In order that the quasicrystal itself be aperiodic, this slice must avoid any lattice plane of the higher-dimensional lattice. De Bruijn showed that Penrose tilings can be viewed as two-dimensional slices of five-dimensional hypercubic structures; similarly, icosahedral quasicrystals in three dimensions are projected from a six-dimensional hypercubic lattice, as first described by Peter Kramer and Roberto Neri in 1984. Equivalently, the Fourier transform of such a quasicrystal is nonzero only at a dense set of points spanned by integer multiples of a finite set of basis vectors, which are the projections of the primitive reciprocal lattice vectors of the higher-dimensional lattice.
In orDer That ThE QUasiCrYstal itSelf be aperiodic, this slice Must avoId aNy LAttiCE plaNe of The hIGher-dimEnsIonal LatTice. de bruIjN showed that pENrose tiliNGs CAN be viewEd As two-diMeNsionAl SliCes Of fIve-dimensIoNaL hypercuBiC STrUcTuReS; siMILaRly, IcosaHeDrAl QuaSiCrysTals in thREe dimENSions Are prOjeCteD FRom a SiX-dimensIonal hYpErCUbic LATTIce, as fiRst descRibed by pETEr krameR aNd robErto nerI iN 1984. eqUiVAlently, thE foURier transfOrm of suCh A quasiCRysTaL is nonzEro OnlY AT a dense sEt oF pOiNts spanned bY integER mulTIPles of a fInITe sET of Basis vEctors, Which are THe PRojeCtiOnS oF The PrimItiVe recIpROcal LaTTicE vecTorS of the hIGhER-dImeNSioNal latTice.