Since this blog is supposed to have a diverse audience, I better provide mathematical background for theoretical stuff as it comes up. Crystallographers are about as diverse a crowd as one can imagine: they range from chemists uncomfortable with vector algebra to physicists familiar with all sorts of computations and occasionally theory to mathematicians who insist on proving stuff before they use it – and even then, we like to generalize things to their most abstract forms.
We just have to be patient with each other. I will presume that readers are at least as familiar with crystallographical nomenclature as I am (since I am relatively ignorant, this shouldn’t be too onerous on the readers) (corrections and comments are welcome); since students may visit, I will occasionally be pedantic. I will put references in these background posts; if you know of good references, feel free to pass them along.
Since future posts are going to presume some knowledge of group theory, perhaps I should say some things about groups. Actually, this is a sort of roadmap to some of the slides presented at the MaThCryst Workshop on Mathematical Crystallography in Banaras, India, last October. I’ll post some other resources as well.
But first, since history and context are important, I’ll start with a brief account of where groups came from. One of the Renaissance obsessions was finding roots of polynomials. The Mesopotamians had figured out how to factor quadratic polynomials, but it wasn’t until the Renaissance that Niccolo Tartaglia figured out how to factor the cubic ax3 + bx2 + cx + d, and shortly after that Lodovico Ferrari figured out how to factor the quartic ax4 + bx3 + cx2 + dx + e. That was 1545, and mathematicians promptly went after the pentic, ax5 + bx4 + cx3 + dx2 + ex + f. No luck. After two centuries of little progress, mathematicians began to just poke at the pentic, trying to figure out what was going on.
Leonard Euler got the idea of classifying the roots of a polynomial by permuting them. This kind of approach helped lead to Paolo Ruffini‘s incomplete proof that no general formula for the pentic exists, and then Neils Abel‘s complete proof. Then Evariste Galois classified polynomials by what kinds of roots they had – specifically, by whether or not a given group has roots one can specify by a finite expression. In order to do this, he invented something he called a group, which consisted of Euler’s permutations in a single package.
Here is a standard example, and in the interests of orneriness, here is a non-crystallographic example that has long bedeviled crystallographers. Consider the pentagon below. What (rigid) motions can we apply to this pentagon so that it’s in an equivalent position (i.e. looks the same)? One possibility is to rotate it about its center by 0, 72, 144, 216, or 288 degrees counterclockwise:
Before going on I should mention that the composition of doing two actions consists of doing one and then the other. So the composition of turning right 90° and then marching ten feet forwards consists of turning right and then marching. Notice that the order of the actions is important: if you start by facing north and then turning right and marching ten feet, you will wind up with a net displacement of ten feet east of where you started. But if you first march ten feet and then turn right, your net displacement will be ten feet north of where you started:
Returning to the pentagon, from the picture above, this cyclic group of five rotations satisfies all the properties of being a group:
- Closure. Each of these rotations is a multiple of a 72°, and the composition of two rotations by multiples of 72° is itself a rotation by a multiple of 72°. Notice that there is no problem going past 360°, for a rotation by 360° is effectively a rotation by 0°.
- Associativity. For any rotations θ°, φ°, and ψ°, all three of them being multiples of 72°, the following two motions have the same result:
- Rotate by θ°, and then by the composition of φ° and ψ°. You could write this down as Rotθ° º (Rotφ° º Rotψ°), where “º” stands for composition. (composition is usually denoted by “º” in math books, although some books just concatenate function symbols).
- Rotate by Rotθ° º Rotφ°, and then by ψ°. You could write this down as (Rotθ° º Rotφ°) º Rotψ°.
You have already seen associativity in arithmetic: for any numbers x, y, and z, (x + y) + z = x + (y + z): addition is associative. On the other hand, there are many numbers x, y, and z such that (x – y) – z ≠ x – (y – z): subtraction is not associative. Similarly, multiplication is associative while division is not.
- Identity. There is a rotation which, when composed with any other rotation, produces just that other rotation. Compose any rotation θ ° by 0° and the result is just θ. We say that the rotation by 0° is the identity of the group.
- Inverse. For every rotation by a multiple of 72°, there is another rotation that undoes the effect of the first so that when you compose the two, you get the identity. 288° is the inverse of 72°, 216° is the inverse of 144°, and 0° is its own inverse. (Remember that we treat 360° and 0° as the same thing.) The inverse of the inverse is the original motion. We often write θ-1 for the inverse of θ, so we have θ º θ-1 = θ-1 º θ = e (mathematicians often use “e” for the identity – don’t ask me why).
Any collection with the properties of closure, associativity, identity, and inverse is a group. In fact, we can have an “abstract” group that just consists of symbols, provided that it obeys the rules. Consider William Hamilton’s granddaddy group in physics: the quaternions. This group has eight elements: 1, -1, i, –i, j, –j, k, and –k, and a binary operator ⋅. The entire multiplication table for this group can be worked out from Hamilton’s equations i ⋅ i = j ⋅ j = k ⋅ k = i ⋅ j ⋅ k = -1(where i ⋅ j ⋅ k is (i ⋅ j) ⋅ k = i ⋅ (j ⋅ k)) and –x = -1 ⋅ x = x ⋅ -1 for each x (and, of course, -1 ⋅ -1 = 1). For example, j ⋅ i = j ⋅ i ⋅ -1 ⋅ -1 = j ⋅ i ⋅ i ⋅ j ⋅ k ⋅ -1 = j ⋅ -1 ⋅ j ⋅ k ⋅ -1 = -1 ⋅ j ⋅ j ⋅ k ⋅ -1 = -1 ⋅ -1 ⋅ k ⋅ -1 = -k. The quaternions started as a sort of poor man’s vector algebra but they are now popular in computer graphics.
Of course, there are other symmetries around. Suppose we looked at a more typical pentagon:
Suppose that the pentagon’s backside is the same from the front, and suppose that we started with two axes around which we could flip the pentagon:
These two axes are 36° apart, and it turns out that if you flip the pentagon along one axis and then the other, the resulting composition is a rotation by 72°:
Composing all these flips and rotations, you obtain the dihedral group of parameter five:
These flip axes are usually regarded as mirrors across which one reflects the pentagon. (We can imagine that when reflected / flipped, the pentagon changed color; when reflected / flipped again, it’s color changed back.) Such a flip is called a reflection. The dihedral group of parameter n consists of n reflections, all across mirrors through a common point angled 180/n degrees apart, and n rotations, all by multiples of 360/n degrees. This group of 2n operations is the group of symmetries of the regular n-gon: a symmetry of a structure is an operation that preserves the structure.
We can look at symmetries of infinite structures. For example, consider this two-dimensional image of a wave function:
Considered as a two-dimensional pattern, the group of rigid actions that move the pattern onto itself are generated by:
- the translation (x, y) → (x + 2π, y),
- the reflection across the line x = 0, and
- the reflection across the line x = y.
Here are these three actions on a contour map of the wave function:
From these three rigid actions, we get an infinite group of actions:
- translations of the form (x, y) → (x + 2kπ, y + 2lπ), where k and l are integers,
- reflections across the lines x = 2kπ and y = 2lπ, where k and l are integers (these lines are the mirrors),
- reflections across the lines x = y + 2kπ and x = –y + 2lπ, where k and l are integers,
- and rotations by 90o, 180o, or 270o degrees counterclockwise about the points (2kπ, 2lπ), where k and l are integers.
We get these actions by composing actions, e.g., by reflecting first across x = 0 and then reflecting across x = y, the result is the same as rotating by 90o counterclockwise about the origin. The group is the set of these actions together with the composition operator.
Crystallographers typically but inaccurately look down at, say, the unit cell bounded by the four points (- π, – π), (- π, π), (π, – π), and (π, π), and say that this is the wallpaper group P4mm; we will say a little later about what the more precise situation is.
For a more extensive but informal introduction to groups of actions, see the first few pages of Nathan Carter’s Visual Group Theory.
To be continued…