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.