During the Twentieth century, mathematicians got in the habit of studying structures by looking at the functions that mapped one structure to another or to itself. Probably the most famous was in topology, where simple objects were made of playdough. Two playdough objects (e.g., a teacup and a donut) were equivalent if one could be deformed into the other without breaking, tearing, or fusing or welding: transform the playdough donut into a teacup by making and expanding a depression that grows into the teacup bowl – making a donut out of the teacup involves reversing the process. But the donut cannot be deformed into a saucer without welding the hole shut, and a saucer cannot be deformed into a donut without tearing open a hole. The notion of genus, how many holes a playdough object had, depended on the transformations available to deform one object into another.

We study groups the same way, looking at transformations from one group to another. Suppose we have two groups, G = (G, ∘) and H = (H, ∗); a homomorphism from G to H is a function f : GH such that for any g1, g2G, f(g1 ∘ g2) = f(g1) ∗ f(g2). Observe that if e is the identity of G, then f(e) is the identity of H, and for any g ∈ G, f(g-1) = f(g)-1.

For example, Let G be the symmetry group of the pentagon illustrated in the 30 January 2015 group theory post. Let Z2 be the group ({0, 1}, ∗) with the following multiplication table:

For each rotation rG, let f(r) = 0, and for each reflection m ∈ G, let f(m) = 1. Thus f outputs 0 if the isometry is direct (i.e. no mirrors) and 1 if it is indirect. This is an example of a homomorphism: recall that while the composition of two direct isometries (or the composition of two indirect isometries: the composition of two reflections across non-parallel mirrors is a rotation) is direct while the composition of a direct isometry and an indirect one is indirect.

One special kind of homomorphism from G to H determines a copy of G inside of H. Say that any kind of function f from G to H is one-to-one (or injective) if, for any g1, g2G, if g1 ≠ g2 then f(g1) ≠ f(g2). For example, the function f1(x) = x3 from the real numbers to the real numbers is one-to-one, while the function f2(x) = x2 is not (as -1 ≠ 1 while f2(-1) = f2(1)). A one-to-one homomorphism is called an embedding (or monomorphism). For example, take the group of integers mod 5, call it Z5, and for each i ∈ {0, 1, 2, 3, 4}, let f(i) be the rotation of the pentagon by 72i degrees counterclockwise. Then f is an embedding from Z5 to the rotation group of the pentagon, and we can say that f embeds Z5 in the symmetry group of the pentagon.

Another special kind of homomorphism from G to H maps G onto all of H. Say that any kind of function f from G to H is onto (or surjective) if, for any h ∈ H, there is a g ∈ G such that f(g) = h. For example, the function f1(x) = tan x maps the reals onto the reals, but the function f2(x) = sin x does not. An onto homomorphism is called an epimorphism. Looking at the previous paragraphs, our mapping of the symmetry group of the pentagon onto Z2 was an epimorphism, while our embedding of Z5 into that symmetry group was not.

If we put these notions together we get: a function f from G to H is bijective if it is one-to-one and onto. A bijective homomorphism is called an isomorphism. For example, the map from Z5 onto the rotation group of the pentagon is an isomorphism defined by f(i) = the rotation by 72i degrees is an isomorphism. An isomorphism is often regarded as a relabeling of the names of group elements. For example, Z5 is a group of integers and the (noncrystallographic) point group 5 of rotations by multiples of 72 ° are not the same thing – one consists of integers and the other of rotations – but there is an isomorphism from one to the other, so we call them isomorphism and often treat them as if they were the same.

One special kind of isomorphism is an isomorphism from a group onto itself: such an isomorphism is called an automorphism. For example, if we mapped the rotation group of the pentagon onto itself by mapping a rotation by 72i degrees to a rotation by 144i degrees, that is an automorphism of that group.

Now that we have some functions from groups to groups, let’s briefly look at groups that are part of bigger groups, much as the rotation group of the pentagon is part of the symmetry group of the pentagon. If all the elements of a group G are elements of a group H, and if they have the same binary operator, we say that G is a subgroup of H, and that H is a supergroup of G. There are two popular ways of specifying a subgroup. In both, we have a group H and a set X of elements of H, and we want the subgroup of H that is “generated” by X. For example, given the symmetry group of the pentagon, the subgroup generated by the rotation by 72 ° is the rotation group of the pentagon, while the subgroup generated by one of the reflections is the group consisting of that reflection and the identity. (And the subgroup generated by any two reflections is the entire group.) What do we mean by “generated by”?

This is the kind of question a mathematician would ask, and here are two answers. First, the answer in all the mathematics texts. Let H be a group. It is a fact (see if you can verify it) that any intersection of subgroups of H is also a subgroup of H. Then one answer is to say that the subgroup generated by X is the intersection of all subgroups of H that contain all the elements of X; this subgroup is often denoted 〈 X 〉. This answer is nice and neat, which is why the texts use it, but it doesn’t tell us how to compute 〈 X 〉.

An alternative is the following recipe for constructing the subgroup generated by X. Let G0 = X. For any nonnegative integer n, given Gn, let Gn+1 = {g, g ∘ h-1 : g, h ∈ Gn}. We get G0, G1, G2, G3, … . If H is finite, this sequence will eventually stop at some n where Gn = Gn+1, and this is (the set of elements of) our desired subgroup generated by X. If H is infinite, the sequence of sets Gn can keep growing forever, and it is their union G that is the subgroup generated by X.

For example, given the symmetry group of the pentagon, and letting X be the set containing the rotation by 72 °, we have G0 = {Rot72 °}, G1 = {Rot0 °, Rot72 °, Rot144 °}, and G2 is the rotation group of the pentagon.

As an example, let’s look at a kind of subgroup that shows up a lot in crystallography. Let G be a group of bijective functions on a set or structure X. For any x ∈ X, let G(x) = {g(x) : g ∈ G}: this is the set of points that x is mapped to by the functions in G. For example, consider the infinite dihedral group (also known as pm) which acts on the number line and consists of two kinds of functions:

• Reflections across integers. For each integer i, let Refi be the function Refi(x) = 2ix. This reflects the number line across the integer i.
• Translations by integers. For each integer i, let Ti be the function Ti (x) = x + i.

Suppose that we had a real number x and we used the infinite dihedral group to move it around. If x was on an orange dot between i and i + 1/2 for some integer i, x would be translated to the left and right by increments of 1 onto something like the orange dots in this picture (where the blue mirrors stand at the integers).

Meanwhile, a mirror would reflect x to a real number between j – 1/2 and j for some integer j, and hence onto a green dot. In this picture, the orange dots are the images of x via translations while the green dots are images of x via compositions of reflection and translation.

If we started with x was between i – 1/2 and i, we would get the same picture, only now the green dots would be the images of x under translations while the orange dots would be the images under reflections + translations. Either way, the green and orange dots together make up the orbit of x.

But if x was an integer, or if x = i + 1/2 for some integer i, then things are a little different. If x was an integer, the translations and reflections map it to other integers, and the orbit of x is just the integers. Or if x = i + 1/2, then the translations and reflections map x to other half-way points, and the orbit of x is {j + 1/2 : j an integer}.

Mirrors and other landmarks appear in crystallography, so one note about these. Let G be a group acting on X, and suppose that the only function in G that maps x to itself is the identity. We then say that G acts freely on x. For example, for all x except the integers and the half-way points i + 1/2, i integer, the infinite dihedral group acts freely on x. On the other hand, suppose that several functions in G map x to itself. For example, in the infinite dihedral group, the reflection across the integer i maps i to itself. The stabilizer of x (which we can denote Stab(G, x)) is the subgroup of G that maps x to itself.

One important kind of subgroup comes from homomorphisms. Let f : GH be a homomorphism from G to H. The kernel of f is the subgroup ker f = {g ∈ G : f(g) = eH}, where eH is the identity of H. Ker f is a subgroup of G, and it has an important property. Call a subgroup N of G normal if, for every g ∈ G and every n ∈ N, g ∘ n ∘ g-1N. This is a peculiar condition, but as we will see in a moment, a useful one.

Suppose that N is a subgroup of G. For any g ∈ G, the left coset of N by g is the set g ∘ N = { g ∘ n : n ∈ N } while the right coset of N by g is the set N ∘ g = { n ∘ g : n ∈ N }. The left cosets form a partition of G, as do the right cosets, and N is a normal subgroup of G if and only if the left cosets form the same partition of G that the right cosets do. The reason is that N is normal in G if and only if g ∘ N ∘ g-1 = { g ∘ n g-1 : n ∈ N } = N, so that multiplying both sides of g ∘ N ∘ g-1 = N by g – on the right hand side of each – produces g ∘ N = N ∘ g.

For example, consider the point group 4m, which consists of four mirrors passing through the origin. The four rotations of 4m (counting the identity as a rotation by zero degrees) form a subgroup 4. And 4 is a normal subgroup of 4m: if r was a rotation (by a multiple of 90°), then for any R in 4m:

• If R was a rotation, then RrR = r is a rotation, and
• If R was a reflection, then RrR = r is a rotation.

On the other hand, consider the wallpaper group p4, with the unit square being a unit cell, and hence 90° rotation centers at all points (i, j), i, j integers. The stabilizer Stab(p4, (0, 0)) is not normal. To see this, let r be the 90° rotation about (1, 0), and we claim that r ° Stab(p4, (0, 0)) ° r-1 ≠ Stab(p4, (0, 0)). Let R be the rotation by 90° rotation about (0, 0), and it suffices to show that r ° R ° r-1 ∉ Stab(p4, (0, 0)). If it was, then r ° R ° r-1(0, 0) would be (0, 0); in fact, r ° R ° r-1(0, 0) = r ° R (1, 1) = r (-1, 1) = (0, -2).

Returning to homomorphisms, a subgroup is normal if and only if it is a kernel of a homomorphism. First of all, given a kernel of a homomorphism f : GH, call it ker f, for any g ∈ G, g ° ker f ° g-1 = ker f. To see this, take any k ∈ ker f, and observe that f(g ° ker f ° g-1) = f(g) * f(k) * f(g-1) = f(g) * f(g-1) (as f(k) is the identity in H) = f(g) * f(g)-1, which is the identity in H.

Secondly, given a normal subgroup N of G, let G/N be the set of left cosets of N in G: G/N = { g ∘ N : g ∈ G }. If N is normal, G/N is a group with binary operator ° as follows: (g ° N) ° (h ° N) = ((g ° (N ° h)) ° N) = ((g ° (h ° N)) ° N) = (g ° h) ° (N ° N) = (g ° h) ° N as N ° N = { n1 ° n2 : n1, n2N } = N. As N is the identity of G/N, and N is the set of elements of G that f maps to N, N = ker f.

G/N is called the quotient group of G and N. As we shall see, the point groups are quotient groups of the space groups.

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.

It was Augustin Cauchy who launched the magic phrase, a group of permutations, which morphed into the groups of actions that crystallographers deal with daily.

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 (xy) – zx – (yz): 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 ii = jj = kk = ijk = -1(where ijk is (ij) ⋅ k = i ⋅ (jk)) and –x = -1 ⋅ x = x ⋅ -1 for each x (and, of course, -1 ⋅ -1 = 1). For example, ji = ji ⋅ -1 ⋅ -1 = jiijk ⋅ -1 = j ⋅ -1 ⋅ jk ⋅ -1 = -1 ⋅ jjk ⋅ -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…

SIAM’s special interest group on Mathematical Aspects of Materials Science has successfully crashed this year’s fall meeting of the Mathematical Research Society with a Symposium NN on Mathematical and Computational Aspects of Materials Science. Symposium NN will run from December 1 to December 4. For more information on this year’s meeting in Boston, see the fall meeting webpage.

As you may have noticed, a comment on the previous post inspired me to adjust the post. (Actually, it raised an issue that will deserve a post of its own.)

Since I would like this blog to be useful to a wide audience, I appreciate comments, corrections, suggestions, criticisms, etc., either posted on this blog or emailed to me. After all, I should be described (to paraphrase Jane Austen) as a partial, prejudiced & ignorant crystallographer, and I would like to keep posts as correct as possible.

Unfortunately, most comments entered come from spambots, programs that surf the web, looking for sites like this one to post ads for X-rated products, post links in order to inflate their own page rank, etc. In order to prevent this blog from being weighed down by this kind of nonsense, comments are moderated. That means that a comment is entered, it sits in a “pending” folder until I get to it and approve it.

I try to check the pending folder daily, but if you posted a comment and it doesn’t appear within a day or so, feel free to email to me. (Feel free to email me about this site in any case: I appreciate the feedback.)

I apologize for the inconvenience, but I think that this is the least inconvenient system for the situation we have.

Let’s start with crystal structure prediction. Suppose that you would like a crystal with various properties (it’s purple, porous with channels two nanometers wide, nonconductive, etc.). Traditionally, finding such a crystal would involve synthesizing many novel chemicals and annealing them, and hopefully a sequence of increasingly successful combinations would ultimately lead to success. This was the method that the alchemists employed in their pursuit of the Philosopher’s Stone, and whose modern, automated incarnation is called Combinatorial Chemistry. It may have been good enough for Paracelsus and Robert Boyle, but it is expensive and frustrating.

More to the point, this is not what engineers and architects do. In construction and industry, someone composes a set of blueprints specifying the final product and (hopefully) intermediate steps, and then someone (often someone else) uses those blueprints to construct the desired product – which, no coincidence, satisfies the original specifications. Chemists should be able to do that.

The general project is crystal engineering, which Wikipedia defines as “the design and synthesis of molecular solid-state structures with desired properties, based on an understanding and exploitation of intermolecular interactions.” Wikipedia says that the oldest reference to the term is G. M. J. Schmidt, Photodimerization in the Solid State in Pure & Applied Chemistry 27:4 (1971), pp. 647 – 678, but Web of Science says oldest reference they have is an abstract for a paper presented to the American Physical Society meeting in Mexico City, Mexico, August, 1955 by R. Pepinsky entitled Crystal Engineering – A New Concept in Crystallography, abstract published (but not posted!) in the Physical Review 100:3 (1955), p. 971. There being nothing new under the sun, the notion probably has been floating around since the beginning of the 20th century.

In the last few decades, crystal engineering has become of field of its own: on October 23, Web of Science listed 2,520 hits from all databases, and now the subject has its own textbook: Crystal Engineeering : A Textbook by Gautam Desiraju, Jagadese Vittal, and Arunachalam Ramanan.

The growth of the field is a little more problematic. Here are the 2,520 hits in the Web of Science by date:

Notice that after Pepinsky’s 1955 talk, Web of Science has no hits until 1976 (Schmidt’s 1971 paper did not have “crystal engineering” in any of the fields Web of Science checks – a warning to people who rely on such databases), and things really didn’t get going until 1991. Most of the growth was during the 1990s: a linear regression shows that publication growth during 2001 to 2013 has been nearly linear, an average additional 3.3 papers a year, or less than 3 % growth per year. Growth in citations looks more exponential, which shows that crystal engineering has been gaining a higher profile lately, but the number of citations has been flattening out during the last few years:

Let’s take a closer look.

In 1988, John Maddox wrote an op-ed in Nature:

 A new calculation of the polymorphs of silica appears to have broken new ground in deriving crystal structure from chemical composition. But X-ray crystallographers need not worry — yet.

and grumped that “[o]ne of the continuing scandals in physical sciences is that it remains in general impossible to predict the structure of even the simplest crystallographic solids from knowledge of their chemical composition.” This is an intermediate position: he is complaining that as of then, a chemist couldn’t choose a random small molecule and predict the structure (or structures!) exhibited by its crystal.

But even if Maddox was taking an intermediate position, he had launched the verb, and it was quickly associated not only with predicting what crystal a given molecule would produce, but what crystal structures could exist. During the past few decades, chemists have explored the possibility of taking the desired properties as a list of specifications, designing a crystal at the molecular or atomic level that will satisfy those specifications, generating a synthesis process from the design, and synthesizing the crystal – which will satisfy the specifications. As of 22 July 2014, Google Scholar listed 3,450 hits for “crystal structure prediction”, and on 25 June 2015, Web of Science reported rapid growth in the field since the mid-nineties:

Notice that unlike crystal engineering, there has been no recent growth slowdown. (Although I have heard chemists and crystallographers use the phrases “crystal enumeration” and “crystal structure enumeration”, and I have seen those phrases in print, Web of Science reports no hits for either one.)

As of 19 October 2014, Web of Science listed 702 hits for either “crystal prediction” or “crystal structure prediction”, and the journals that had published at least twenty articles in the subject were:

 Journal Hits Crystal Growth & Design 64 CrystEngComm 47 Physical Chemisty / Chemical Physics 33 Physical Review B 30 Journal of Chemical Physics 25 Journal of Physical Chemistry 25

And this is just those articles for which one of the two phrases is picked up by the Web of Science topics field.

Crystal structure prediction is new enough so that one can’t say what the fundamental mathematical issue is, but one could start with the work of Alexander Wells, whose book on Three dimensional nets and polyhedra presents the notion of a net, i.e. a finite or infinite graph embedded in three dimensional space. The vertices of this graph are points in space, while the edges are line segments (or curves) whose endpoints are vertices; typically, we all edge intersections should be at vertices.

(“Nets” might also be called “Euclidean graphs” or “geometric graphs” as they are graphs embedded in a Euclidean space.)

A net could represent a material structure by having the vertices represent atoms or molecular building blocks and having the edges represent chemical bonds or linkers. (We sidestep the issue of whether we believe in chemical bonds.) Then a (classical) crystal may be represented by a periodic graph, i.e. a graph with translational symmetries in three axial directions. We have reached the formulation of “crystal nets” as described in Michael O’Keeffe and Bruce Hyde’s Crystal Structures I : Patterns and Symmetry.

In mathematics, once you get your paws on a definition you can do things with it. We have a definition of the word “periodic graph”, which is the central mathematical notion in what Omar Yaghi calls reticular chemistry. A mathematician then mimics Stewart Robertson’s agenda and proposes the following:

• Let P be the set of all periodic graphs. Since P itself has a geometric and topological structure, we might call P the space of all periodic graphs.
• We usually regard two periodic graphs as being the “same” if one is the result rigidly moving the other around, so we can define the equivalence relation ≅, where “AB” means that the periodic graph A is the result of moving B. Given a periodic graph A, its equivalence class is the set [A] = { B : AB}. It is this equivalence class that fixes the size and shape of the structure, so once could say that the space of these equivalence classes, which we denote P/≅, are the models of (classical) crystal structures.
• This is not quite how periodic graphs are currently specified in, say, the Reticular Chemistry Structure Resource (RCSR). Current practice is to specify periodic graphs by isomorphism type. Call two periodic graphs A and B isomorphic if there is a one-to-one correspondence between their vertices such that that there is an edge connecting two vertices of A if and only if there is an edge connecting the corresponding two vertices of B. If two periodic graphs A and B are isomorphic, write AB, and the isomorphism type of A is [A] = {B : AB}. We could denote the entire space of periodic graph isomorphism types by P/∼, and RCSR has a catalogue of periodic graph isomorphism types.

As Igor Barburin observed (see his comment below, which inspired a revision of this post), the situation is a bit more subtle than this. While two periodic graphs are often considered to be the “same” if they are isomorphic, a database like RCSR may have, for each periodic graph listed, a particular periodic graph in its catalogue, listed as a geometric object with geometric properties. For example, while pcu comes in many shapes and sizes, RCSR lists a particular geometric realization of pcu (of maximal symmetry). The infinitely many other geometric realizations of pcu are themselves classifiable by geometric properties (including symmetries) – and topological issues like chirality.

We will discuss issues in classifying and cataloguing periodic graphs in later posts…beginning with the question of exactly why Dr. Baburin’s comment inspired me to replace the word “classify” with the word “specify”.

Anyway, two of the central problems of crystal structure prediction is the generation and classification of periodic graphs by rigid motion equivalence relations, by isomorphism types, and by other criteria. But even that is not enough. The Atlas of Prospective Zeolites Structures has over five million nets, but few of them have been synthesized. If crystal structure prediction is to be more than a recreational activity, it must include designing the synthesis process. After all, the purpose of a blueprint is to provide a roadmap for building the structure.

But at present, most of the activity seems to be in generating periodic graphs. But before we survey that activity, we should follow Socrates’ advice and get a handle on what it is we are talking about. So what do we know about periodic graphs…?

The Society of Industrial and Applied Mathematics’ Special Interest Activity Group on Mathematical Aspects of Materials Science has just launched a Facebook page. This seems to be the informal penumbra around plans to create a wiki at the group’s website.

A bit over a century ago, the scientific community had decided what a crystal was. A crystal was a material whose atomic or molecular arrangement (this was the same era during which atoms and molecules were finally accepted) repeated periodically in three axial directions. Sir William Bragg and his son developed x-ray crystallography, and crystallographers could develop good descriptions of what these repeating “unit cells” looked like.

This was probably a necessary step. Socrates would say that if we are going to study crystals, we must first decide what a “crystal” is.

Socrates’ is not a universal sentiment. To paraphrase Ludwig Wittgenstein, to teach a student what a crystal is, one presents the student with a diamond and say, “crystal”, and then with a large salt cube and say, “crystal”, and then with a lump of amethyst and say, “crystal”, and then the student starts getting the idea. In real life, Wittgenstein is right: definitions (and food fights over definitions) emerge from catalogues of examples and counterexamples.

That does not mean that definitions are a waste of the taxpayer’s money. Consider my current obsession: predicting crystals. Crystal prediction requires software, software requires theory, and theory requires definition. If one is to predict crystals, one needs to know precisely what crystals are. For crystals (as they were understood over most of the Twentieth century), one will probably wind up doing a variant of one of the following:

• Design a crystal by assembling a structure within the space of a unit cell. One takes a generic parallelopiped, with side (vectors) labeled x, y, and z, as in this picture…

…and then one identifies the three pairs of opposing faces of the unit cell, so that a fly buzzing into one face will then buzz out of the opposing face in the same direction. Within this unit cell, one assembles a structure, possibly adjusting the shape of the cell (i.e. adjusting x, y and z) en route. (References for this sort of topology includes Michael Henle’s Combinatorial Course in Topology and Hajime Sato’s Algebraic Topology : An Intuitive Approach.)

• Assembling a structure by taking some kind of fragment or collection of fragments, and then attaching them one to another to another, all monitored by a device that can recognize when a unit cell or equivalent has been assembled. (References for this sort of group theory include John Meier’s Groups, Graphs and Trees.) This is the approach I proposed in my presentation to the MathCryst commission.

(All this also requires linear algebra – see, e.g. Hoffman & Kunze’s Linear Algebra – and abstract algebra – see, e.g. Israel Herstein’s Topics in Algebra.)

Both of the above approaches presumes a definition of “crystal” that is somewhat like this:

• A crystal is a material composed of a finite number of types of constituents, and whose structure admits a symmetry from any constituent to any other constituent of the same type.

This is the fundamental classical definition based on nanoscopic structure, and it is the one that a mathematician might start with. But this definition is not the definition that emerged from Eighteenth and Nineteenth centuries and held sway until the 1980s. For the more popular definition, I’ll quote from Charles Kittel’s Introduction to Solid State Physics (2nd ed.):

• A perfect crystal is considered to be constructed by the infinite regular repetition in space of identical structural units or building blocks.

This definition is at least as old as Kepler, and may go back to the Greek atomists. Mathematically, these two definitions are equivalent, a fact that one might regard as the Fundamental Theorem of [Classical] Mathematical Crystallography: A material is composed of a finite number of types of constituents such that its structure admits a symmetry from any constituent to any other constituent of the same type if and only if it is constructed by the infinite regular repetition in space of identical structural units or building blocks.

Yet Kittel was typical in starting with the second definition, and the first definition – if it is mentioned at all – is mentioned as a rationale for the second. In practice, these two classical definitions above are quite different.

1. The first definition arises from the apparent homogeneity of crystals, that is, it is about an observable property of crystals. Thus it is somewhat like what computer scientists call a specification: given a crystal, this is the “spec” that it has to satisfy. A specification may not say very much about what the object is so much as how it behaves.
2. The second definition is closer to what applied mathematicians call a model. It is both descriptive (giving a better idea of how to recognize a crystal if you encounter one) and prescriptive (giving a better idea of how to construct one, if only out of styrofoam balls and toothpicks).

(Of course, the first definition is rather model-ish. We will see more pure specifications in a moment. In general, there is a spectrum from specification-ish to model-ish.)

Perhaps the main theme of the 2014 IUCr Congress is that old definitions have been replaced by new ones, thanks to quasicrystals and the like. Very roughly, the new definitions can be associated with the work of Dan Schechtman (who won the 2011 Nobel Prize in Chemistry) and of Aloysio Janner and Ted Janssen (who shared the 2014 Ewald Prize), respectively:

1. From the IUCr Online Dictionary: A material is a crystal if it has essentially a sharp diffraction pattern… (the rest of the entry devoted to what “essentially” means). In Volume C of the IUCr tables, Janner, Janssen, Looijenga-Vos and Wolff restrict this definition to require that “… its diffraction pattern is characterized by a discrete set of resolved Bragg peaks, which can be indexed accordingly by a set of n integers …”. These definitions are specifications, pure and simple. They impose criteria that must be satisfied in order for an object to be a quasicrystal, but they does not tell us what quasicrystals are.
2. There are a number of mathematical models of crystals. For example…
• One model is the cut-and-slice model, which I can oversimplify as follows. Given an n-dimensional lattice L, and one creates a slice consisting of an k-dimensional subspace S and a n- k – dimensional “window” W, to get the n-dimensional slice W × S. Project all points of L in the slice orthogonally onto S, and the projected points give the positions of the atoms (see, e.g. Marjorie Senechal’s Quasicrystals and Geometry).
• Another popular model is called inflation, which is a higher dimensional (and geometric) analogue of what computer scientists call a grammar. A grammar consists of several rules for replacing individual letters with strings of letters. For example, the grammar defined by a → a, a → bab generates strings from a of the form b…bab…b, with equal numbers of b’s on both sides of the a. Notice there is no rule for replacing b’s: in inflation, many sets of substitution rules have at least one rule for each letter. One can go beyond strings of letters to geometric shapes, replacing a shape (or tile) with some configuration of several tiles, and then “inflating” the configuration until the new tiles are the same size as the original ones, and repeating (see, e.g. Michael Baake and Uwe Grimm’s Aperiodic Order I: A Mathematical Invitation).

Such models give us visualizations of what a quasicrystal is. Cognitive scientists claim that we think in metaphors, and that is what makes these definitions valuable.

Having a lot of definitions suggests that a field is new and practitioners have not settled on a definition to inflict on students. We can ask mathematicians for a new Fundamental Theorem, but it is possible that the situation is more complicated. For example, in 2000, Jeffrey Lagarias asked eleven questions about the relations between various definitions. At the 2014 IUCr Congress, Lorenzo Sadun (with Johannes Kellendock) announced that the answer to Problem 4.10 was “no”, suggesting that the world is a little more complicated than anticipated. (See also Lagarias’s short paper on these definitions. I’d like to thank Lorenzo for helping me with some of these definitions.)

If we have several definitions, and if they are not equivalent, then we have a problem. Outside of encouraging food fights, definitions provide a methodological anchor. But it would be helpful if we could settle on what the subject of our endeavor is.

Often (as in classical crystallography), the goal is a single widely usable definition. In logic, a notion is robust if it is expressible in many different but straightforward ways. In mathematics, a representation theorem says that two different notions are actually equivalent. The Fundamental Theorem of [Classical] Mathematical Crystallography is such a representation theorem – and a particularly important one, since it connects a specification with a model. So some of us may hope for a demonstrably robust notion, whose robustness can be demonstrated by a representation theorem.

But that may not be in the cards. Sometimes the universe is messy, and what we really need is a catalog. We may then hope that organizing principles will arise out of the mounds of data, like quarks arising from the heaps of subatomic particles in the early 1960s.

Either way, we don’t seem to be there yet. And that means that the paradigm shift presided over by Shechtman, Janssen and Janner is still underway.

The Twenty-third Congress of the International Union of Crystallography met in downtown Montreal, one or two kilometers south of McGill University, whose medical school peeks over the stadium towards downtown:

Downtown Montreal sits right on the St. Lawrence River, and amidst the tall buildings there squats the Palais des Congres…

…whose southern entrance opens to a small park…

…a few blocks from Rue Wellington (sorry, as an Anglophile, I had to mention that).

Not being good at estimates, I don’t know how many people attended, but there were enough people to fill a very large auditorium and keep eight parallel sessions going through the last (seventh) day – the IUCr estimates over 2,200.

Like many attendees, I cherry-picked events, but I did have a revelation. Suppose you go to a conference on a sprawling, multidisciplinary field, where the plenary talks focus on The Latest Hot Topic (more about that below, and in the next post), and where there are approximately a zillion parallel microsymposia on various subfields. Where would you go to get a general impression of what is going on?

To the poster sessions, of course.

Posters and exhibits were relegated to a large multipurpose room tucked beneath the multi-story complex that makes up the conference space. To someone used to mathematics conferences, where the exhibits are dominated by publishers hawking monographs and textbooks, the IUCr exhibits seemed to consist largely of software and hardware. Almost like a (gasp) trade show. There was, so I heard (but I did not see it on exhibition), one book on display (the Little Dictionary of Crystallography, which quickly sold out). And surrounding the exhibits were hundreds of posters.

Poster presenters are an odd lot. My father preferred presenting posters over giving talks: he wasn’t convinced that people actually listened to the talks (!). And with a poster you could have a one-on-one conversation with passersby – which, as educational psychologists will tell you, is a more reliable way to impart information that talking at lots of people from a distance. Another thing about posters is that screening committees are less picky: you can take risks and talk about odd topics without getting hurled out the door. The result is that some posters talked about funky or fun stuff that would never have made it into a microsymposium. And considering how many graduate students present posters at a meeting this large, you can get an impression of what is going on at major research centers simply by walking along the aisles of posters.

Schechtman mentioned four things a scientist needs to effect a successful paradigm shift.

• One has to be very good at transmission electron microscopy (“TEM”). Schechtman focused on one particular skill that made his discovery possible. It takes many years to become any good at TEM, and few succeed. This may be generally true: sociologists and psychologists are increasingly remarking on the need for a great skill to accomplish a great deed: a great idea, in of itself, is often not enough.
• One needs tenacity. Schechtman told the story of a student who saw a quasiperiodic diffraction pattern before he did, recognized its significance, but (possibly reflecting on Semmelweis’ fate) said nothing. Schechtman said that an odd observation may be an artifact (it probably is), but you will never know if you don’t pursue it. Schechtman also mentioned resilience, which is not quite the same thing. (On the other hand, there is the tenacious and resilient example of Louis Agassiz – the fellow who discovered ice ages and later showed great tenacity and resilience as the last great holdout against evolution.)
• Believe in yourself. Schechtman actually had an arch-enemy, Linus Pauling, who was convinced that quasicrystals were classical crystals with very large units. But Schechtman persevered. There are two ways of looking at this. One is that this is the basic advice for aspiring writers: you know you aren’t a real writer until you can paper your wall with rejection slips (Lord of the Flies, which won Golding a Nobel, got 20 rejections, as did Frank Herbert’s Dune, while Harry Potter and the Philosopher’s Stone garnered only sixteen – and A Wrinkle in Time got 26 rejections while Gone with the Wind got 38).

There are two things that Schechtman did not mention: recognition and communication.

• When I was a graduate student at UCLA, Paul Erdös (one of the great eccentrics in mathematical history) visited us, and told us the following story (which I have not checked to see if it is true). Before Marie Curie’s work on radioactivity, workers in a lab observed that if you left pitchblende and undeveloped film together, the film would get fogged. In reaction, the lab workers put a rule in their books: pitchblende and undeveloped film should not be stored together. Henri Becquerel later observed the same phenomenon and, paying more attention, told his co-workers (the Curies) about it. From Alexander Fleming’s chance observation of penicillium’s chemical warfare against germs to Charles Goodyear dropping rubber on a stove, some of the greatest discoveries were made by people who were paying attention.
• In the mid-Nineteenth century, the physicists knew that they needed an algebra of three-dimensional space. In fact, William Hamilton knew that they needed an algebra of umpteen dimensional space (and Hamilton’s quaternions did not fit the bill). Hermann Grassmann developed such an algebra, but being of a philosophical bent, communicated his discoveries in books that Hamilton and other physicists were unable to decipher. Decades later, Willard Gibbs independently developed a similar algebra – which we now call vector algebra – and Gibbs was much better at marketing: he delivered his discoveries in bite-sized articles aimed at the audience he had.

And as far as reception goes, it may be wise to follow the advice from the Yes, Prime Minister episode, The Ministerial Broadcast. When a politician is to announce something routine, the opening music should be Stravinsky, the politician should wear a modern suit, and the background should feature abstract art. When a politician is to announce something truly revolutionary, the opening music should be Bach, the politician should wear a dark suit, and the setting should be oak paneling, leather volumes and Eighteenth century portraits.

Schechtman was merely the most notable of the speakers on aperiodicity. Marjorie Senechal gave a keynote address on Mathematical Crystallography in the 21st Century, and started by saying that major areas for future research include folding and flexing, diffraction and imaging, superspaces, symmetry, self-assembly and self-organization, and mapping the aperiodic landscape. She then focused on aperiodic structures from two points of view. From the point of view of the final structure, she said that under certain conditions, a Delaunay set must periodic, and asked if there was a (nice) set of conditions for a Delaunay set to have a discrete diffraction pattern. Then from the point of view of the formation of the structure, she observed that icosahedral shell structures of less than two thousand or so atoms are more stable than cuboctahedral structures, and asked how the transition from icosahedral to cuboctahedral takes place. She concluded by mentioning the Defense Advanced Research Projects Agency (DARPA), which frequently posts “challenge” problems. In 2007, DARPA posted 23 challenge problems for mathematicians (which for some reason DARPA took down, but Professor Vasilios Alexiades of the University of Tennessee archived the list), and Challenge Number Eleven was:

• Optimal Nanostructures. Develop new mathematics for constructing optimal globally symmetric structures by following simple local rules via the process of nanoscale self-assembly.

Senechal said that she asked someone at DARPA what all these terms meant, and was told that DARPA hadn’t assigned specific meanings to the terms. So the challenge is somewhat open-ended.

Officially, the big event was the Ewald Prize, which was awarded to Aloysio Janner and Ted Janssen for developing the “superspace model” of quasicrystals, and that leads into my next (curmudgeonly) posting…

The Commission will meet on August 11, 12:15, in room 441.

Recall from the 17 August 2013 post that Luis Bettencourt, David Kaiser, Jasleen Kaur, Carlos Castillo-Chavez, and David Wojick had constructed and tested a model of emergence, and the results suggested that for an emerging field, recruitment of new participants is critical.

When recruiting, there are two audiences one may have in mind.

• Of course, there are students. Students are energetic, ambitious, and have yet to develop that conservatism that keeps middle class academics in their own fields.
• There are also colleagues. Colleagues usually do not change fields unless impelled to do so (by necessity or by wanderlust). Colleagues know how the game is played, and they bring expertise.

These are two different audiences. I don’t know how much is known about recruitment of students versus colleagues (perhaps very little, as “emerging fields” as a branch of the sociology of science is itself an emerging field), so all I can do is engage in armchair theorizing. The advantages and disadvantages of joining an emerging field can be assessed differently by students and researchers.

• A new field is an opportunity. Economists and ecologists have long observed that when individuals find a new environment they can exploit, they can more readily succeed than they would in established terrain. In a new field, there is less competition, there are basic and tractable problems to be solved – and considerable rewards for doing so.
• A new field is a gamble. Most new businesses fail, most revolutions fail, and probably most new academic fields implode; we remember the successes while the failures are quietly forgotten and rarely make it into the history books. Some do succeed, but fail to deliver on their promises. Even successful ventures may not produce timely success, and the original revolutionaries may not live to see the success of their revolution.

Let’s consider two issues: attraction and access.

Mathematical crystallography faces the same problem that small-town boosters have: how to attract new enterprise. We are all familiar with Chamber of Commerce boosters (the old-fashioned non-ideological kind) who appeared in the business section of the newspaper, talking up the potential of the town. That’s us. So how do we attract enterprise?

It pays to advertise, or so the poem goes. In academia, one can advertise with accessible, semi-popular works. The power of advertising may be over-rated: in assessing the impact of Richard Feynman’s 1959 speech, There’s Plenty of Room at the Bottom in inspiring nanoscience, Chris Toumey found that while many people later remembered Feynman’s article being discussed, there was little documentable evidence of impact prior to the 1980s.

Still, books like Herman Weyl’s Symmetry, Marjorie Senechal and George Fleck’s Shaping Space, and John Conway, Heidi Burgiel and Chaim Goodman-Strauss’s Symmetries of Things may be useful in attracting students to the field.

Papers might also be useful advertising, although the ones that come to mind look more like materials for recruiting colleagues. For example, in 2003, Omar Yaghi et al published a manifesto, Reticular synthesis and the design of new materials, advocating the development of a design process to facilitate the synthesis of novel crystals. On the other hand, in 2008, Massimo Nespolo published a manifesto, Does mathematical crystallography still have a role in the XXI century? which introduced readers to some of the active research areas. And I have just published an article, Prospects for mathematical crystallography, allied to postings in this website.

One possible selling point is that mathematical crystallography is an interdisciplinary field. Unfortunately, this could be a liability. Interdisciplinary fields have been all the rage for some time, but institutional support tends to be a bit thin. In 1995, the National Research Council issued a report on Mathematical Challenges from Theoretical / Computational Chemistry, which addressed institutional obstacles to recruiting for a closely related interdisciplinary field:

• For faculty, the institutional support and reward structures are designed for within-disciplinary efforts. For example, faculty are evaluated by departments, which often focus on their own disciplines. In addition (though the report did not go very much into this), internal funding is often made available through departments, and credit for external funding is usually assigned to departments.
• For students, the curriculum is often determined by departments, usually independently of each other. A student in one department could face a long sequence of prerequisites before being prepared to take a relevant course in another department.

The NRC did not mention another problem: external funding agencies like their grant proposals to go into the correct pigeonhole, which is a problem for interdisciplinary programs. So by all means, talk up the interdisciplinary aspects of mathematical crystallography. But we may have to support some institutional reform.

This brings us to a major obstacle: access. Chemists who want to join need to learn physics and mathematics, physicists who want to join have to learn chemistry and mathematics, and mathematicians who want to join need to learn chemistry and physics. I am most familiar with problems learning mathematics, so let’s look at those.

In 1992, Mattel’s Barbie doll said that Math class is tough. Mattel got in trouble with the American Association of University Women because Barbie’s audience was little girls, but there is a feeling that mathematics is an unusually difficult subject. There are a number of theories as to why.

There are even a few people who claim that math is not hard, or at least shouldn’t be. But considering that mathematics has a reputation that chemistry and even physics does not, participants in a mathematical field need to think about how to make their field accessible to novices.

Meetings (like the upcoming IUCr 2014 meeting; see the previous entry for details) are invigorating and fun and provide opportunities to meet and recruit people, but the work of learning a new field is a more lonely business and it involves a lot of reading. We have a strong interest in having a lot of accessible material. And accessibility is a problem in academia.

Etymological dictionaries tell us that arcane is a descendent of the Latin arcere, to contain or maintain, to keep or ward off, and perhaps to the Greek arkein, to keep off. It is associated with the Latin arcana, or mysterious secrets. These are all members of the family of words led by the Latin arca, or ark, as in Noah’s Ark and the Ark of the Covenant.

As a new word, arcane bubbled up in the Sixteenth century, in the midst of the explosion of books created by Gutenberg’s press. Books on just about everything suddenly appeared, and they sold. Just as the Internet not only made everything available to everyone, and made a virtue of universal access, so the Gutenberg revolution made all knowledge available to any plowman able to afford a book.

But some works seemed to remain out of reach, hence the notion of arcane works. Perhaps the printing press was partially responsible for the Plato boom that inspired Renaissance scientists: Plato was a natural novelist (see, e.g., his great literary invention, Atlantis). Perhaps the printing press was partially responsible for Aristotle’s poor reputation in that era: the Complete Works are a jumble of often unintelligible notes and fragments of dubious provenance. And Aristotle was hardly unique. As Great Works became available to the Common Man, common men found many of them inaccessible.

Communication theorists tell us that arcana can serve a purpose. I don’t mean unintentional arcana, like a bunch of moldy papers somehow associated by a major philosopher. I mean (often unconsciously) intentional arcana, like books by German metaphysicians we could name. Consider this grumpy passage by the greatest of them all, Immanuel Kant. In his Prolegomena to Any Future Metaphysics, a sort of Kant-for-Dummies-by-Kant-Himself, Kant writes that his major work, the Critique of Pure Reason

 … will be misjudged because it is misunderstood, and misunderstood because men choose to skim through the book, and not to think through it – a disagreeable task, because the work is dry, obscure, opposed to all ordinary notions, and moreover long-winded.

Some communication theorists might detect a little pride in that sentence. Still,

 I confess, however, I did not expect, to hear from philosophers complaints of want of popularity, entertainment, and facility, when the existence of a highly prized and indispensable cognition is at stake, which cannot be established otherwise, than by the strictest rules of methodic precision.

There are two interesting points about this passage. First, Kant was irritated that philosophers, much less educated laymen, were unwilling to invest the vast amount of time and energy required to read the Critique. But second, he did write the Prolegomena after all, and that was consistent with what some commentators have claimed about Kant: he had an agenda, and that agenda required readers and students. Kant started his introduction to the Prolegomena with the words, “These Prolegomena are destined for the use, not of pupils, but of future teachers …”; perhaps concerned by the criticism of his colleagues, he wanted to make sure that teachers, at least, understood what he was trying to say.

Academics aren’t the only ones who make the difficulty of the material into a gatekeeper, but that is one of our bad habits. Especially when our goal is accessibility. It all boils down to who the audience is. New Yorker archivist Joshua Rothman compared journalists, whose text is relatively undemanding because it is intended for a mass audience, to academics.

 In academia, by contrast, all the forces are pushing things the other way, toward insularity. As in journalism, good jobs are scarce—but, unlike in journalism, professors are their own audience. This means that, since the liberal-arts job market peaked, in the mid-seventies, the audience for academic work has been shrinking. Increasingly, to build a successful academic career you must serially impress very small groups of people (departmental colleagues, journal and book editors, tenure committees). Often, an academic writer is trying to fill a niche. Now, the niches are getting smaller. Academics may write for large audiences on their blogs or as journalists. But when it comes to their academic writing, and to the research that underpins it—to the main activities, in other words, of academic life—they have no choice but to aim for very small targets. Writing a first book, you may have in mind particular professors on a tenure committee; miss that mark and you may not have a job. Academics know which audiences—and, sometimes, which audience members—matter.

But mathematical crystallographers are not trying to fill a niche. We are trying to get people involved (or if you prefer, we are trying to create an array of niches). That means that we should be writing papers and books for as wide an audience as possible. So here are some “best practices” we probably should engage in:

• Papers should be self-contained. In an interdisciplinary field, one’s readers may be unfamiliar with some jargon and some concepts. It may be useful to define them in the paper if that does not take up too much space. Of course, sometimes that would be distracting, so another thing one needs is:
• Papers should have primary references. Where does the novice reader go to find out what actions, entropy, and carboxyl groups are? It takes very little space to give a citation to a general reference known to be accessible.

It may be a good idea to imagine a graduate student when writing a paper.