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 and :
Because of the geometric nature of optical imaging, the ratio of to is equal to the ratio of to . In equation form, we may write
To put this another way, an object at distance will appear to be the same size as an object at distance if
This, in turn, implies that an object moved from position to position will appear to be narrower by a factor
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 ; the apparent area of the second image will be . Continuing the process, we have the total area of all the images:
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 , and the length :
The horn is defined such that, at any point along its length, its radius is given by . The circumference through any slice of the horn is , while the area of the interior of any slice of the horn is .
This is the connection to the infinity mirror! Just as the areas of the images decrease as and the widths decay as , the area of the horn decreases as but the circumference decays 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!