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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?

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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.

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    $\begingroup$ Welcome to the site! Note that you can use LaTeX notation (so that e.g. $x(t_n)=x(0)+Ʃ_{k} a \cos(k t_n+\phi)$ will render as $x(t_n)=x(0)+Ʃ_{k} a \cos(k t_n+\phi)$), it will make your post much easier to read. $\endgroup$ – Emilio Pisanty Feb 19 '15 at 11:23
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The QSN and Its Mapping to E8

The Quasicrystalline Spin Network (QSN) is a 3D quasicrystalline point space on which we model physics. The QSN is deeply related to the E8 crystal. The following is a brief explanation of the relationship between the various related objects.

We begin with an 8-dimensional crystal called the E8 lattice. The E8 lattice is an 8D point set representing the densest packing of spheres in 8D. The basic cell of the E8 lattice, the Gosset polytope, has 240 vertices and accurately corresponds to all particles and forces in our (3D) reality and their interactions, specifically the way they can all transform from one to another through a process called gauge symmetry transformation.

FOR GRAPHICS AND FULL ARTICLE:

http://www.quantumgravityresearch.org/portfolio/a-deep-link-between-3d-and-8d

There are many research papers on this website that relate to quasicrystalline mathematics.

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  • $\begingroup$ Visualizing a Hypothetical Planck Scale Substructure of Reality: Geometer, animator, and QGR research scientist Dugan Hammock displays some of his work visualizing the quasicrystalline point space on which we model our physics. This 3D point space which we call the QSN (Quasicrystalline Spin Network) is derived from a 4D quasicrystalline point space called the Elser-Sloane quasicrystal, which is a projection to 4D of the E8 lattice at a particular angle. youtu.be/tMg0-9Vc6V8 $\endgroup$ – Giovanna Brandi Oct 2 '17 at 18:31

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