Posted by: Dave Richeson | January 24, 2009

## 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.

Right
17. a. = RIGHT HAND 2.

Right hand
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 $\mathbb{R}^3$, consider this procedure. Take a vector pointing out its top $\mathbf{v}_{\text{top}}$ and a vector pointing out its front $\mathbf{v}_{\text{front}}$. Then the cross product of $\mathbf{v}_{\text{front}}$ and $\mathbf{v}_{\text{top}}$ is a vector pointing to its right, $\mathbf{v}_{\text{right}}$. That is:

$\mathbf{v}_{\text{right}}=\mathbf{v}_{\text{front}}\times\mathbf{v}_{\text{top}}$.
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.

## Responses

1. Thanks for describing the mirror paradox; the reversal of the coordinate system is a very elegant description of the effect. It does not, however, explain the effect. We are still left with why does a mirror reverse the coordinate system. Optics can explain why but if I were teaching Physics now, I would use your reversal of cooordinates in defining the problem.

I wouldn’t be too hard on the OED. As Gödel demonstrated, Whitehead and Russell had the same problem with their magnificent Principia Mathematica. We all are complete in our own minds only.

• The above was great!

David Freeman is incorrect though.

Whereas Goedel was correct, that within a system, one can not prove its axioms/premises; it does not however stand to reason that “therefore” reality/everything is “in the mind”.

The correct way to express reality is “object-as-perceived” [Objectivism, the philosophy of Ayn Rand, the only complete and correct philosophy]

So this is incorrect: “reality is OUT there” [irrespective of the mind, subjectivism]
This is also incorrect: “reality is IN HERE” [all in the mind, intrincist]

If “reality” was merely “out there” then we can never know it, no one can, and all our assumptions would be mere assumptions, and not “the truth”, so 1+1 will not neccessarily equal 2.

If “reality” was merely “in here” then “anything goes”: any mad man like Hitler/Bin-Laden would be rational for the word “rational” would imply that whatever one thinks subjectively can or may be correct.

However as reality is “object-as-perceived” by you and THEN must be identified by you using the method of reason and logic, one can grasp “reality”, otherwise known as “facts”, distinguished from mere “opinion”.

1+1=2 , only 2 and nothing but 2. We can prove that.
———————–
Whereas Goedel said that one can not prove axioms/premises within a system; it doesn’t mean de facto that axioms therefore can not be validated, that reality can not be validated. This is a crucial distinction.

Firstly “reality” exists, otherwise better expressed as “existence exists”.

How do you “know” that existence exists? Because you validate it by your senses [even if you were blind, mute and dumb – which alas some people are by birth; still the person realizes that there ‘is’ existence; that they are not ‘some dream’].

To summarize:

i. Existence exsts as opposed to “no existence” and

ii. The purpose of your mind is to grasp existence [as a human to survive firstly and pursue happiness through growth secondly]

iii. The above two axiomatic concept imply Aristotle’s law of identity: that something is either right or wrong, true or false , with no in-betweens. That loigc is the way to identify the truth, and the truth is absolute, immutable.

Some more notes:

a. Humans are not omniscient and the purpose of our minds, is to grasp existence, to grasp “facts, the truth” and the way to do that is to use reason and logic, to integrate and differentiate concepts, to find the truth wherever possible. Summary: the lack of omniscience does not prevent us from grasping the facts of reality with the passage of time; by building upon concepts. Newton, for example grasped and labeled “gravity”

b. Einstein grasped errors in Newtons reasoning , using Newtonian physics as his referential base to extend the concept and label it “relativity”.

In summary: humans extend concepts by the process of integrating and differentiating other existing concepts of reality. Then one can “test” these ideas such as scientists conducting experiments.

Thanks!

2. The easiest way to explain it is as follows. Mirrors do not reverse left to right. We see ourselves in the mirror, and in our mind, we walk around behind the mirror to put ourselves in “his” (that is, our mirror image self’s) place. It is we who do the reversing as we mentally walk behind the mirror, trying to imagine “his” left and right.

So instead of using the big word “enantiomorph” for the observed effect, try the simpler word,

Empathy.

The ability to put ourselves in other people’s shoes. Love comes before logic!

We gotta remember that the math doesn’t drive the concept. The concept drives the math. That’s why Dirac’s delta function was accepted by physicsts before there was a legitimate mathematical notion of a distribution function to put it on a “sound” mathematical foundation.

3. […] In Ed Does Math on 26 January 09 at 11:05 am I loved this post at Division By Zero and recommend it here for all my math and physics students. Dave Richeson answers the question: […]

4. […] 27, 2009 by jedward706 I loved this post at Division By Zero and recommend it here for all my math and physics students. Dave Richeson answers the question: […]

5. Above I wrote some interesting comments. Below I will summarize the entire problem.

Answer [ii]: Mirrors do NOT reverse anything! It is the mind of man that perceives that the mirror is reversing things because s/he is looking at the mirror from ONE perspective; which is diffferent to the perspective from the mirror’s position!

Metaphysically [the nature of reality as it “is”] the mirror is not reversing anything as explained above;

Epistemological [how does a human get to “know” anything to be right/wrong, true/false, fact?]: the human must use their mind using logic and reason to determine facts!

The “facts” can be gleaned by understanding:
[a] perspective of the observer in contrast to the mirror
[b] vector math
——————————————–

6. Don’t forget when you see an object like a chair, light photons hit the object and then reflect directly back to your eyes; but when you see a reflection of an object then light photons hit the object and “then” the mirror and “then” reflect back to your eyes, a different situation between the earlier and the latter!

7. […] [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. […]

8. This doesn’t really explain why the mirror specifically reverses left-right and not up-down.

9. I recently saw the puzzle about left/right vs up/down reflection in mirror.

At first I thought it was because human eyes were laid out horizontally. But then when you close one eye, it does not change the way mirrors reflect.

However, let’s say we look at a mirror while lying down sideways on a bed. If your right ear is facing the ceiling, in the mirror, it will look like your right ear is facing the bed. Isn’t that up/down reversal?

So, the mirror doesn’t choose left/right or up/down. It is our perspective that decides that.

Eddy Chang.

10. Please allow me to offer the following:

A New Approach to an Old Problem

The mirror reversal puzzle is not so much a matter of the science of light rays, optics, and the Physics behind mirror reflection, but is far more a PEOPLE thing.

I’d say it can best be solved by a science of human activities, a science that deals with how people interact with complicated or subtle things. The structure of “reversedness‭” ‬in a human being’s mental imagery is far more subtle than the reversal of a photon’s trajectory.

When people want to compare any two nearly identical objects for some subtle difference there are two common strategies they can use. These two strategies are often useful, but they are also oddly contradictory.

***

In one strategy, the two objects are lined up to face in the same direction before they are compared. For instance, if two nearly identical pens are to be compared, no one I know of would ever hold one pen horizontally, the other vertically, and then proceed to compare them. People commonly want hold them facing in the same direction for such a task.

When a person uses this strategy in the mirror situation they like to imagine themselves rotating about a vertical axis for a comparison with their image. This rotation brings them to face the same direction as their earlier image was pointing. This also requires them to mentally freeze their image as it was when they faced the mirror. When ALL this is done, the Left/Right reversal is obvious.

It can get a little complicated, but many people seem to have the mental circuitry to perform all this in a near subconscious flash of imagery. Verbalizing it is far more difficult and hardly anyone can hold on to the images long enough to do that.

***

The second common human strategy for dealing with complicated situations, is to freeze EVERYTHING, avoid disturbing the scene, and look at the situation “as is” in order to perform an analysis of the subtle differences between two objects. This is the classic Sherlock Holmes approach to a crime scene.

In the mirror setup, performing this “as is” strategy means NOT rotating anything at all before doing the comparison between image and object, and the Left/Right reversal fails to show up. Instead, a Front/Back reversal is apparent in this “as is” comparison.

***

Of the two strategies, the second is FAR simpler. The first is almost too complicated to be done mentally, and it’s often impossible to document in any way. Yet, it can happen in a flash of mental imagery, only to fade as soon as words are brought in to capture it.

Often a person, deep in the throws of the mirror riddle, will drift from one strategy to the other subconsciously. This is often described as “magical” by those who delight in that sort of thing, while others will literally complain of the headache it causes them.

When each strategy is carefully thought through, ONE AT A TIME, clarity results.

***

It’s unlikely, but it might possibly occur to a gymnast to apply the first strategy by rotating about an horizontal axis, instead of the vertical, in order to face the same direction.

This means performing a hand stand to face in the same direction as the earlier frozen image, and then Up and Down would be seen to be reversed. But who is prone to do this difficult action or even to imagine it? And not only is it difficult, but it risks injury and even seven years of bad luck if the mirror is broken?

So, it’s not the mirror so much that does any reversing of perceived images, as it is people who do that by their selection of comparison strategies.

The three strategies mentioned “reverse” all three dimensions: Left/Right, Front/Back, and Up/Down. Mirrors reverse the direction of light rays, but people decide how they are going to compare image to object. It’s in this decision that the perceived reversal occurs.

***

I’ve been examining, collecting and analyzing answers to this riddle for many years. It’s been puzzling people for centuries. Even Nobel physicist Richard Feynman has tried his hand at it, and less than successfully, in my opinion. Recently I set up a website to discuss this subject at length.