Thursday, 25 June 2009

Ask Squark: Mirror

This is a part of the Ask Squark series.

Thx Assaf for submitting the first question to “Ask Squark”!

Question:

Why does the mirror reverse left and right but not up and down?

Answer:

Firstly, there is a hidden assumption here that the mirror is hanged conventionally, i.e. vertically. A horizontal mirror (like in Hotel California) does reverse up and down.

It might seem that a vertical mirror displays some sort of asymmetry: left and right (which are perceived as horizontal directions) are reversed, whereas up and down (the vertical directions) are not reversed. However, let me assure you that there is perfect rotational symmetry with respect to the axis orthogonal to the mirror plane. The apparent paradox is mostly semantic.

Let us remember how an ideal planar mirror works. The real mirrors are not ideal (alas), but it’s irrelevant for the discussion.

Suppose the mirror lies in the plane M. Consider a point P in space. The mirror image of P is the point P’ satisfying the following criteria:

  • The line PP’ is orthogonal to M
  • The distance of P to M is equal to the distance of P’ to M

In reality, the mirror is usually reflective from one side only, hence we have to assume P lies in one of the two half-spaces defined by M. However, this is inessential.

A direction in space can be specified by two points: the beginning and the end of a vector. Thus, given a direction v = PQ, the mirror image direction is v’ = P’Q’ where P’ is the mirror image of P and Q’ is the mirror image of Q. It is easy to see that when v is parallel to M, v’ = v but when v is orthogonal to M, v’ = –v.* For example, if M is the plane spanned by the up-down and north-south directions, the mirror preserves up, down, north and south but reverses east and west.

The illustration is by the courtesy of SurDin.

But what about left and right? These notions are more complicated. While north, south, east, west, up and down are absolute directions, left and right are relative. For instance, if you face another person, her right is your left and vice versa.

Mathematically, we can describe left and right as follows. Consider ordered triples of mutually orthogonal vectors of unit length
(u, v, w). It can be shown that such triples fall into two classes R and L such that

  • A triple in R can be rotated into any other triple in R.
  • A triple in L can be rotated into any other triple in L.
  • A triple in R cannot be rotated into any triple in L.
  • A triple in L cannot be rotated into any triple in R.

Suppose a Cartesian coordinate system S is given. Then any triple (u, v, w) can be decomposed into components with respect to S and represented by a 3 x 3 matrix. Then, for some triples the determinant of this matrix is 1 whereas for others it is –1. These are R and L.

We can derive the following interesting property of R and L. Suppose (u, v, w) is a triple in R (L). Then (-u, v, w), (u, –v, w) and
(u, v, –w) are triples in L (R). It follows that (-u, –v, w), (-u, v, –w) and (u, –v, –w) are triples in R (L). Also, (-u, –v, –w) is a triple in L (R). This property follows, for instance, from the fact that the sign of the determinant is reversed when we reverse the sign of one of the matrix columns.

The relation to the familiar notions of right and left is as follows. Consider X is a person and (u, v, w) a triple as above. Suppose u is a vector pointing in the direction from X’s legs to her head. Suppose v is a vector pointing in the direction from X’s back to her front. Then w points either to X’s right or to X’s left, according to the class to which (u, v, w) belongs!

Consider (u, v, w) a triple as above. Consider (u’, v’, w’) the mirror image triple. That is, u’ is the mirror image of u, v’ is the mirror image of v and w’ is the mirror image of w. Then, if (u, v, w) is in R then (u’, v’, w’) is in L and vice versa. In this sense, the mirror reverses left and right. For instance, suppose u and v are parallel to M. Then w is orthogonal to M. We get u’ = u, v’ = v, w’ = –w. Thus (u’, v’, w’) = (u, v, –w) belongs to the class opposite to that of
(u, v, w). The general case can be proven e.g. using determinants: if we choose S such that M is parallel to two of the axes, mirror imaging amounts to changing the sign of one of the matrix rows.

The illustration is by the courtesy of SurDin.

* Remember that two vectors v = AB and w = CD are considered equal when

  • The line AB is parallel to the line CD (or A = B and C = D which means both vectors are the equal to the 0 vector).
  • The line AC is parallel to the line BD (or A = C and B = D which means both vectors coincide in the trivial sense).

Also, remember that, by definition, if v = AB then –v = BA.

9 comments:

Assaf said...

Thanks for tackling the issue.

I do raise two objections, though:

1. My brain hasn't hurt this much since 1st year Math at TAU. I asked for a simple intuitive explanation! :) I guess I should have defined "intuitive" more precisely.
2. I get left/right hand coordinate systems (thank god for that Computer Graphics course...). I just don't see how this explains why the mirror doesn't reverse up and down.

SurDin said...

It doesn't reverse your hands either, but it reverses back and front. So, because your back and front changed, you must change the names of your hands to fit the coordinates system's orientation.

Squark said...

0. SurDin kindly provided illustrations that I added into the post. The border looks crappy because the crappy editor of blogger wouldn't let me configure it.

1. They say that if you go to the gym after a long time without exercise and your muscles hurt, it's a good thing... ;)

2. The key point is that up/down are absolute directions, whereas left/right are relative. This is the important difference, not vertical vs. horizontal. To see this, suppose you lie down on your side while looking at the mirror. Left/right are still reversed even though they becomes vertical. On the other hand, if the plane of your mirror is up-down-north-south, the directions north/south aren't reversed even though they are horizontal.

Eli said...

Dear Squark,

While we're on the topic of mirrors, I'd like to ask you something that has been bothering me for some time now. Why do horses (note that donkeys, mules and other equines are not affected by this) tend to appear in mirrors like a certain variety of the sea sponge (the Acarnus Erithacus, to be precise), while the same species of sea sponge tends to appear in mirrors like form #106 of the Israeli IRS?

And on a completely unrelated subject - what can you tell me about the biochemical processes behind the psychoactive effects of the Fly Agaric? The Wikipedia article is long and technical, which is obviously a problem (words crawling/flying away etc.)

P.S.

Have you checked out The Straight Dope?

And if so, what did you think about the Schrodinger's cat poem?

Squark said...

Dear Eli,

If you want to submit a question about Amanita Muscaria to "Ask Squark", e-mail me, although the subject is not very interesting for me (I'm mostly interested in unicellular organisms and natural processes), so I'll only consider to accept very specific questions.

Eli said...
This comment has been removed by the author.
Eli said...

A.. ha

I would say "whoosh" but I'm not sure that's even true

Anonymous said...

Here is the intuitive explanation (it seems you somewhat missed the point Mr.squark):
The reason for the mirror not inversing up and down has to do with the positioning of the mirror relative to the person looking at it - for example if the person was to stand on the mirror it WOULD actually inverse up and down.
All the mirror does is change the direction (+ to - or vise versa) of the axis perpendicular to the mirror plane

Squark said...

Please be kind enough to read the text you are commenting on. In the first paragraph of my answer I explain that a horizontal mirror reverses up and down. In the paragraph above the illustration I explain that the mirror preserves directions parallel to it but reverses directions orthogonal to it.