Quasiperiodic tilings with low-order rotational symmetry

Abstract

Using a projection method of de Bruijn [Proc. K. Ned. Akad. Wet. Ser. A (1981), 43, 39-66], Whittaker & Whittaker [Acta Crysr (1988), A44, 105-112] obtained nonperiodic tilings of the plane with n-fold rotational symmetry, n=5, 7, 8, 9, 10 and 12. However, when their method was applied to the cases of 3-, 4-and 6-fold rotational symmetry it produced periodic tilings. This might be taken as circumstantial evidence that 3-, 4-and 6-fold rotational symmetry is incompatible with nonperiodicity. It is demonstrated that they are compatible by constructing quasiperiodic tilings with only 3-, 4-and 6-fold rotational symmetry. This approach uses basic eigenvalue methods of matrix theory. AbstractThe effects of polarization on X-ray multiple diffraction are investigated experimentally. Polarized and unpolarized incident beams are used in multiple 0108-7673/91/050502-09503.00 diffraction experiments for GaAs and Ge crystals. All the three-beam diffractions in a 360 ° azimuthal rotation of the crystals are analyzed for diffracted intensities and phase determination, based on the article by Chang

A three tile 6-fold golden-mean tiling

“…[4][5][6][7][8]), and the initial intrigue of the exhibition of perfect long-range order in structures with 'forbidden' rotational symmetry, or, orders of symmetry not associated with periodicity. However, neither quasicrystals nor aperiodic tilings are restricted to displaying non-periodic rotational symmetries [9], and indeed, there are examples of aperiodic tilings which share symmetries with periodic systems [10][11][12][13][14]. Further definitions of similar tilings represents an exciting arena of research: both the exploration of the physical properties of these stand-alone tilings, and the possibility of investigating interfacial periodic-aperiodic systems which share rotational symmetries.…”

Quasiperiodic tilings with fourfold symmetry

“…The dual map to the multigrid constructs the quasiperiodic tiling. Recently, Clark & Suryanarayan (1991) presented a nonperiodic fourfold-symmetry tiling constructed by self-similarity. A discussion of the grid method has been provided by Socolar (1989) for tilings with eight-, ten-and twelvefold symmetries.…”

“…3 is a tiling corresponding to Fig. It is therefore reinforced that not all quasiperiodic tilings have self-similarity as an essential property and therefore the self-similarity property cannot be used to establish the quasiperiodicity of a tiling, in contrast to the view of Clark & Suryanarayan (1991). Alternatively, this tiling may be obtained by the projection method in which the four-dimensional hyperlattice spanned by four orthonormal basis vectors (el, 2e2, e3, 2e4) is divided into two subspaces (el, e3 and 2e2, 2e4) each of two dimensions.…”

High-resolution transmission electron microscopyedited by P. Buseck, J. Cowley and L. Eyring

“…3(b) does not possess the self-similarity property. It is therefore reinforced that not all quasiperiodic tilings have self-similarity as an essential property and therefore the self-similarity property cannot be used to establish the quasiperiodicity of a tiling, in contrast to the view of Clark & Suryanarayan (1991). Since quasiperiodicity implies a continuum in a higher-dimensional space, one should 'lift' (Levitov, 1988) a tiling obtained by self-similarity to a higher space to prove its quasiperiodicity.…”