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(The question was why a mirror appears to reverse left and right,
not up and down)

If we apply a certain type of linear transformation (a reflection) to
certain asymmetrical objects, the result may not be congruent to the
original object.  A reflected glove, for example, is not congruent to
the original glove.  These objects are said to possess "handedness",
and the handedness is reversed by the reflection.

We use the words "left" and "right" as handedness-dependent directions
relative to certain asymmetrical objects.  The position of the steering
wheel of a car, for example, defines directions called "left" and
"right" relative to the car.

Now, just as a mirror does not "reverse up and down", it does not
"reverse left and right" as directions.  If I look into a mirror and
point up, my image points up.  If I look into a mirror and point to my
right, my image points to my right.

The confusion arises because my image has opposite handedness from me,
and although we agree on what is "up", my "right" is different from its
"right" because we have different handedness.

To sum up, a mirror appears to reverse right and left because the words
"right" and "left" are defined relative to the handedness of a real or
virtual object, and my reflection has opposite handedness.



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