Recently I was sitting in front of my 3-year-old daughter, Sonia,
and helping her to put on her shoes. I held one up and asked, "Which
foot does this go on?" She enthusiastically responded, "The right
one!" Then she added "And this is my right hand!", holding up a hand
that was indeed her right hand. For a moment I basked in the glow of
my daughter's impressive intelligence. But then that bubble was
deflated as she added, "And that's Daddy's right hand!", pointing to
my hand which was directly in front of her right hand. But since I
was facing her, the hand she pointed to was actually my left hand. I
tried to correct her, but she seemed upset. "Why is right and left
different for me and you, Daddy?" I tried to dismiss the question,
but she insisted. I started to articulate an answer... but realized
that I didn't really have a good one. Her up and down were the same
as my up and down-- why were our rights and lefts different?
After thinking about it for a while, and reading a nice online
article by someone named Eric Schmidt, I was able to clarify my
thoughts and come up with an answer. Left and right are simply a
different type of direction than up and down. Up and down are clearly
defined in relation to some reference point, usually the center of the
earth: anything going towards that is going down, and anything going
away from that is going up. If you stand in the center of a crowded
Math Mutation fan club party and ask everyone to point up, they will
all point the same direction. But left and right are relative
directions: if you ask everyone at the same crowded party to point
to their right, some will point north, others will point east, etc.
You need to define both an 'up' and a 'front' first, and only then can
you define left and right, in relation to those two directions. So if
two people are in a room facing different directions, they have a
different front, which naturally changes their definition of the
relative directions known as left and right.
Related to this issue is the question of why, when you look in a
mirror, your mirror image has its left and right sides reversed. In
one sense, we can say that this statement is false: the image does
*not* have its left and right sides reversed. When you are looking in
the mirror, there is only one sentient being in the room, you, and
your 'up' and 'front' are used to define which way is left and right.
If you hold up your right arm, then the image is holding up its right
arm-- where right is defined according to your orientation. Unless
you are part of an Alice in Wonderland novel, the image is not a real
creature, just a set of light waves that happen to be bouncing around
the room. If you choose to anthropomorphize the mirror image,
imagining that there is an actual being in there just like you, then
you are subconsciously turning around the concept of 'front', since
your mirror image is facing out of the mirror. Then, since right and
left are only relative directions defined in terms of a front, you can
label the hand that is on your right side to be the image-creature's
left hand. But you're really comparing apples to oranges here: once
you change the conceptual orientation, you expect the relative
directions of right and left to be different.
Unfortunately, all these nice explanations were lost on Sonia. By
the time I was ready to answer her question, she was no longer
interested in the concepts of relative and absolute directions, and
had moved on to play with her toy tweety-bird.
And this has been your math mutation for today.