Oct 202018

The Kungliga Vetenskapsakademien (the Royal Swedish Academy of Sciences) has announced that the 2019 Gregori Aminoff Prize in Crystallography will be awarded to Arizona State University Professor Michael O’Keeffe and University of California – Berkeley Professor Omar Yaghi.

While the announcement stressed metal-organic frameworks, the prize is “for their fundamental contributions to the development of reticular chemistry,” and the press release said:

“Michael O’Keeffe and Omar M. Yaghi have laid the foundation of reticular chemistry, where the whole idea is to create well-ordered coordination polymers. These materials are built according to simple structural principles; by varying the size of the ligands or functional groups bonded with the ligands, for example, series of materials with similar but regulated variable properties can be synthesized.”

The statement on the prize in general is: “The intention of the Gregori Aminoff Prize is to reward a documented, individual contribution in the field of crystallography, including areas relating to the dynamics of the formation and dissolution of crystal structures. Some preference shall be given to work that demonstrates elegance in its approach to the problem.”

May 092018

The American Crystallographic Association is meeting on July 20 – 24 in Toronto, Canada. The keynote speaker is John Polanyi, who shared the 1986 Nobel Prize in Chemistry “for their contributions concerning the dynamics of chemical elementary processes.”

In addition to a special session on Crystallization on the International Space Station (!), there will be a session on Theoretical and Computational Crystallography – Present and Future Opportunities at the Structural Interface of Experiment and Theory 1, chaired by Branton Campbell (and no matter what the site says, I am merely chief kibbitzer) and Theoretical and Computational Crystallography – Present and Future Opportunities at the Structural Interface of Experiment and Theory 2, chaired by Peter Khalifah and Wenhao Sun.

Jan 022017

In my previous life, I was a mathematical logician specializing in theoretical computer science. I got my Ph.D. in 1986, so I saw the aftermath of the Soviet-Western thaw in science, when scientific articles in Russian were translated into English. One consequence was that a number of major discoveries were made independently in the USSR and in the West; the former published in Russian and the latter in English (or occasionally French – by that time, the German mathematicians were giving up and publishing routinely in English). Thus the large number of articles whose citations of previous work leads to two primary threads, one in English and one in translations from Russian.

With the growing research activity in East Asia and Latin America (and perhaps with activity to come in the Middle East), we can expect a recurrence of this phenomenon: Brazilians are unlikely to learn Chinese, who in turn are unlikely to learn Portuguese, and of course Americans know only their own language (assuming that we Americans actually know English – a controversial assumption).

In East Asia, India Today reports that less than two thirds of all scientific reports are published in English. Considering all the MOF and other crystallographic activity in China and elsewhere in Asia, this probably affects us as well.

No doubt the problem is complicated by the fact that unlike Chinese and Portuguese, English is not so much a language as a Frankenstein monster of ancient Danish and German in conflict with French, with Arabic cream and a Latin cherry on top. (And that doesn’t even count the Americanisms.) This can lead to communication problems. Last October, BBC reported that Native English speakers are the world’s worst communicators. “Non-native speakers generally use more limited vocabulary and simpler expressions, without flowery language or slang. Because of that, they understand one another at face value,” reported BBC, while native speakers are more prone to confusing embellishments.

For those readers who are not familiar with the language wars, the alleged cure is William Strunk and E. B. White’s The Elements of Style (this link is to Wikipedia’s page; Strunk’s original notes are posted on Gutenberg and someone has posted the joint version on Google Documents; but it is a cheap paperback you can get anywhere). Journals that prefer English submissions may want to recommend using it.

Dealing with English as a lingua franca – at least for the sciences – is different from protecting, preserving, and strengthening languages around the world. Languages reflect the societies that created them, and when UNESCO reports that many languages are dying out, that means that major parts of our heritage are disappearing. And fluency in foreign languages provides some familiarity with different modes of thought. That is different from a lingua franca, which provides a common basis for communication and reduces the chance of important discoveries being ignored.

Nov 202016

During the last week, two interesting news items on physics appeared and their juxtaposition may contain a message for mathematical crystallography.

First, Quanta Magazine, “illuminating science,” ran a feature article on how An unexpected connection has emerged between the results of physics experiments and an important, seemingly unrelated set of numbers in pure mathematics. The point of the article is the explosion in the size of Feynman Diagrams used to analyze particle interactions as the number of particles increases. (Crystal structure prediction encounters a similar problem in surveying new structures as the number of “kinds” of components increase.) The point of the article is that it appears that some of the results correspond to parameters generated in algebraic geometry.

Algebraic geometry is concerned with surfaces that can be defined by polynomial equations. For example, a sphere of radius 3 can be defined by the equation x2 + y2 + z2 = 32 = 9. (Algebraic geometry can work in arbitrary dimensions, and a surface-like structure in a higher-dimensional space is called a manifold. Algebraic geometry is concerned with manifolds defined by polynomial equations.)

This heads up suggests that mathematical crystallography might also benefit from using other bits of mathematics that we normally don’t think of. But there is a problem.

Andrew Higginson and Tim Fawcett are back. Four years ago, they published a study on how Heavy use of equations impedes communication among biologists. That paper asserted that empirical biology did not seem to rely much on theoretical biology, and that formal mathematics in theoretical biology papers negatively impacted their citation rates. Last year, three physicists claimed that statistical observations about biologists could also be made about physicists, and suggested that there might be other reasons for what Higginson and Fawcett observed. (The physicists observed that, “much anxiety and pain also seems to be related to doing math,” although the paper that they cited claimed that the anxiety and pain was associated with the anticipation of doing math – and not the actual doing math.) Higginson and Fawcett have just conducted a study of physicists and concluded that “equation density” negatively affected the impact of physics papers as well.

No doubt the food fight will continue, but realistically, whether one reads or scans or drops a paper could depend on whether flipping through the paper it does not look like much fun to read. It would not be surprising if Higginson and Fawcett were on to something. For mathematical crystallographers, the question is how much it affects our field; a rough guess would be somewhere between biology and physics.

Jun 122016

I was a lot less organized this year, so this report is a lot skimpier.

The Society of Industrial and Applied Mathematics‘s activity group on Mathematical Aspects of Materials Science held its 2016 conference in May in Philadelphia, about a half a mile east of Independence Hall. SIAM conferences consist of a handful of plenary speakers plus two-hour minisymposia, each consisting of four presentations.

Four of us got together and organized four minisymposia. Here they are.

The first minisymposium focused on Tilings, Packings, Graphs, and Other Discrete Models.

Left to right: Davide Proserpio gave a talk on Entanglement in 2-Periodic Coordination Networks, Ileana Streinu gave a talk on Polyhedral Origami, Ma. Louise N. de las Penas gave a talk on On Algebraic and Geometric Properties of Hyperbolic Tilings, and Ciprian S. Borcea gave a talk on Auxetics and Spectrahedra.

The second minisymposium focused on Polyhedra, Cluster Models, and Assembly.

Left to right: Erin Teich gave a talk on Clusters of Polyhedra in Spherical Confinement, Jean Taylor gave a talk on What Role for Entropy in Stability and Growth of Quasicrystals? , Miranda Holmes-Cerfon gave talk on The Statistical Mechanics of Singular Sphere Packings, and Natasha Jonoska gave a talk on Algorithmic Self-Assembly and Self-Similar Structures.

The third minisymposium focused on Groups, Lattices, Spaces and Superspaces.

Left to right: Bernd Souvignier gave a talk on Recognizing the Lattice Type to Which a Unit Cell (almost) Belongs, Massimo Nespolo gave a talk on Applications of Groupoids to the Description and Interpretation of Crystal Structures: The Example of Pyxorenes, Mois I. Aroyo gave a talk on Materials Studies by the Bilbao Crystallographic Server, Jeffrey Lagarias and gave a talk on The 12 Spheres Problem.

The fourt minisymposium focused on Beyond Crystallography.

Left to right: Peter Zeiner gave a talk on Comparing Coincidence Rotations and Similarity Transformations of Lattices and Modules, Uwe Grimm gave a talk on Diffraction and Dynamical Spectra in Aperiodic Order, Gregory Chirikjian gave a talk on Molecular Packing Problems and Quotients of the Euclidean Group by Space Groups, and Egon Schulte gave a talk on Skeletal Polyhedral Complexes and Nets with High Symmetry.

And here are the four organizers:

Left to right: Jean-Guillaume Eon, Greg McColm, Marjorie Senechal, and Mile Krajcevski.

Most of the above photos were taken by Massimo Nespolo, who very kindly shared them with me; there are more posted at the IUCr page for this conference. The next Mathematical Aspects of Materials Science conference is in 2019.

Jun 042016

Some time ago, I stopped putting up regular posts on this blog. Part of the problem was that things happen, and part of it was a technical problem. Hopefully, both issues have been resolved, and it can be started up again.

The IUCr very kindly gave me this space in 2012, and it was up and running for the 2013 SIAM Mathematical Aspects of Materials Science meeting, where there were minisymposia on Geometric Foundations, Beyond Crystal Symmetry, and Structure-Building Principles.

After the SIAM meeting, some people expressed interest in the blog, and it ran for a while. The SIAM meeting led to a Virtual Issue on Mathematical Crystallography in Acta Crystallographica A, and I got the idea of a sort of survey of mathematical crystallography – or of mathematics in crystallography, which turns out not to be quote the same thing. I published a paper in that issue on what the mathematics + crystallography community (communities) look like, and started a sort of survey of the subject on the blog.

This led to two complications.

  • I am a mathematician, and like Lewis Carroll’s White Rabbit, I am therefore prone to begin at the (very) beginning. The result were twelve Mapping the Community posts, from 24 June 2013 to 3 July 2015. Clearly I have just gotten started, but it is not clear how useful these posts are.
  • Blogs tend to function best if there is a working comment function. Comments were quickly overwhelmed by spam, and most people were unwilling to have comments moderated.

There were posts on a few other things (I have been a bit disorganized lately, and will not get up posts on last month’s SIAM conference until another week or so), but lacking feedback, that was it.

As I mentioned in both the Acta paper and in early posts, the experts on whether Newly Emerging Science and Technology efforts succeed or fail depends on recruitment and communication. So the question is how this blog can facilitate recruitment and communication.

  • I am inclined to continue occasional postings on Mapping the Community, unless people ask me to stop. I am not sure what the pace will be. But I suspect that that more topical posts might be more useful.
  • The IUCr webmaster has very kindly installed an Askimet filter (using Captcha) in the comments section, and I have enabled comments. So you can enter a comment, type in the phrase that Askimet produces, and it should appear. We’ll see how this works: after four weeks with the Comments section on, Askimet blocked 5,108 spams, put 17 spams in the spam queue for me to look at, and accepted two (which got posted).

Anyway, comments are now open. To comment on any posting, click on the headline of the post and it should go to that post, with space for comments at the bottom.

The big question is how this blog could be useful. Let’s see how this goes.

Dec 242015

From a certain point of view, this blog concerns one of the mundane areas of science and engineering, doing our routine work under the twin shadows of Cosmology and High Energy Physics, where scientists work to uncover the meaning of the universe – or, as Stephen Hawking humbly put it, read the mind of God.

Curmudgeons have taken a dimmer view, especially with physicists publishing books and narrating TV shows about strings, dark energy, multiverses, and other critters that look like escapees from Professor Snape’s lab. Such a jaundiced view of transcendental physics seems especially justified by comparing the cosmological effort to two decades of exoplanet search, the latter of which has produced a daunting list of real planets using good old-fashioned science. Meanwhile, cosmologists and high energy physicists are presenting us growing array of theories whose support occasionally seems to rest on esthetic arguments.

As Star War fans might put it, the Curmudgeons have struck back. Like the computer scientists and mathematicians (and even biologists!) before them, the physicists are consulting the philosophers. One year after George Ellis and Joe Silk wrote that “Physicists, philosophers and other scientists should hammer out a new narrative for the scientific method that can deal with the scope of modern physics,” a workshop on Testing and Trusting in Physics at Ludwig Maximilian University addressed the awkward question of whether what the participants were doing were, ahem, science.

The old view of science popularized by Karl Popper was that a theory was not “scientific” if it was not falsifiable: if there was no experiment or sequence of experiments which could impeach the theory, then it was not a “scientific” theory. This standard seemed useful against pseudoscience: astrology is readily falsified (although devotees cling to it anyway) while creationism (which is rather slippery about how to deal with evidence) cannot be falsified and is therefore not scientific.

Falsifiability seemed to be a good way to keep honest people honest, but there were problems. Curmudgeons may be delighted to see Freudianism and monetarism exploded as pseudoscience, but since no one seems to have the imagination to dream up anything that could falsify evolution or the theory that the Roman Empire once existed, philosophers started looking for alternatives. One is Bayesian epistemology, which simply asks: if you are going to claim that cosmological inflation actually occurred, how much are you willing to bet?

Quanta Magazine reported that the philosophers reassured the physicists that (other than a few incorrigible curmudgeons) we are all Bayesians now. And the Bayesian bet turns out to be: do we believe in cosmological inflation enough to pursue it further? The answer to that question is, of course, yes.

Chemists and materials scientists should not be overly smug about all this: as Quanta Magazine observed, no one has seen an atom – but we’ve taken “photographs” of them. And as physicists are prone to observe, chemists and materials scientists are prone to believe in convenient fictions like chemical bonds and molecules. Considering the dangerous territory nanoscience is taking us, we should not be surprised if some time soon chemists and materials scientists find themselves, hats in hand, at the door of the friendly neighborhood Philosophy department.

Meanwhile, the pseudo-science problem remains. For example, two decades ago, a poll of Americans concluded that more Americans believed that the U. S. government was concealing evidence of extraterrestrials than believed that there were extraterrestrials to conceal. (Poll results vary, but it appears that about a third of the public seem willing to tell pollsters that aliens are abducting people and conducting experiments on them.) How to deal with this sort of pseudoscience seems beyond the philosophers.

Jul 032015

Crystallographers use group theory in order to get their paws on the (global) symmetries of crystals. Most of the time, they look at groups of “isometries”: if Rd is d-dimensional space, then an isometry on Rd is a function f : RdRd such that for any x, yRd, |f(y) – f(x)| = |yx|. Since the composition of two isometries is an isometry, and the inverse of an isometry is an isometry, and the identity is an isometry, the isometries on Rd form a group.

We would like to have formulas for isometries, and it is an exercise in linear algebra that for every isometry f on Rd, there is a unique vector m and a unique matrix M such that for every xRd, f(x) = m + Mx (such a function is called affine). There are no restrictions on m, but there are restrictions on M.

Recall that a basis of Rd is a set of nonzero vectors {b1, …, bd} from Rd such that every vector xRd may be uniquely expressed as a linear combination of b1, …, bd, i.e., in the form x = a1b1 + … + adbd, where a1, …, ad are real numbers. A basis is orthogonal if the vectors in it are mutually perpendicular, i.e., for any i, j, if ij then bibj = 0. An orthogonal basis is orthonormal if every vector in it is of norm 1.

A matrix is orthogonal if its columns form an orthonormal basis, or, equivalently, its rows form an orthonormal basis. Orthogonal matrices have all kinds of nice properties. The product of two orthogonal matrices is an orthogonal matrix, and the inverse of an orthogonal matrix is an orthogonal matrix: since matrix multiplication is associative, the orthogonal matrices form a group, often called the orthogonal group of dimension d. In addition, the determinant of an orthogonal matrix is either 1 or -1, and the inverse of an orthogonal matrix is its transpose (the transpose of a matrix is the result of reflecting it across its main diagonal, so that the i, j-entry of the transpose is the i, j-entry of the original matrix: ai,jt = aj,i).

So here we have it: an affine function f(x) = m + Mx is an isometry if and only if M is orthogonal.

There are several ways to enumerate all the kinds of isometries on R3. Yale uses a geometric construction while Giacovazzo uses algebraic machinery. Let’s follow Yale’s enumeration since this lets us look at geometry.: what Yale did was to look at fixed points: recall that a fixed point of a function f : RdRd is a vector x such that f(x) = x.

We need a notion. An affine space or affine flat in R3 is a point, a line, a plane, or the entire space. It is an exercise in linear algebra to prove that for any affine function from Rd to Rd, the set of its fixed points forms an affine space. We use this fact to classify the isometries of R3. But first, the main characters (see the figure below).

  • Given a plane, a reflection across that plane is an isometry.
  • Given two planes, if they are parallel, then the composition of the two reflections is a translation; if they are not parallel, then the composition is a rotation about the line of intersection.
  • Given three planes, if they all intersect at a point, it is an roto-reflection: a rotation about the line of intersection of the first to planes followed by a reflection across the third. If the first two of them are parallel, it is a glide reflection: a translation through the parallel mirrors and a reflection across the third.
  • Given four planes, the first two intersecting on a line perpendicular to the last two, which are parallel, it is a screw.

That’s all there is. We classify isometries by counting fixed points: p is a fixed point of f if f(p) = p.

  • If f has four fixed points, not all on a plane, then the set of fixed points must be the entire space and f is the identity.
  • If f has three fixed points, not all on a line, but f is not the identity, then the set of fixed points must be a plane. For any x, if y was the point on the fixed point plane closest to x, the line through x and y is perpendicular to that plane. As f is not the identity and f must preserve the distances between x and all points on the plane, f(x) must be the point on the line of distance |yx| from y but opposite x. Repeating for all x, f must be a reflection and the plane of fixed points must be its mirror.
  • If f has two fixed points and is not a reflection or the identity, its set of fixed points is a line. Choose a point x not on the line and let y = (f(x) + x)/2. Let P be the plane containing f’s fixed points and also y, and let r be the reflection across P. Then r ° f is an isometry that fixes P, so as we have already seen, r ° f is a reflection, call it h. So f = r-1 ° h is a composition of two reflections, and hence a rotation as translations have no fixed points.
  • If f has but one fixed point p, choose xp, and let y = (f(x) + x)/2, and let P be the plane through p and y perpendicular to the line through f(x) and x and let r be the reflection across P. Then h = r ° f is an isometry fixing p and x, and hence is the identity, a reflection, or a rotation. So f = r-1 ° h is a reflection, a rotation, or a roto-reflection; as it has only one fixed point, it must be a roto-reflection.
  • If f has no fixed points, choose any p and let g be the translation xx + (p – f(p)). Then p is a fixed point of h = g ° f, and we have four subcases.
    • If h has four fixed points, not all on a line, then h is the identity and f = g-1 ° h is a translation.
    • If h has a plane of fixed points but is not the identity, then f = g-1 ° h is the composition of a translation and a reflection and is a reflection (if all three mirrors are parallel) or a glide reflection.
    • If h has a line of fixed points but not a plane, then f = g-1 ° h is a composition of a translation and a rotation and is a rotation (if two mirrors coincide) or a screw rotation.
    • If h has only one fixed point p, then f = g-1 ° h is a composition of a translation and a roto-inversion and is a glide reflection or a roto-inversion or a reflection.

And that’s it.

One additional point. One of the more useful functions from matrices to real numbers is the determinant. Two important facts. First, the determinant of a product of matrices is the product of the determinants. Second, while the determinant of the identity matrix is 1, the determinant of a matrix of a reflection isometry is -1. That means that the matrix of a composition of an even number of reflections (the identity, rotations, translations, and screws) are 1 while the composition of an odd number of reflections (reflections, roto-reflections, and glides) is -1; noticed that the matrix is 1 if and only if the isometry is a direct motion, and otherwise it is -1 if it is an indirect motion.

And now for groups of isometries. A group of isometries is a set of isometries closed under composition and inverse. If f(x) = a + Mx and g(x) = b + Nx, then (g ∘ f)(x) = b + Na + NMx and thus f-1(x) = –M-1a + –M-1x.

First of all, a point group is a group of isometries that fixes a common point. The most important point group is the orthogonal group, which is the group of all isometries that fix the origin.

Then there are groups of translations. The most popular of these is the lattice group: in d-dimensional space, one has a basis b1, …, bd such that the isometries xb1 + x, …, xbd + x generate a subgroup of isometries (see the 20 June 2015 posting for details on generating subgroups). One important fact. Let G be a group of isometries, and let T be the group of translations in G. Then T is normal in G (see the 20 June 2015 posting for details on normal subgroups): if xa + Mx (which we denote [a, M]) and xb + x (which we denote [b, I]) are in G, then so is the composition [a, M] ∘ [b, I] ∘ [a, M]-1 = [a + MbM-1a, I], which is a translation and hence in T.

Now for the reason why crystallographers are interested in all this stuff. In the Nineteenth century, a few daring souls considered the then politically incorrect idea that solids were made of atoms, and they followed up on Kepler‘s idea that a crystal may be composed of a regular array of atoms. (This idea was politically incorrect because as of 1800, all right-thinking folk knew that the atomic theory was wrong even if it was useful for chemical bookkeeping.) If a crystal is composed of, say, identical atoms, and if the crystal is symmetric in a way that it looks the same from each atom, what does that say about the structures of a crystal? It took much of the Nineteenth Century to distill a notion of symmetry for crystals that would have the following criteria.

As a simplification, imagine an infinite crystal filling all space. Its arrangement of atoms would satisfy:

  • The crystal would look the same from any atom in the crystal.
  • There is a minimal distance between any two atoms.
  • For any plane in space, there are atoms on both sides of the plane.

We can distill this into group theoretic language; recall from the 20 June 2015 posting that the orbit of a point p under a group G is the set G(p) = {g(p) : g ∈ G}. We say that a group G of isometries on d-space is crystallographic if:

  • For some (or any – either quantifier will do) point p, we look at the orbit G(p) as follows…
  • There is a number ε > 0 such that for any two distinct points g(p), h(p) ∈ G, | g(p) – h(p)| > ε. This property is called uniform discreteness.
  • For any hyperplane (a line in 2-space, a plane in 3-space), there are points of G(p) on both sides of the hyperplane.

The FyodorovSchoenfliesBieberbach theorem states that:

  1. A group G is crystallographic in d-space if and only if its subgroup T of translations is generated by a basis of d translations, and G/T is finite (see the 20 January 2015 posting for quotient groups).
  2. For any two crystallographic groups G and H, if G is isomorphic to H, then for some affine function f, H = f ∘ G ∘ f-1.
  3. For each d, there are finitely many isomorphism classes of crystallographic groups on d-space.

This is probably the closest thing to a fundamental theorem of mathematical crystallography – at least for classical crystallography; adapting this to cover quasicrystals is one of the great challenges of contemporary mathematical crystallography. At any rate, probably the most accessible account is in R. L. E. Schwarzenberger’s N-dimensional crystallography; see also E. B. Vinberg’s Geometry II: Spaces of Constant Curvature. There are other references, but they embed the theorem in heavy-duty Riemannian geometry.

Jun 232015

The U. K. Guardian – the former Manchester Guardian – reminds us of what impresses ordinary people about crystals. Ten Crystals with Magic Powers is a David Letterman-type list, starting with fluorite’s colors, giant selenite crystals, Viking navigation using Iceland spar, quartz watches, semi-conduction with galena, superhard carbon crystals Tristan Ferroir and company found in a meteorite, radioactive fluorescing autunite, getting sugar to sparkle, biophotonic crystals, and volcanic ice crystals. Useful list to have around the next time you talk to a bunch of high school students about what you do all day – especially autunite, which is almost perfect for the next Marvel movie.

Jun 202015

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.