1991
DOI: 10.1107/s0108767391003392
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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 demonstra… Show more

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Cited by 6 publications
(4 citation statements)
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“…[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.…”
Section: Introductionmentioning
confidence: 99%
“…[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.…”
Section: Introductionmentioning
confidence: 99%
“…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.…”
mentioning
confidence: 99%
“…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.…”
mentioning
confidence: 99%
“…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.…”
mentioning
confidence: 99%