Mirror - Solution
Mirrors invert front to back, not left to right.
The popular misconception of the inversion is caused by the fact that a person when looking at another person expects him/her to face her/him, so with the left-hand side to the right. When facing oneself (in the mirror) one sees an ‘uninverted’ person.
See Martin Gardner, “Hexaflexagons and other mathematical diversions,” University of Chicago Press 1988, Chapter 16. A letter by R.D. Tschigi and J.L. Taylor published in this book states that the fundamental reason is: “Human beings are superficially and grossly bilaterally symmetrical, but subjectively and behaviorally they are relatively asymmetrical. The very fact that we can distinguish our right from our left side implies an asymettry of the perceiving system, as noted by Ernst Mach in 1900. We are thus, to a certain extent, an asymmetrical mind dwelling in a bilaterally symmetrical body, at least with respect to a casual visual inspection of our external form.”
Martin Gardner has also written the book “The Ambidextrous Universe.”
Casper has sent in the following:
-
‘The riddle isn’t 100% true. If one turns his head 90 degrees to the right before looking in a mirror, one will notice that up and down are being reversed! Left, right, up and down are relative expressions, therefore the meaning of these expressions depend on how they are used. In fact, the problem of the riddle is that the mirror seems to only change the image in one direction.
The reason for this phenomenon is quite easy. A mirror is a two-dimensional field and can change an image in two directions. But because a human has got only two eyes, the image is only mirrored in one direction. If a human would have three eyes (or more), the image would be mirrored completely in two directions.
Furthermore, I wonder what would be the effect for some of us who’ve got only one eye. If such a person is used looking with one eye, I expect that the image won’t change at all. Maybe someone else can give an answer to this.’
I don’t agree with him, especially on that one eye (try it out yourself, closing one eye). His way of thinking is however creative, so I thought I could not withhold his reaction.
The answer to this is very simple. It has nothing to do with the number of eyes you have. No matter how many eyes a person has the results would be the same. The answer lies with the fact that humans are symmetrical. What we perceive in a mirror appears to be ourselves if we have turned around 180 degrees. We are turning left to right or right to left attempting to match the orientation of the image we have perceived in the mirror. By doing this we are inverting our left and right, but not our up and down. If you look at the mirror with your head turned 90 degrees to the right, we have also rotated our axis of symmetry. To match the image seen in the mirror you would have to turn your head 180 degrees on that new axis of symmetry and therefore, you would be inverting up and down.
Really up, down, left, and right are not inverted. For example if your mirror runs west to east and you point east facing a mirror, the direction your mirror image points will also be east. Though it will appear that he is using the opposite hand since to match his orientation you would have to turn 180 degrees and switch hands.
The only direction the mirror does invert is front and back. These directions are “toward the mirror” and “away from the mirror”. Let’s continue our previous example of the mirror running west to east. If you pointed north toward the mirror, your mirror image would be pointing north (toward the real you).
To simplify this, you can see a diagram in two dimensions from an aerial view. (We are right now ignoring the up/down dimension, since we can agree that it is not reflected.)
W W
S+N N+S
E E
real mirror
/\ /\
> <
In a refection over a line in two dimensions, only one dimension is inverted: the dimension perpendicular to the line of reflection.
In three dimensions, the mirror is the plane over which the object is reflected and similarly, only one direction is inverted: the dimension perpendicular to the plane of reflection
It also helps to know facts about mirrors. What a person sees is the way light reacts to objects. When light hits a mirror, it bounces back at the same angle. While looking at an object directly it appears normal, but the mirror changes reverses the angle of light, thus creating the “backward” image.
A cool experiment though. Place two mirrors, reflective sides in the same direction, perpendicular and look at the edges that join. The image is a true rotation of the body.