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As is well known, if you put three mirrors at right angles to each other, a ray of light entering the corner will be reflected back in the direction it came from.  The people we sent to the moon left some devices of that kind there, with the result that we can do lidar on the moon and measure its motions exquisitely.

I have wanted for a long time to put such a thing in an upper corner of my room, so that if you look at it from anywhere, you'll see your face upside down.  1-ft mirror tiles, available on the Web from various suppliers, are just the thing, but I worried about how to mount them.  Then it occurred to me that they wouldn't have to be mounted on the ceiling; they could rest on one of my tall bookcases.  So I ordered some, and put them together with duct tape, in such a way that the bottoms & tops of the vertical mirrors are flush.  The top mirror, being unsupported at the outer corner, sags a little, so that it looks as if you had four eyes; but an inconspicuous strut, consisting of a 12-in. stretch cut from a coathanger, takes care of that.

The tiles come in boxes of six, so I have three left over.  Anyone in the vicinity who wants to make a retroreflector is welcome to them.

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Sounds like fun.

Is the image reversed?

1 mirror: reversed orientation

2 mirrors (dihedral right-angle groove): original orientation, because the image bounces twice, the second one undoing the first one's reversal

3 mirrors (trihedral right-angle [? I guess so] nook): re-reversed orientation?

Actually, I don't doubt that it's not in the original orientation, but multiple reversals get hard to parse and you've got an actual sample there.

I am confused about that too. However, it is easy to check. I am looking at it with my good (left) eye, and consequently its image is at the vertex of the retroreflector. It is on my right, which means it is on the left in the image (because the image is facing me). But the image is upside down, so it is the image face's right eye. Thus, the image is reversed -- same as it would be in an ordinary mirror.

I know you've been answered but I want to add:

If a mirror reverses left and right, why doesn't it reverse up and down? If I look in a (normal wall-mounted) mirror while lying on the floor, why doesn't my mirror image appear to be lying on his room's ceiling?

These things are easier to work out if you think of a normal mirror as reversing front and back rather than "left" and "right". For example, if I am standing and facing north while looking into a mirror, the mirror is reversing north and south, while leaving east/west (and up/down) unchanged. The part of me that is furthest north (my front) appears in the mirror to be furthest south.

The illusion of left-right reversal comes from the fact that people are usually upright when looking at mirrors and also have a superficial bilateral symmetry. But we are not symmetric on the inside. If my mirror image were able to come through the looking-glass to join me, he would be unable to read my books or digest the glucose in my bread; when eventually we went to the hospital the doctors would report that his heart was on the "wrong" side of his body and he would respond incredulously, "No it isn't, it's still on my left side like it has always been!".

So returning to the trihedral retro-reflector, the answer becomes: One mirror reverses the north-south, a second mirror reverses east-west, and the third reverses up-down. The thrice-reflected image has had its chirality reversed three times, and so its right eye is on what we would call its left, even though it thinks that is its right.

- Robert Munafo

For further elucidation of the left-right "paradox", one may note:

1. If you stand on, or under, a horizontal mirror, it does indeed show you upside down.

2. Suppose we were not (nearly) left-right symmetrical, but instead (nearly) top-bottom symmetrical. Suppose we had legs on both ends, but only one arm. (Presumably, our faces would be in our bellies. The task of making them symmetrical is left as an exercise.) While standing on (say) our Republican feet, we would have only a right arm; on our Democratic feet, only a left arm. Now, standing on your Republican feet and looking in a mirror, you would see an image standing on its Democratic feet -- that is, yourself turned upside down.

3. When you hold a book up to a mirror, you usually flip it around a vertical axis, so left & right are reversed. You could just as well flip it around a horizontal axis, and then top & bottom would be reversed. If you hold a transparent page up to the mirror, you don't need to flip it to see the print, and then the print is readable as usual in the mirror.

Strut really necessary?

It sounds like you supported the front corner of the top mirror with a strut oriented like a vertical column.

But if these mirror tiles are made of glass, like the ones I've seen, then it seems that you could instead place a few pennies on the back corner of the top mirror, and the weight would then be enough to keep it aligned -- unless these mirrors aren't actually square or unless you're viewing from really far away. Am I missing something?

Re: Strut really necessary?

The back corner of the top mirror is at the vertex. It is resting on the top edges of the vertical mirrors, which are resting on the shelf. Weighting it would not budge it. The effect, AFAICT, is due to slight bending of the front half of the top mirror, which is cantilevered. You are right in suggesting that the effect is largest at long distances. It is also largest at oblique angles. I do not pretend to understand the details, but at any rate the strut makes it go away, pretty near.

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