of images are hidden “behind” the first image. We will nevertheless investigate the area and total width of the hidden images of a single person: the mathematics is easier, and as far as I can tell the results don’t change when one considers a real off-center observation.
Let us now carefully set up and analyze the problem. We imagine a person standing exactly between two parallel mirrors separated by a distance d:
The first image will, of course, appear a distance d away from the observer, behind the mirror he is facing:
The next image facing the observer will be produced by the reflection of the first image in the mirror behind him:
The second image will appear a distance 3d away from the observer. One can continue this chain of reasoning to argue that images will appear facing the observer at d, 3d, 5d, 7d, and so on, in principle to infinity.
In order to calculate the total area of the images, however, we also need to know how the side of the images depend on the distance from the observer. Here we can borrow a little intuition from an old post on anamorphic imaging, and consider the following figure with two objects at distances 
Because of the geometric nature of optical imaging, the ratio of 
To put this another way, an object at distance
This, in turn, implies that an object moved from position
If the first image a distance d has a width W, then an image at a distance 2W will appear half as wide, and an image at a distance 3W will appear one third as wide, and so on. The same argument applies for the height H of an image at any distance.
Let’s put all this together. The apparent area of the first image will just be 
This is known in mathematics as an infinite series — an infinite sum of terms that may or may not add to a finite value. It can be shown — using more math than I want to get into in this post — that the sum of this particular infinite series of terms is finite. That is, the total area of the infinite collection of images is finite.
Now here’s where things get a bit odd — suppose, instead of considering the total area of all the images, we consider the combined width of all the images. We then get the following summation:
This series is comparable to one known as the harmonic series, discussed in a previous post, given by
The harmonic series diverges; that is, the sum of all the terms is infinite, even though the terms get progressively smaller. Similarly, because it is comparable to the harmonic series, the combined width of all the images in the infinity mirror is infinite.
We therefore have an unusual situation: the combined area of all the images in an infinity mirror is finite, but the combined width of the images is infinite!
This mathematical peculiarity is closely related to a geometric object known as Gabriel’s Horn, pictured below:
This horn, which extends off infinitely to the right, is an object with an infinite surface area, but which encloses a finite volume! To see the connection to our infinity mirror images, let us label the radius of the horn 
The horn is defined such that, at any point along its length, its radius is given by 
This is the connection to the infinity mirror! Just as the areas of the images decrease as 
The odd properties of Gabriel’s Horn were first studied by Italian mathematician Evangelista Torricelli (1608-1647), and it is also known as “Torricelli’s trumpet”. To understand how truly weird this mathematical object is, it could be filled with a finite amount of paint, but it would be impossible to completely decorate the outside of the horn with this paint, no matter how thin a coat!
Of course, we cannot make an infinite version of Gabriel’s Horn, and similarly we never see an infinite number of images in an infinity mirror. The images either drift off the visible surface of the mirror after a finite number of repetitions, or become too dim to see — less and less light contributes to the more distant images.
The lesson here, I suppose, is that unusual mathematics can be found even looking at relatively simple physical phenomena such as an infinity mirror. The secondary lesson is that weird things happen when one considers problems involving infinity!
I was thinking about the math changes is you go off center. The image locations and sizes remain the same and so it’s just their apparent size that changes. A little uglier but not that bad, I would think. That made me wonder about the actual image you showed. If the image locations shouldn’t change, I was wondering about the curve seen as you look further “into” the mirror. Does that curve tell you about the un-parallel-ness (how about that for a word!) of the mirrors?
The math isn’t *too* bad for the off-center case: one has to take into account both the diagonal distances to the image and the fact that the different images will have a different orientation. The curve in the image I think may simply be the 1/n dependence of the image size on distance! However, some distortion that is present in the image is likely due to the “unparallelness” and even non-flatness of the mirrors.
Quick question: is there a simple relation for how bright each successive image is?
Not that I know of! One factor is the reflectivity of the mirrors — some small fraction of the light gets absorbed at each bounce. Also, more distant images should appear dimmer simply due to propagation of light, though I’m not sure exactly how to account for that in this case.
It’s the diffusion factor I was most interested in, and I don’t know how to treat that. The absorption is a simple relationship, if my analysis is correct: each reflection removes a certain fraction of the intensity. I wonder which is more significant then — absorption or diffusion.
I’m not sure how to do the diffraction/diffusion term either. One might naively think that the intensity of a radiation source goes as 1/R^2, and therefore the image brightness would do the same, but I suspect it’s a bit more subtle than that.
“To understand how truly weird this mathematical object is, it could be filled with a finite amount of paint, but it would be impossible to completely decorate the outside of the horn with this paint, no matter how thin a coat!”
“You mean, it’s bigger on the outside!”
In my hotel room last night (in Omaha for the AAPT meeting) I used two bathroom mirrors and a closet mirror to do a 3 mirror version of this.
It would be fun to do the math on that!
How is that possible skulls?
If I were to assume that mathematics is a ‘true language’ of the universe, and that the mathematical presumptions for this definition are correct, taken from our universe, then it should be true, should it not?
Is there any real application possible from this definition?
And if not, can it then be said to be true to us, or should I consider it a definition true in a ‘sandbox’, defined by axioms that are not fitting our reality?
Or do you have an alternative opinion?
Not quite sure what you’re asking — are you concerned about the infinite width of the images? This is only an “in principle” calculation; as noted, in practice the images are cut off by the edges of the mirror or fade into darkness with a decrease in light.
You can’t fill the horn using any amount of paint, because the paint will take an infinite amount of time to reach the tip
Ha! Nice point.
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If area = a function of width and height, then surely there’s a contradiction here. If the object were a rectangle, area = width x height, and if the width or height is infinite, the area can’t be finite. I’m sure I’m missing something here.
Well, if it were a rectangle, then of course the argument wouldn’t apply! But we’re not dealing with a rectangle, but rather an unusual geometric shape. The rough idea is that both the width and the height of the images get smaller and smaller. The width of the images gets smaller too slowly to converge to a finite number, but the product of width X height decreases fast enough (since it is the product of two decreasing numbers) to converge to a finite number.
It’s very non-intuitive, but this is exactly the problem whenever one is dealing with infinity — infinite concepts are inherently non-intuitive!
Infinite images??
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A B
No. The images cannot be infinite. This is because light has finite speed.
Justification
Images are due to the passage of light between the mirrors. Light from A moves to B, covers the distance ‘d’ between them, bounces back & returns. An image of A is created in B, which is at apparent distance of d from B due to symmetry. But this will require some time as the light speed ‘c’ is not infinite. Now the same is applicable for B. Thus now light has covered a total distance of 2d.
The process continues and we get some number of images denoted by N at the end of time t. In the process, light travels a two way distance of 2ct.
Because of symmetry, the apparent distances between the images are the same: d.
Since light is just bouncing between A & B, saying that “light travelled a distance of 2ct between A & B in time t” is equivalent to saying “the apparent distance between the extreme images is 2ct”. But this distance includes a distance of 2d i.e. distance travelled by light (1st displacement) between the mirrors, which should not be considered for N. Thus
N=2(ct-d)/d
Well, yes. As I noted in the post:
The point isn’t that there is a perfect infinity of images in the mirrors, but that the set of mirrors is a “reflection” of a mathematical infinity.
Dear skullinthestars,
I notice that you are using my photograph at the top of this thread (lady in the mirror), probably obtained from Flickr. The image is, in fact, copyrighted so I would be grateful if you would either remove it or contact me through Flickr to discuss terms.
Many thanks,
R-V-P
Changed. I actually found it on another site via a Google search — http://infinitymirror.org/infinity-mirror — who I’m sure you’ll want to contact. I avoid taking images from photosharing sites, in particular due to copyright issues.
P.S. Apologies!
thanks for this post. I was wondering what you thought of a set of mirrors in which was planted a small but powerful light source. As point of view of the observer changed ever so slightly from dead on (parallel) to some small angle, the photons would be bouncing such a long distance (as Hernish the Mathemetician was suggesting) that they momentarily would be outside the “event cone” — would have too far to travel to be seen.
It’s an interesting thought! Assuming the light source was bright enough (and the mirrors perfect enough) that images a few light-seconds away in distance could be seen, one would actually detect a time lag, where images “further away” would respond to changes a few seconds later. Of course, these images would be extremely far away (a light-second is 186,000 miles), so one wouldn’t be able to see much anyway!
Interesting Post! Thank you!
You’re welcome!