When looking at (the mathematics of) symmetric structures in space, it doesn’t matter much whether those structures are ensembles of points (representing locations of atoms or molecular building blocks), or graphs embedded in space, or even continuous functions: the theories of each of these is roughly equivalent. The crystallographic restriction remains the crystallographic restriction.
The feature of the November 12 Notices of the American Mathematical Society – the glossy magazine the AMS sends to all 30,000+ members – looks at the crystallographic restriction applied to continuous functions. A little review for visitors a bit hazy on their Calculus III. If we represented a function f : R^{d} → R as an infinite sum of vectors
f(x) = ∑_{n ∈ Zd} a_{n} e^{2 π i n ⋅ x},
where R is the set of reals and Z is the set of integers, we get a periodic function as follows. Write n = (n_{1}, …, n_{d}). Recall that in R^{d}, the standard basis is the set of vectors e_{i} = (0, 0, …, 0, 1, 0, …, 0), with the 1 in the ith position. Then for any term a_{n} e^{2 π i n ⋅ x},
a_{n} e^{2 π i n ⋅ (x + ei)} = a_{n} e^{2 π i n ⋅ x} e^{2 π i n ⋅ ei} = a_{n} e^{2 π i n ⋅ x} e^{2 π i ni} = a_{n} e^{2 π i n ⋅ x}
as e^{2 π i} = 1. Thus f is periodic, with the standard basis as its geometric lattice. A symmetry of f is a function g : R^{d} → R^{d} such that fg = f, and if Π is some subspace of R^{d}, then a symmetry of Π is a function g : Π → Π such that fg = f on Π.
In this article, Frank Farris looks at rotations on planes Π in R^{d}, specifically rotations of order 3 and 5. The point is that you can always get a rotation of order d in a plane in R^{d} (use the rotation
(x_{1}, x_{2}, …, x_{d}) → (x_{d}, x_{1}, …, x_{d – 1}));
the problem is the tilt of the resulting plane.
- For d = 3, Farris obtains a plane parallel to a pair of lattice vectors, so f restricted to that plane has all of its translational symmetries – as well as the rotational symmetry of order 3.
- For d = 5, the plane is not parallel to any lattice vector, so it has no translational symmetries. But it does have a rotational symmetry of order 5 – and there are lattice vectors arbitrarily close to parallel to the plane, so there are translations that are “nearly” symmetries.
This is a variant of one of the standard procedures for obtaining “quasi-periodic” structures.
This is a fairly accessible introduction to basic issues and basic nomenclature.