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Mathematics of Mirrors

Mathematics of Mirrors

Hold a ball and stand in front of a mirror. Now pretend the mirror does not exist and what you are seeing is real. Is the mirror image of the ball any different than the real ball? No.

Now hold up your right hand. The mirror image holds up its left hand. So it's not the same, which means you are chiral. An object is chiral if it's different than its mirror image.

Mirrors have fascinated people for thousands of years. But what exactly do they do?

All objects are exactly the same after two reflections, but with a single mirror there is only one reflection. Is an object the same after just one reflection? Some are, but many are not. This is true even for molecules in chemistry and elementary particles in physics. Despite what we would like to think, nature is not mirror symmetric. Chirality rules!

Think of a mirror reflection as a transformation R on an object. Two applications of R always gives the original, so..

R*R=I where I is the identity transformation which corresponds to doing nothing to the object.

But this equation has two solutions R=I and R=-I. It's the latter solution that's chiral.

Let's get back to the ball. Now spin it clockwise and look at the mirror image. The mirror image is spinning counter-clockwise. So it's not the same. A ball is not chiral, but a spinning ball is chiral!

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Content written and posted by Ken Abbott