Abstract
We define new tilings of the plane with Robinson triangles, by means of generalized inflation rules, and study their Fourier spectrum. Penrose's matching rules are not obeyed; hence the tilings exhibit new local environments, such as three different bond lengths, as well as new patterns at all length scales. Several kinds of such generalized tilings are considered. A large class of deterministic tilings, including chiral tilings, is strictly quasiperiodic, with a tenfold rotationally symmetric Fourier spectrum. Random tilings, either locally (with extensive entropy) or globally random (without extensive entropy), exhibit a mixed (discrete+continuous) diffraction spectrum, implying a partial perfect long-range order.
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Godrèche, C., Luck, J.M. Quasiperiodicity and randomness in tilings of the plane. J Stat Phys 55, 1–28 (1989). https://doi.org/10.1007/BF01042590
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DOI: https://doi.org/10.1007/BF01042590