condensed matter - Quasicrystals - Projections from higher dimensional regular crystal lattices

Why are quasicrystals projections from higher dimensional regular crystal lattices?

See for example wikipedia:

»Mathematically, quasicrystals have been shown to be derivable from a general method, which treats them as projections of a higher-dimensional lattice.«

Mathematics aside, is there a physical reason, why this has to be the case?

Answer

In the modern crystallography there is a notation of aperiodic crystals (or quasicrystals). They are crystals with normal basis $\mathbf a,\mathbf b,\mathbf c$ and a set of propagation (or wave) $\mathbf k$-vectors that are incommensurate with the metric $\mathbf a,\mathbf b,\mathbf c$. The atomic positions (or/and occupancies) are modulated according to $$\vec{x}(t_n)=\vec x_0+\sum_k a\cos(k t_n),$$where $x_0$ is the position in zeroth cell, $\mathbf t$ is a vector pointing to an $n$th cell. The extra dimension is simply the phase $x_4=k t_n/(2\pi)$ for one propagation $\mathbf k$-vector. The idea is that you can recover translational invariance by applying a returning translation along the 4th dimension $x_4$.

For each symmetry operator $A$ of the 3D-space group you can define a returning $\mathrm{phase}(A)/(2\pi)$ which runs from 0 to 1 (for each $\mathbf k$-vector), similar to the 3D-fractional coordinates of atoms. One can then construct a 3D+1 (for one-$\mathbf k$ case) superspace group that fully describes the aperiodic crystal symmetry and structure. There are many experimental examples of aperiodic structures and you can also look at the math behind at e.g. this page , this one and references therein.

There is no principal difference between the situation with one $\mathbf k$-vector and the case with two or three $\mathbf k$-vectors that occurs in quasicrystals. For instance icosahedral phase of $\mathrm{AlMn}$ has 3 $\mathbf k$-vectors that corresponds to 3+3 Bragg indices, i.e. 3D+3=6D space.