When we face a mirror, why does it reverse left and right, but not up and down?
Like many argument-provoking chestnuts, this one conceals misdirection in its deceptively innocent wording. The book The New Ambidextrous Universe by author and recreational mathematician Martin Gardner addresses this mirror “paradox” and a range of phenomena from mathematics, physics, chemistry, and biology related to geometric reflection. This post merely introduces a few mathematical basics of reflection.
Reflections
Assume \(n\) is a positive integer. In \(n\)-space, there exist \(n\) mutually perpendicular lines but no more. Fixing a set of \(n\) mutually perpendicular lines through a point \(O\) and viewing each as a number line with its zero at \(O\) gives coordinate axes, and sets up a dictionary between locations in \(n\) space and ordered \(n\)-tuples of real numbers.
Each set of \((n – 1)\) coordinate axes determines a coordinate hyperplane in \(n\)-space. (In ordinary language, hyper connotes excess; for example, hyperspace often refers to \(4\)-space. Somewhat ironically, in geometry the term hyperplane has the specific meaning of one lower dimension. The same is true of hypersurface.)
To each hyperplane in \(n\)-space, there corresponds a unique reflection: A rigid motion of \(n\)-space that fixes each point of the hyperplane and multiplies the perpendicular coordinate by \(-1\).
In ordinary \(3\)-space, which can easily be mistaken for the universe we inhabit, a “hyperplane” may be viewed as a flat mirror. Introduce spatial coordinates \((x, y, z)\) whose \(x\)-and \(y\) axes lie in the plane of the mirror, and whose \(z\)-axis is perpendicular. For each point with coordinates \((x, y, z)\) on one side of the mirror, there is a reflected point with coordinates \((x, y, -z)\) lying on the opposite side of the mirror.
What about mirror images? On one hand, as it were, a left-handed glove is indistinguishable from its matching right-handed glove using only measurements of distance and angle. On the other hand, something about the gloves is different: Experience shows we cannot rotate one into the other. Mathematicians say matching gloves have opposite handedness. This gives a (pleasantly self-descriptive) name, but does not yet define a distinction.
Mathematically, \(n\)-space has a group of symmetries called motions. In a precise sense, this group has two pieces, or components. The identity component comprises proper motions, which are connectable by a one-parameter family to the identity transformation. The other component consists of improper motions, which are not connectable by a one-parameter family to the identity transformation.
Qualitatively, we can continuously transform an object to its image under a proper motion, but cannot continuously transform an object to its image under an improper motion. Or can we?
In (some fonts of) the Roman alphabet, the letters b and q can be rotated into each other. The same is not true of b and d: These letters have the same shape, but each is a reflection of the other. If, however, we write the letter b on a thin piece of paper with a marker that bleeds through and turn the paper over, we see the letter d on the other side.
Doesn’t this simple experiment contradict our claim about components? Happily it does not, though it does exemplify the care required when speaking about reflection. To make our b into d, we did not rotate the paper in the plane of the paper. By giving ourselves an extra dimension, we were able to effect a plane reflection by spatial rotation.
Generally, we can place \(n\)-space inside \((n+1)\)-space as a hyperplane. An improper motion of \(n\)-space can be effected as a proper motion of \((n+1)\)-space that also reflects the perpendicular direction to \(n\)-space in \((n+1)\)-space.
To paraphrase Gardner’s cogent resolution of the mirror chestnut, a mirror does not exchange left and right at all: It exchanges front and back. The appearance of exchanging left and right happens only if we follow this front-to-back reflection with a mental half-turn of our mirror reflection about a vertical axis, using bilateral symmetry of the human body to align ourselves with our rotated reflection.
Handedness
According to Gardner, the German philosopher Immanuel Kant believed for a time that handedness was an intrinsic property of \(3\)-space: Imagine a human form placed into \(3\)-space. Because the body is mirror-symmetric in its sagittal plane, it appears we have no way to label its hands as left and right. Now introduce a pair of gloves. One glove is the right-hand glove, and each glove fits only one hand; therefore the torso itself had a well-defined right-hand before the gloves were introduced.
Kant later (correctly) reversed this belief. You may enjoy pondering the error before reading further.
To use the framework of sets, we have three sets of two elements: the human body’s pair of hands, a pair of gloves, and the labels left-right. If a hand and glove are placed in \(3\)-space, geometry alone does determine whether or not they match. The problem is, geometry alone does not uniquely label the gloves: If we mirror-reflect the body in its plane of symmetry, its hands and gloves are swapped, so what was “right” becomes “left.” The assemblage of hands-and-gloves is invariant under reflection, however, so the labels themselves do not swap.
Algebraic Examples
The remainder of this post assumes more mathematical background than usual. If you have not seen vectors, determinants, and the cross product, earlier posts on high-dimensional geometry and symmetry provide some background. Additional material on vectors, matrices, and their algebra and geometry may be found in my book Linear Algebra, and in a post at math.stackexchange on orientation of a vector space.
The Cross Product
There are at least two ways to specify the handedness of three-space. Algebraically, we specify the ordering of the coordinates, or of a triple of basis vectors. Geometrically, we use the right-hand rule. These conventions are logically independent, a fact easily overlooked.
\(\DeclareMathOperator{\Rot}{Rot}\DeclareMathOperator{\Ref}{Ref}\newcommand{\Vector}[1]{\mathbf{#1}}\newcommand{\e}{\Vector{e}}\renewcommand{\u}{\Vector{u}}\renewcommand{\v}{\Vector{v}}\)Confusion about handedness tends to arise from a common pair of definitions for the cross product of two vectors. Algebraically, the cross product is defined by a formula involving determinants. Geometrically, the cross product \(\u \times \v\) is defined to be the unique vector whose magnitude is the area of the parallelogram spanned by \(\u\) and \(\v\); that is perpendicular to the plane spanned by \(\u\) and \(\v\); and such that \((\u, \v, \u \times \v)\) is a right-handed triple. These definitions are only equivalent if we use a right-handed coordinate system; otherwise they differ by a sign.
Complex Differentiability
Although not usually conveyed in these terms, the distinction of handedness is crucial to calculus over the complex numbers. To an algebraist, there are two complex square roots of \(-1\), indistinguishable by algebra. (Precisely, there is a field isomorphism that exchanges the two imaginary units.) By contrast, before we can define “complex-differentiability” we must fix one imaginary unit to be \(i\), leaving the other to be \(-i\). All of complex analysis is anchored to this choice. For example, complex conjugation is not complex-differentiable.
Motions and Matrices
Motions may be described algebraically using matrices. Particularly, a motion of \(n\)-space that fixes the origin \(O\) may be identified with multiplication by an \(n \times n\) orthogonal matrix, one whose columns are an orthonormal basis. An arbitrary motion of \(n\)-space may be viewed as a type of linear transformation of \((n+1)\)-space: Identify a point \(\v\) of \(n\)-space with \((\v, 1)\) in \((n+1)\)-space. If \(A\) is an \(n \times n\) orthogonal matrix and \(\v_{0}\) is a vector, then \[ \left[\begin{array}{@{}cc@{}} A & \v_{0} \\ 0 & 1 \\ \end{array}\right] \left[\begin{array}{@{}c@{}} \v \\ 1 \\ \end{array}\right] = \left[\begin{array}{@{}c@{}} A\v + \v_{0} \\ 1 \\ \end{array}\right]. \] In words, the motion \(T(\v) = A\v + \v_{0}\), encoded by the pair \((A, \v_{0})\), may be viewed as multiplication by a particular \((n+1) \times (n+1)\) block matrix built from \(A\) and \(\v_{0}\).
The set of all \(n \times n\) real matrices may be identified with \(n^{2}\)-space. The group of motions of \(n\)-space may therefore be viewed as a subset of \((n^{2} + n)\)-space. In principle, at least, the group of motions is a concrete geometric set, which we might hope to visualize.
(Incidentally, the two components of the group of motions are themselves congruent in \((n^{2} + n)\)-space: If \(\rho\) is an arbitrary reflection of \(n\)-space, then for each motion \(T\), pre-composition \(\rho \circ T\) (first do \(T\), then do \(\rho\)) defines a motion of the mapping space that exchanges the sets of proper and improper motions. Despite this symmetry of point sets, and unlike the situation with left and right gloves, the group structure singles out the proper motions: The identity mapping is a proper motion.)
One obstacle to visualization is, the dimension \(n^{2} + n = n(n + 1)\) grows quadratically with \(n\). The group of motions of the number line (\(n = 1\)) sits inside \(2\)-space, an ordinary plane. By contrast, the group of motions of the plane (\(n = 2\)) sits inside \(6\)-space, or if we neglect translations, inside \(4\)-space. To do the same for rotations of \(3\)-space using matrices, we must consider subsets of \(9\)-space (since \(9 = 3^{2}\)). Let’s see what we can see.
Assume \(n = 1\), so our “space” is a number line. To reiterate our dimension-counting, a \(1 \times 1\) matrix amounts to a real number \(A\), and the translation vector \(\v_{0}\) is a real number. The assemblage may be viewed as an ordinary plane with rectangular coordinates \((A, \v_{0})\). Further, an orthogonal matrix is either the identity transformation \([1]\) or the reflection \([-1]\). The group of motions of a number line consists, therefore, of two vertical lines: Points of the form \((1, \v_{0})\) comprise the proper motions, while points of the form \((-1, \v_{0})\) comprise the improper motions. As claimed, the group of motions has two components, and these components are congruent in \((n^{2} + n)\)-space.
Next assume \(n = 2\). For simplicity let’s ignore translations, considering only rotations and reflections. Assume \(A\) is an orthogonal \(2 \times 2\) matrix. The images of the standard basis, namely the columns of \(A\), form an orthonormal basis of the plane. Since \(A\e_{1}\) is a unit vector, there exists a real \(\theta\) such that \[ A\e_{1} = \left[\begin{array}{@{}c@{}} \cos\theta \\ \sin\theta \\ \end{array} \right]. \] Since \(A\e_{2}\) is a unit vector orthogonal to \(A\e_{1}\), we have either \[ A\e_{2} = \left[\begin{array}{@{}r@{}} -\sin\theta \\ \cos\theta \\ \end{array} \right] \] or \[ A\e_{2} = \left[\begin{array}{@{}r@{}} \sin\theta \\ -\cos\theta \\ \end{array} \right]. \]
With the first choice, \(A\) rotates each standard basis vector, and therefore the entire plane, about the origin by angle \(\theta\); put \[ \Rot_{\theta} := \left[\begin{array}{@{}rr} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \\ \end{array} \right]. \] The identity transformation is rotation by \(0\) (or, for each integer \(k\), by angle \(2\pi k\)).
With the second choice, \(A\) reflects the plane across the line with polar angle \(\theta/2\); put \[ \Ref_{\theta/2} := \left[\begin{array}{@{}rr} \cos\theta & \sin\theta \\ \sin\theta & -\cos\theta \\ \end{array} \right]. \]
Every linear motion of the plane has one of the preceding forms, rotation or reflection. A rotation is uniquely determined by its first column. The set of first columns is the unit circle in the plane. Consequently, the set of plane rotations is abstractly a circle. The same is true of reflections. In the group of plane rotations, each component is abstractly a circle.
These abstract circles are subsets of \(4\)-space, where they are honest circles: Plane curves lying at fixed distance from a point. Further, they can be visualized in \(3\)-space with the help of (our old friend) stereographic projection: Each orthogonal \(2 \times 2\) matrix has two orthogonal unit vectors as columns, so has magnitude \(\sqrt{1^{2} + 1^{2}} = \sqrt{2}\) in \(4\)-space. That is, the set \(O(2)\) of orthogonal \(2 \times 2\) real matrices is a pair of great circles in the sphere of radius \(\sqrt{2}\). In the language of the blog post on conformal torus knots, each circle comes from a line of slope \(1\).
Alternatively, if \(z\) denotes a unit complex number, the identity component of \(O(2)\) is identified with the set of points \((z, iz)\), while the non-identity component is the set of points \((z, -iz)\). Under stereographic projection to \(3\)-space, these circles in \(4\)-space map to a pair of linked circles. A half-turn about a suitable axis exchanges the circles.
With satisfying concreteness, our abstract claims are realized in each example we could visualize: The group of motions has two components. One component is distinguished by containing the identity transformation. Each proper motion can be joined to the identity by a continuous path. No improper motion can be joined to the identity by a continuous path. As sets of points the components are congruent.
Describing motions in detail if \(n \geq 3\) is beyond the scope of this post, but one fact should be emphasized: If \(n \geq 3\), an improper motion fixing the origin is usually not a reflection. In \(3\)-space, for example, reflection in a plane followed by rotation in that plane is an improper motion, but is a reflection if and only if the rotation angle is zero.
Further Reading
Gardner’s The New Ambidextrous Universe is an excellent read, as are The Road to Reality by Roger Penrose and Origin of Life by David Deamer.