Today I'd like to talk about tesselations of the plane. With regards to regular polygons, it is not hard to see that only triangles, squares, and hexagons work, and of course one may extend this to any shearing or stretching of those shapes (parallelograms). What happens if we don't restrict ourselves to regular polygons? Still if we confine ourselves to a single repeating unit, we find that the symmetries obeyed by the resulting pattern continue to be 2-fold, 3-fold, 4-fold, or 6-fold.

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For example, the above Escher drawing exhibits 3-fold symmetry. This simple observation turns out to be fundamental in describing the structure of crystals, Nature's very own tesselations of space.

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The above depicts the crystal structure of diamond, which is cubic; specifically, it's known among crystallographers as the face-centered cubic Bravais lattice. Up to shearing and stretching, the crystal systems on which Bravais lattices are based are either cubic or hexagonal. Why do no other crystal systems (and hence no other Bravais lattices) appear? Why, for example, are there no icosahedral lattices?

To investigate this question, we'll make the notion of a lattice more precise as follows and end up proving the result known as the crystallographic restriction theorem. Define a lattice in to be a discrete subgroup of that spans it as a vector space. More concretely, we can describe a lattice in by specifying linearly independent vectors and considering the set



For example, the set of points with integer coordinates forms a lattice, and is in some sense the only lattice (but not for our purposes). We relate lattices to tesselations as follows: in any tesselation of a single repeating unit, we can identify a point in that repeating unit. The set of all points in the entire tesselation should form a lattice because we want tesselations to have translational symmetry in independent directions. Now that we have mathematically formulated the notion of a lattice, what can we say about its symmetries?

By "symmetries" we mean orientation-preserving isometries that map every point in the lattice to another point in the lattice; this is intuitive. The isometries of are always linear, so we are talking about a subgroup of the Euclidean group that we can exhibit as a m atrix group. A lattice is said to have -fold symmetry there exists an orientation-preserving isometry (in two and three dimensions, rotations about a point and an axis, respectively) such that , the identity map. We now wish to show that the only possible orders in dimensions and occur when .

Proof. Written in the basis , every symmetry of the lattice must take each basis vector to an integer combination of the other basis vectors, hence any such symmetry must have integer coordinates in this basis, in other words, must be an element of .

In dimensions and , the minimal polynomial of such a matrix has degree at most , has integer coefficients, and must divide the polynomial . On the other hand, the monic irreducible factors of are precisely the cyclotomic polynomials. The cyclotomic polynomials of degree at most (in fact, exactly ) are precisely . Hence the only nontrivial finite orders an element of can have are as desired.

The beauty of the above approach is that it is absolutely general. For example, the possible symmetries of a lattice in dimensions occur when .

Corollary: Let be positive integers. If is rational, then it can only be . (See also MellowMelon's demonstration using Chebyshev polynomials.) This is not, strictly speaking, a corollary; one needs the additional observation that the irreducible factors of are still the cyclotomic polynomials over as well as over , that is, Gauss's lemma.

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There are a few interesting questions to ponder from here. One is purely mathematical: what interest are lattices to a mathematician? I hope I have already convinced you of the power of lattice methods in number theory. Indeed, you may have spotted that the lattices with the symmetries described above could be said to include the Gaussian and Eisenstein integers, which have shown up more than once in this blog.



In Lie theory, lattices play the role of root systems, a method by which Lie groups and Lie algebras are analyzed and classified. Certain extremely symmetrical lattices correspond to what are called the exceptional Lie groups, and these extremely symmetrical objects occur in physical theories; for example, has some tantalizing connections to theoretical physics.

Lattices in the complex plane also occur in the theory of elliptic curves via a type of doubly periodic function called an elliptic function. Elliptic functions parameterize elliptic curves over and have periods which are two complex numbers whose ratio is not real. Since a doubly periodic function has the same value if its domain is taken "modulo" the lattice spanned by its periods, we can identify the range of the elliptic function - that is, the curve itself - with a fundamental parallelogram of the lattice with its sides identified - which is a torus!

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And now you know why all those specials on Fermat's Last Theorem tell you that elliptic curves are doughnuts.

The other question we could ask is scientific: we assumed that crystals are described by lattices with independent translational symmetries. Do there exist crystals that do not have this property? Scientists did not discover physical evidence of such quasicrystals until the 80s, but

- the mathematical community was aware that these structures could, in principle, exist decades earlier, and
- Islamic architects were already using aperiodic tilings centuries earlier!



Some aperiodic tilings occur with -fold, -fold, or -fold symmetry, and that's because they are projections down from -dimensional lattices along non-lattice planes, which explains both the large amount of structure and the lack of translational symmetry.

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This scientific point brings up another mathematical question. Crystals tend to organize themselves into lattices because that is the structure that maximizes interaction among the atoms and therefore minimizes energy and excess volume. In other words, the relationship between crystals and lattice structures is related to the circle-packing problem. Is it always the case, then, that the densest packing in dimensions is a lattice packing?

This seems a silly question to ask at first. Intuitively, any irregularity in a packing correlates to some excess that could be removed by increasing the regularity of the packing. However, even the statement for dimension was only proven very recently, and this question is actually open in dimensions and higher.

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Practice Problem 1: Prove the general form of Minkowski's theorem. (Try to find a proof other than the one given in the Wikipedia article! In two dimensions, there is a proof using Pick's theorem.)

Practice Problem 2: Let and . Show that for all . Show, on the other hand, that .

Practice Problem 3: The generic example of a potential function gives the energy of two spherical charges with charges at a distance from each other as . Compute the average potential energy per charge of an infinite lattice of alternately charged circles of radius and charges in the plane (with circles centered at lattice points with even coordinates). This is essentially a Madelung constant. (This sum obviously doesn't converge absolutely, so be careful. Also, I haven't done this problem and I'm not sure the answer comes out nicely at all )