Monday, August 6, 2007
Symmetry: Reflection or Translation?

If you have shoes that lace up, look at your feet. Your shoes have reflection symmetry, but almost surely your bow knot is not reflection symmetric.

By reflection symmetric, I mean that if you put your left shoe side by side with a mirror, the image in the mirror would look just like your right shoe looks.

What about the knot? Well, the left and right hands have very different roles in tying the simple shoe lace knot. It takes most kids days or weeks to learn how to tie the knot, and once they learn how to do it one way, they aren't going to go back and teach the left hand to do the right hand's role and vice versa.

So, watch yourself tying your shoes - the knot is "translation symmetric" - i.e., the knot on your one shoe (at least logically, though not exactly) looks just like the knot on your other shoe, but shifted over (or "translated").

Just for fun, you might want to try tying your shoes with the roles of the right and left hands reversed. Of course you can learn it - you learned it the way you do it, so you can flip. It just takes some time, because the two sides of your brain need to teach each other what they know.

So if you tied your shoes reversing the roles of your hands for each shoe, the knots would then be "reflection symmetric." But why on Earth would anyone bother to put all this effort into creating "reflection symmetric" shoe lace knots?!

Ah, but when you play the kalimba, once you learn a cool riff where the left and right thumbs have different roles, it DOES make sense to take the time to learn how to totally flip sides and make something that is reflection symmetric.

;