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Conference rusure::math

Title:Mathematics at DEC
Moderator:RUSURE::EDP
Created:Mon Feb 03 1986
Last Modified:Fri Jun 06 1997
Last Successful Update:Fri Jun 06 1997
Number of topics:2083
Total number of notes:14613

1890.0. "Mirrors? Call me Dirty Tricky" by EVTSG8::ESANU () Wed Aug 24 1994 16:55

	You have all looked in a mirror. Why do people say that
	mirrors reverse left and right? And not up and down, for
	instance?

This seems to have originated in the physicists' folklore. Perhaps you know it
already? In any case, please contribute with reasonable answers.

Thank you,
Mihai.

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1890.1RUSURE::EDPAlways mount a scratch monkey.Wed Aug 24 1994 18:2639
    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
    
    
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1890.2WRKSYS::ROTHGeometry is the real life!Thu Aug 25 1994 20:2710
   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
1890.3Yup + some commentsEVTSG8::ESANUAu temps pour moiTue Sep 06 1994 11:5556
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.

1890.4More briefly...ASDG::BELLTue Sep 06 1994 16:526
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