[I apologize to those of you who have been reading my blog for more than a year. I’m reposting something I wrote last year at this time. I was then, and am now, teaching Calculus III, and we just finished discussing the cross product. I ended the conversation by telling my classes how the cross product helps us answer the question: why do mirrors reverse right and left but not up and down?]
Stand in front of a mirror and hold up your right hand. The person standing in the mirror holds up her left hand. Why is that? Why does a mirror reverse left and right? After all, it does not reverse up and down.
Before we answer that question, we have to ask a more basic one: what is “the right”?
Looking for an answer, I turned to the venerable Oxford English Dictionary (subscription required). I was disappointed to discover that the OED does not give a definition of the term “right”! Instead it gives the circular path shown below.
17. a. = RIGHT HAND 2.
2. a. The right side. b. The direction towards the right. = RIGHT n.1 17a.
“Right” and “left” are a slippery concepts that are hard to define. In fact, you need to know other things about an object before you can determine its right and left. For example, if I handed you a blob-like sea creature and asked you which side is its right side, you may not be able to answer me. If I told you where the top and front sides of the critter were, then you could quickly identify the right side.
Here’s a mathematical explanation of what you would be doing mentally. Take the coordinate axes shown below, point the z-axis out of the top of the creature and the y-axis out its front, then the x-axis will point to its right.
If you are familiar with vector operations in , consider this procedure. Take a vector pointing out its top and a vector pointing out its front . Then the cross product of and is a vector pointing to its right, . That is:
“Right equals front cross top“
The three directions, top, front, and right are mutually perpendicular and that if you know two of them, you know the third. For a person, a car, an animal, etc, the top and the front are unambiguous and intuitive. Then we use them to determine which side is the right side.
Now let us go back to the mirror. What does it really reverse? If you raise your arm, your reflection raises her arm. If you stick your right arm out to the side, the reflection sticks an arm out in that same direction. However, if you point at the mirror, then the reflection points in the opposite direction—she points back at you. In other words, the mirror reverses front and back!
Here’s where things start going wrong. Your brain does not have to do any work to recognize the top and front sides of your reflected image. Then it uses them to calculate your reflection’s right side. More specifically, your reflection’s z-axis points in the same direction as yours, but her y-axis points in the opposite direction (yours points into the mirror and hers points out). Consequently, if we use the xyz-coordinate system above, her x-axis points in the opposite direction as yours. Thus your right arm corresponds to her left arm, and we perceive the mirror reversing left and right.
Here another way to describe what is happening. The xyz-coordinate system shown above is often called a right-handed coordinate system because if you take your right hand, point your fingers in the x-direction and curl them in the y-direction, then your thumb will be pointing in the z-direction. What a mirror truly does is changes a right-handed coordinate system into a left-handed one—that is to say, the reflection of a right-handed system is a left-handed system. When we look into a mirror, our brain, which is accustomed to using a right-handed coordinate system to tell right from left, errs because the mirror world actually has a left-handed coordinate system.
When we see words in a mirror, they look like they are written from right to left, but that is because we are imposing our right-handed coordinate system on a left-handed mirror world.
I’ll end this post with some assorted thoughts about the left and the right.
- One thing that occurred to me while writing this post is that we treat different objects differently. Suppose I was holding a piece of paper out in front of me with my two hands and you were facing me. If I told you to point to the right hand side of the paper, then to point to my right hand, you would point to two opposite sides of the paper! There are certain objects (people, animals, cars, boats, etc.) in which right and left refer to the right and left sides from the objects’ perspective. However, there are other objects (pieces of paper, buildings, etc.) in which you use your right and left side to reference it. I assume this has to do with whether we can mentally substitute ourselves in place of the object—we can do that with other living things or with vehicles in which we can ride, but not inanimate objects like a piece of paper.
- Perspective is important. As a child, I was always confused about where right field was on a baseball diamond. Is it on the batter’s right or on the fielders’ right?
- It is no wonder that so many people confuse their right and their left. They have to compute a cross product in their head each time.
- It is a good thing that right and left are relative quantities, otherwise cars driving in opposite directions on a two-way road, both driving on the right-hand side would be in the same lane!
- I just came across these two articles about Andrew Hicks’ work with mirrors.