Physics Puzzle: the Inverting Mirror
Posted on May 2, 2008
Filed Under Physics Puzzles |
Here’s a nice puzzle that left me scratching my head for a while:
If you stand in front of the mirror with a shirt that has something written on it, you will see it “inverted” (left-to-right becomes right-to-left). Informally, we can say that a mirror inverts left and right. However, note that up & down are not inverted! That is, you do not appear “upside-down”. Why is this so? What is so special about left & right?
Good luck & have fun! 
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4 Responses to “Physics Puzzle: the Inverting Mirror”
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We have bodies that are mirror-symmetric through a vertical plane that bisects us. And this includes our brain and visual apparatus. It’s our hands that are not symmetric through their own vertical plane, and that is what is transformed by being “pushed through” the mirror. We are not symmetric through any horizontal plane that bisects us.
Would aliens constructed of a spherical, translucent brain and a single, central, omnidirectional eye even understand handedness until they started thinking about screws, helices, and parity conservation?
Hasn’t this been discussed by Weyl, Feynman, and that great amateur, Martin Gardner, among others?
I’ll take a stab at it…
The way you’re presenting the letters to the mirror are reversed from your point of view, but are normal from the man in the mirror’s perspective.
(1) You look at your T-shirt, and all the letters are presented in normal order
(2) You ROTATE the T-shirt 180 degrees around the vertical axis to put it on, so the direction of the letters have reversed
(3) The letters REFLECT onto the mirror and REFLECT back to you in the same direction as they were presented, which is in reversed order.
So, there are three operations: rotation, reflection, and reflection. Meanwhile, since you never rotated the shirt along the horizontal axis, the letters won’t be upside-down.
Good point, Amol.
This question can be answered on many different levels. Optically, one can draw a ray diagram for every point in our body (a ray of light originating at that point & ending up in our eye) and examine the virtual image in the mirror. This, by the way, would show the effect does not depend on us having two eyes - it would work equally well for a cyclops ;). Such a ray diagram would show that no “upside-down” inversion occurs.
The word “invert” is, in itself, somewhat problematic. Realize that, if our mirror is the zy plane and we are standing on x>0 and have some spatial density D(x,y,z), then our virtual mirror image would be D(-x,y,z), which goes to show that not only z, but y as well would be unaffected (y being unaffected meaning that left gets mapped to left & right to right, just as up gets mapped to up & down to down).
This thinking shows, however, another interesting point (not directly related to the original question, mind you!) - a mirror changes the handedness of our image. Even if our “twin in the mirror” would step out of the mirror and stand by our side we would not get an exact copy of ourselves - our “left side” would get mapped to our “right side”, and no amount of rotation would invert him/her back. A mathematician would say that our mirror reflection took us from one partition of the orthogonal group O(3) to another (O(3) is subdivided into two SO(3) groups, connected by a reflection). But group theory will perhaps be discussed at another time :).
Nice blog, full of good infos, keep the good work going.