| Mirrors reverse front and back.
Stand in front of a mirror with an object. Move the object left, and
the image moves left. Move it right, and the image moves right. Move
it up, and the image moves up. Move it down, and the image moves down.
Clearly the mirror does not alter left, right, up, or down.
Move the object in the direction that points from your front to your
back. The mirror image goes the opposite way. Move the object in the
direction that is from your back to your front. The image goes the
opposite way.
Front and back is the only dimension reversed by the mirror.
Why do people think the mirror reverses left and right? When we walk
around, we keep our bodies vertical. We turn left and right, but
rarely turn upside-down. When we interact with another person, their
body is generally in the position ours would be in if we rotated 180
degrees (and moved a few feet). If we rotated 180 degrees, our former
left would be on our new right, and vice-versa, and our former back
would be toward our front, and vice-versa.
Rotation about a vertical axis reverses two-dimensions: left-right and
front-back.
In the mirror, the image is only reversed front-back. But it is in the
position where we are accustomed to seeing a person whose front-back
AND left-right are reversed with respect to ours. So the left-right
looks wrong from our expectations. If you had rotated a person into
that position, their left-right would be reversed. But it isn't
reversed in the mirror, so we think it is odd.
-- edp
Public key fingerprint: 8e ad 63 61 ba 0c 26 86 32 0a 7d 28 db e7 6f 75
To find PGP, read note 2688.4 in Humane::IBMPC_Shareware.
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| Feynman had an amusing explanation of this phenomena in terms of
pulling your face through to the back of your head and vice versa!
Actually, a reflection mathematically has a determinant of -1, and
that explains it all :-)
Now, how about that Spinor Spanner ? (Another note is on this somewhere
in here!)
- Jim
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| Yup.
I'd like to add some comments.
(Let us restrict ourselves to R^3). For each reference class consider
the sign of the determinant of its linear mapping on the standard
reference system (i.e. (1,0,0),(0,1,0),(0,0,1) ). Obviously, "having
the same determinant sign..." is an equivalence relation on the
reference systems. The orientation of a reference system is defined as
being its equivalence class. So we have right and left (or positive
and negative) reference systems, depending on their having or not the
same orientation as the standard reference system.
Reflections change the equivalence class (called orientation), while
translations and rotations conserve it.
When considering a reference system Oxyz and its reflection O'x'y'z',
if you identify two by two two of the axes with their , say Ox with
Ox' and Oy with Oy', you must inverse the third axis (Oz with the
opposite of O'z'), otherwise the two reference systems would have the
same orientation.
When looking in a mirror, we identify ourselves with the reflected
image, and generally we do it by identifying the back-front axes and
the down-up axes - so what's left? Inversion of the left-right axes.
Try to imagine that you are a happy dolphin, more accustomed (let's
say) to move head over heels than with our human rotations on earth,
and try to use, while identifying yourself with your reflected image,
the pairing of the back-front axes and of the left-right axes.
How are you feeling ? :-)
I hope that you do feel that the mirror inverses up and down, and not
left and right.
Now the question is why do we use instinctively in our mental
representations only translations and rotations, and only after a
little bit of thinking (and education?) are we able to grasp the
reality of reflections.
I suppose that the animal experiences of moving around (only by
rotations and translations) preceded for a long time the spark of
understanding that experienced the first humans by looking in a pond.
Young children understand translations and rotations before
reflections.
And have you tried the question 1890.0 on people without mathematical
or physical education? Try it, and enjoy!
Mihai.
P.S. What's this spinor spanner story? Can somebody tell us it?
Thank you.
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| As noted in .1, the only direction reversed is the direction perpendicular to
the mirror. Since most mirrors are mounted vertically, up and down are indeed
not reversed. However, even with a mirror mounted on the ceiling, right handed
shapes would still be turned into left handed shapes.
/David
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