School of Physics & AstronomyTel Aviv University 



Symmetry of Quasicrystalsby Ron LifshitzWe saw in the introduction that the
facets of a (quasi)crystal as well as its diffraction diagram clearly reveal
a certain kind of symmetry. This symmetry is expressed by the set of rotations
that leave the directions of the facets unchanged (Figure 1), or the set
of rotations that leave the positions of the Bragg peaks in the diffraction
diagram unchanged (Figure 2). But what exactly is the nature of the symmetry
exhibited by the microscopic arrangement of atoms in the crystal itself?
What do we really mean when we say that a crystal has the symmetry of a
certain rotation?




In the case of a periodic crystal we mean that the rotation leaves the density of the crystal invariant to within a translation. This means that after rotating we may need to apply a translation but after that the points of the rotated crystal will exactly coincide with the points of the original crystal. Consider for example a 4fold rotation of Escher's angel and devil drawing (Figure 3). The densities of quasiperiodic crystals, however, in general possess no such symmetries. In fact, it is a nice exercise to show that if a twodimensional crystal contains more than a single point, about which an nfold rotation (n>2) brings it into perfect coincidence with itself, then the crystal is necessarily periodic. So a crystal with, say, 5fold symmetry cannot contain more than a single point of "exact" 5fold symmetry, and if we were given a finite chunk of that crystal there is a good chance that we would never find that point. We can therefore no longer rely on the criterion of "invariance to within a translation" as a definition of crystal symmetry. Nevertheless, certain rotations, when applied to a quasiperiodic crystal, take it into one which looks very much like the original unrotated crystal. 



To demonstrate this we shall use the Penrose
tiling (Figure 4) as our model of a twodimensional quasicrystal. The tiling
contains two kinds of rhombic (diamond shaped) tiles — a thin one with
a small angle of 36 degrees and a thick one with a small angle of 72 degrees.
These two types of tiles are arranged in a very specific and ordered manner
to produce a tiling which is quasiperiodic. If we were to place atoms at,
say, the vertices of the tiling and perform a diffraction experiment, we
would see a diffraction pattern similar to the one in Figure 2 above. In
the next section you can read a little
more about generating these kinds of tilings and even create your own tilings
using an interactive Java program.
This is a good point to clear up a widespread misconception about the Penrose tiling. If you wait long enough for your browser to download the entire image file you will see the Penrose tiling in Figure 4 beginning to rotate. [Clicking your browser's "stop" button will freeze the animation. Clicking the "reload" button will get it going again.] The point of rotation is the one and only point in the whole infinite tiling which has "exact" 5fold symmetry. Nevertheless, due to the existence of this single point many people tend to think that the Penrose tiling has only 5fold symmetry, whereas in fact it has 10fold symmetry. It is a decagonal and not a pentagonal tiling, as we shall immediately explain, even though it contains not even a single point of "exact" 10fold symmetry. 



To see the effect of a 10fold rotation on the Penrose tiling
take a look at the animation in Figure 5. A red Penrose tiling is placed
over an identical blue one. The red tiling is then rotated counterclockwise.
Nothing interesting happens until a 10fold (36 degree) rotation is completed
and a fair fraction of the vertices of the two tilings coincide. The red
tiling is then translated by some amount until whole regions, of about
10 to 15 tiles across, coincide. Finally, the red tiling is translated
even further until regions of the order of the full patch coincide. Between
the coinciding regions there always remain strips, often called "worms,"
containing tiles that do not match. If we could see the entire infinite
tiling we would observe that any bounded region in the rotated tiling can
be found in the unrotated tiling, but the larger the region the further
away we have to look in order to find it. Even so, there is no translation
that will bring the whole infinite red tiling into full coincidence with
the infinite blue one.
The reason for this observation is that the two tilings contain the same statistical distribution of bounded substructures of arbitrary size. This property is demonstrated by Figure 6 below which shows patches of two Penrose tilings that differ by a 10fold rotation. Both patches contain roughly the same number of 5fold stars pointing to the right (highlighted in yellow) and roughly the same number of such stars that are surrounded by five thin tiles (highlighted in magenta). To observe the statistics of larger structures we would need to look at larger patches, but if we could do that we would find that the two tilings contain the same statistical distribution of bounded structures at all scales. 



Two crystals whose densities are statistically the same in this sense are called indistinguishable, a term which was coined by Dan Rokhsar, David Wright, and David Mermin. The key to understanding the symmetry of quasicrystals is to realize that a symmetry operation is one that takes the crystal into an indistinguishable one, just like the 10fold rotation does to the Penrose tiling. Only when the crystal is periodic does the notion of indistinguishability reduce back to the traditional notion of "invariance to within a translation."  
Point groups and space groups 

The collection of all rotations and reflections that are symmetries
of a given crystal form the point group of the crystal. The word
"group" refers to a particular mathematical structure, which in this context
simply states that (1) If a clockwise nfold rotation is a symmetry
then so is a counterclockwise nfold rotation; and that (2) The
effective operation you get by applying
two symmetry operations one after the other is also a symmetry operation. To fully specify the symmetry of the crystal — given by what
is called the space group of the crystal — we need to indicate exactly
in what way the original and rotated versions of the crystal differ. If
the crystal is periodic, where indistinguishability reduces to invariance
to within a translation, all we need is to specify exactly what translation
needs to follow each rotation to leave the crystal invariant. If the crystal
is quasiperiodic a simple translations are insufficient and the required
information is given by a set of very simple functions, called "phase functions."
Alternatively, one may also express the information contained in a phase
function by a "super"translation — a translation in an abstract highdimensional
space called "superspace." We prefer the first, more direct, approach but
there are also advantages to the highdimensional description. For more
detail please refer to the suggested reading below.


Please visit the next section on quasiperiodic tilings, where you can create your own tiling using an interactive Java program, and the section after that which deals with color symmetry and magnetic symmetry of quasicrystals.  


Suggested Further Reading
M. Senechal, "Crystalline symmetries: An informal mathematical introduction" (Adam Hilger, Bristol, 1990). R. Lifshitz, "The symmetry of quasiperiodic crystals," Physica A 232 (1996) 633647. [Click here to download.] N. D. Mermin, "The space groups of icosahedral quasicrystals and cubic, orthorhombic, monoclinic, and triclinic crystals," Reviews of Modern Physics 64 (1992) 349. C. Janot, "Quasicrystals: A primer," 2nd edition (Clarendon Press, Oxford, 1994), ch. 3. [Click here for a book review by Walter Steurer.] 



Some of My Related Publications
R. Lifshitz. Physica A 232 (1996) 633647. R. Lifshitz. Rev. Mod. Phys. 69 (1997) 11811218. R. Lifshitz. Phys. Rev. Lett. 80 (1998) 27172720. R. Lifshitz. Proceedings of the 6th International Conference on Quasicrystals, Ed. S. Takeuchi and T. Fujiwara (World Scientific, Singapore, 1998) 103107. R. Lifshitz. Lecture notes for the International Workshop on Application of symmetry analysis to diffraction investigations, July 69, 1996, Krakow, Poland. 



Other Sites on Crystal Symmetry
Note that Baake uses the term "Local Isomorphism" instead of "Indistinguishability." Note that Radin uses the term "Statistical Symmetry" instead of "Indistinguishability." 


Last modified: October
22, 1998 by Ron Lifshitz.
This page has been visited
times since February 20, 1999.