When Periodicities Enforce Aperiodicity
When Periodicities Enforce Aperiodicity
复制标题
当周期性强制非周期性时
DOI：
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发表时间：
2013
影响因子：
2.4
通讯作者：
Thomas Fernique
中科院分区：
文献类型：
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作者：
Nicolas Bédaride;Thomas Fernique
非周期瓦片和局部规则通常用于模拟准晶体的长范围非周期秩序和稳定准晶体的有限范围能量相互作用。本文的重点是平面菱形平铺，它是欧几里德平面的平铺，可以看作是嵌入在高维空间中的实平面的近似。我们的主要结果是当嵌入空间为四维时，这种平铺存在局部规则的表征。证明是代数和几何的相互作用，它利用了嵌入平面坐标之间的理性依赖关系。我们还将这一结果应用于高维嵌入空间中的某些情况，特别是具有n倍旋转对称的平铺。
Non-periodic tilings and local rules are commonly used to model the long range aperiodic order of quasicrystals and the finite-range energetic interactions that stabilize them. This paper focuses on planar rhombus tilings, which are tilings of the Euclidean plane, which can be seen as an approximation of a real plane embedded in a higher dimensional space. Our main result is a characterization of the existence of local rules for such tilings when the embedding space is four-dimensional. The proof is an interplay of algebra and geometry that makes use of the rational dependencies between the coordinates of the embedded plane. We also apply this result to some cases in a higher dimensional embedding space, notably tilings with n-fold rotational symmetry.