# Mirrors into Infinity

Could someone please name the phenomenon regarding the stretch of reflections into infinity between two opposing mirrors, and also explain why the reflections curve away instead of meeting at a perspective point within the reflections?

The effect you note - the curving off instead of converging to a point at infinity - is due to the imperfect alignment of the mirrors. While they may be nearly parallel, they will always be off by some small angle $\theta$. This angle gets added up reflection on reflection: if the angle between the two mirrors is $\theta$, then the angle between mirror 1 and its first image in mirror 2 will be $2\theta$, its image inside that first image will be angled at $4\theta$, and so on. This then causes further reflections to shift off to one side (or up, or down) until they are no longer visible.

In the image below, each 'mirror' is off from the vertical by $\theta=2^\circ$, the image of the room in each successive mirror is out by an extra angle of $\theta$, and this quickly accumulates.

As an application, by placing your head at the top of one mirror and counting how many copies of the room are visible, you can estimate the tilt angle between the mirrors. Say the room has width $L$ and the mirrors are a height $h$, and you can see $n$ copies of the room. Since the vertices in the image are part of a (quasi) regular polygon, the bottom of the $m$th reflection of the right mirror is at a height of $R(\cos\theta-\cos(2m+1)\theta)$ above the original, for $\tan\frac\theta2=\frac L{2R}$, and when this passes $h$ you get $n=m$:

$$L(\cos\theta-\cos(2n+1)\theta)=2h\tan\frac\theta2,$$ or its small-$\theta$ version $$2n(n+1)\theta=\frac{h}{L}.$$

Why am I going into so much detail? Notice that the dependence on $n$ is quadratic whereas the equation is only linear in $\theta$. This means that to see twice as many copies of the rooms, you need the alignment of the two mirrors to be four times as good. This reflects something known well to experimental phycisists: aligning optics is hard. Hence the fact that mirrors casually mounted on walls are very rarely aligned well enough that you can see more than five to ten copies of the room.

• + Nice, and you get a similar effect if you have 1) a TV camera, connected to 2) a closed-circuit TV monitor, and 3) the camera is looking at the monitor. In the 70s this was an example of electronics "freaking out". – Mike Dunlavey Jan 13 '14 at 21:55
• What a great explanation, with mathematical evidence. Superb, this was the type of technical answer I was looking for. – Travis J Oct 17 '16 at 6:07

I do not know the name of the phenomenon (maybe there is none), but assuming that you are standing at some distance from the centre of 2 mirrors perpendicularly away from the line joining the 2 mirrors, you can see that after each reflection the mirrors would form the mirror both perpendicularly away as well as away from the center which appears to be curving away.

Suppose the distance between mirrors is 2 m and you are standing 2 m away from the line joining their center, the first set of images would be formed 4 m away from you and 1 m inside each mirror, the next set of kmages would be even further from you, and each would be 3 m inside the mirrors, this would continue and the images would continue to go both away from you and more and more inside the mirrors, this seems like curving away.

A simple point: If your head or eye level is higher than the mid point of the mirrors, immediately a small angle comes into play. Now if the mirrors are spaced far apart, the visual effect of curving gets magnified, which further increases with subsequent images. It is plain geometry actually. Draw two parallel mirrors and place an object between them such that it is slightly higher than the midpoint of the two mirrors. Draw the resulting images and you will get your answer.