Infinity is weird… even in infinity mirrors!

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 z_1 and z_2:

Because of the geometric nature of optical imaging, the ratio of y_1 to y_2 is equal to the ratio of z_1 to z_2.  In equation form, we may write

\displaystyle \frac{y_1}{y_2}=\frac{z_1}{z_2}.

To put this another way, an object at distance z_2 will appear to be the same size as an object at distance z_1 if

\displaystyle y_2 = \frac{z_2}{z_1}y_1.

This, in turn, implies that an object moved from position z_1 to position z_2 will appear to be narrower by a factor

\displaystyle \Delta = \frac{z_1}{z_2}.

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 WH; the apparent area of the second image will be WH/3^2.  Continuing the process, we have the total area of all the images:

\mbox{area} = WH \left(1+1/3^2+1/5^2+1/7^2+\ldots\right).

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:

\mbox{width}= W\left(1+1/3+1/5+1/7+1/9+\ldots\right).

This series is comparable to one known as the harmonic series, discussed in a previous post, given by

H=1+1/2+1/3+1/4+\ldots.

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 y, and the length x:

The horn is defined such that, at any point along its length, its radius is given by y=1/x.  The circumference through any slice of the horn is 2\pi y = 2\pi/x, while the area of the interior of any slice of the horn is \pi y^2= \pi /x^2.

This is the connection to the infinity mirror!  Just as the areas of the images decrease as 1/n^2 and the widths decay as 1/n, the area of the horn decreases as 1/x^2 but the circumference decays as 1/x.

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!

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43 Responses to Infinity is weird… even in infinity mirrors!

  1. Andy "SuperFly" Rundquist says:

    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.

  2. 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.

  3. Tony Sidaway says:

    “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!”

    • Naham says:

      well no its not bigger on the outside the whole thing is infinite and so therefore a finite amount of paint will not by any means be enough to paint even a small fraction of infinity. Any fraction of infinity is still infinity, so 1/2 of infinity = 2/1 kind of like when you snap a worm in half and you have two worms but the worms would have to have no beginning and no end for this analogy to work. So if you have an infinite worm and you snap it in half you have two infinite worms. If you try to eat the entire worm, you will just be creating more and more infinite worms with every bite you take due to the nature of the infinite worm.
      No, infinite worms do not exist. At least, not in this dimension I hope.

  4. Andy Rundquist says:

    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.

  5. Yoron says:

    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.

  6. Melf_Himself says:

    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 🙂

  7. Pingback: Monday links | MathBlog

  8. andybarss says:

    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.

    • If the object were a rectangle, area = width x height, and if the width or height is infinite, the area can’t be finite.

      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!

  9. Infinite images??
    | |
    | |
    | |
    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:

      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 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.

  10. R-V-P says:

    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

  11. 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!

  12. Gargi says:

    Interesting Post! Thank you!

  13. barytheblade says:

    i am desigining an event where i would like to construct infinity mirrors and place large floral and chandeliers in front that then reflect to infinity in the mirrors, any advise

    • paulgibby says:

      I would advise placing flowers and chandelier as close to a mirror as possible as the eye is a wide-angle lens and the images will rapidly get smaller in the reflections. Good luck!

  14. Really interesting to come across this article today. I just posted this youtube video yesterday. It is a video of me using my usb webcam as it records its own live feed on my computer creating a similar effect as the two mirrors facing each other. Interestingly, the video is always white in the center, so though you can see the images duplicating, as it were, they eventually disappear into white light. You can check it out for yourself here —> http://www.youtube.com/watch?v=7r4DJG4xPTU&feature=youtu.be

  15. Sush says:

    What do you think is the mathematical concept connected to it if it were to be explained to a primary school child? Will it be “repeated symetry”? Do reply at the soonest I have a student waiting to find out.
    🙂

  16. Felix says:

    Hello.
    I am no scientist, i dont have degrees or talent with maths so what i am about to suggest might not make any sense at all, i ask only for you to consider the possibility.
    I recently experienced the davinci mirror chamber at an exhibition and this made me wonder about possible relationships between einsteins relativity theory and the way the infinity mirror reflects light. As i understand the theory one would need a machine travelling faster than the speed of light away from earth in order to see time change. One might need a strong telescope to look back at the earth and as the speed increases past the speed of light one might see time standing still and gradually go backward.
    Since the infinity mirror reflects light repeatedly and according to your calculations duplicate into infinity – somewhere along the gabriels horn the rate at witch images are reproduced relative to our vantage point might cause light to lag behind so i think my question is: is the speed of light relative to the rate of reflection(rate of images being reproduced)?
    The one problem you have with being able to see far enough is the angle at witch an object is being reflected, what if you placed a two way mirror opposite a normal one and then placed a telescope centered behind it? Using this method, if you were to place a lit object off centre, zoom in with the telescope and switch the light on and off, i wonder if you might see a delay of light?
    If this doesnt make any sense do not reply otherwise please explain how the infinity mirror cannot be part of the riddle of time travel.

  17. sebastian says:

    How many mirrors would it take to create a five second delay from you to the last mirror?

  18. teacherblack says:

    Hey–I am actually writing a blog post about narcissism, but wanted to use this metaphor for the generational effect–would you mind if i copied your image?

  19. philip smith says:

    I invented the infinity mirror system in 1963 with my buddy maurice vigon and you aint seen nothing yet.

  20. Pingback: Caleidoscopicamente | Eva lo sapeva

  21. Megha says:

    hey can u tell me the real life uses of this mirror ? nt ableto find it anywhere

    • As a set of image-forming mirrors, there isn’t much use other than being artistic. In physics, however, parallel mirrors play a role in the Fabry-Perot interferometer, where the same sort of effect acts as a very effective light filter.

  22. Deepanshu balani says:

    I have made the project infinity mirror for 12th class but I am not able to know right ray diagram for infinity mirror. What are the factors affecting infinite depth ?

  23. James says:

    I am lost in the explanation up to point where you conclude: “This, in turn, implies that an object moved from position z_1 to position z_2 will appear to be narrower by a factor (z1/z2)”. I don’t understand the notion of the object being ‘narrower’. All that is getting though to me is simply that if this proportion is satisfied, in viewing two different sizes objects at the relative distances that the proportion describes, they would appear the same size. Could you explain where this concept that the reflected downfield image will become narrower?

  24. Martine says:

    I am an architect and I am trying to AVOID the infinity mirror effect in a master bathroom. I hear it can be reorienting. The vanities are designed face to face with 5′ space between, each have a mirror and they face each other. Can I angle the Mirrors slightly – thinking of 2 degrees – to avoid Infinity effect? We will make a mock up but wanted to see if there are any other suggestions we can try.

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