Three-periodic nets, tilings and surfaces. A short review and new results.

IF 1.9 4区材料科学 Q3 CHEMISTRY, MULTIDISCIPLINARY

Acta Crystallographica Section A: Foundations and Advances Pub Date : 2023-03-01 DOI:10.1107/S2053273323000414

Olaf Delgado-Friedrichs, Michael O'Keeffe, Davide M Proserpio, Michael M J Treacy

引用次数: 0

Abstract

A brief introductory review is provided of the theory of tilings of 3-periodic nets and related periodic surfaces. Tilings have a transitivity [p q r s] indicating the vertex, edge, face and tile transitivity. Proper, natural and minimal-transitivity tilings of nets are described. Essential rings are used for finding the minimal-transitivity tiling for a given net. Tiling theory is used to find all edge- and face-transitive tilings (q = r = 1) and to find seven, one, one and 12 examples of tilings with transitivity [1 1 1 1], [1 1 1 2], [2 1 1 1] and [2 1 1 2], respectively. These are all minimal-transitivity tilings. This work identifies the 3-periodic surfaces defined by the nets of the tiling and its dual and indicates how 3-periodic nets arise from tilings of those surfaces.

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三周期网，瓷砖和表面。一个简短的回顾和新的结果。

简要介绍了三周期网和相关周期曲面的铺层理论。贴图具有传递性[p q r s]，表示顶点、边缘、面和贴图的传递性。描述了适当的、自然的和最小传递性的网砖。基本环用于寻找给定网的最小传递性平铺。用平铺理论求出所有边与面可传递的平铺(q = r = 1)，并分别求出7个、1个、1个和12个具有可传递性的平铺例子[1 1 1 1 1 1 1 1 12]、[1 1 1 1 1 1 1]和[2 1 1 12]。这些都是最小传递性贴图。这项工作确定了由瓷砖网及其对偶定义的3周期表面，并指出了这些表面的瓷砖如何产生3周期网。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Acta Crystallographica Section A: Foundations and Advances CHEMISTRY, MULTIDISCIPLINARYCRYSTALLOGRAPH-CRYSTALLOGRAPHY

CiteScore

2.60

自引率

11.10%

发文量

419

期刊介绍： Acta Crystallographica Section A: Foundations and Advances publishes articles reporting advances in the theory and practice of all areas of crystallography in the broadest sense. As well as traditional crystallography, this includes nanocrystals, metacrystals, amorphous materials, quasicrystals, synchrotron and XFEL studies, coherent scattering, diffraction imaging, time-resolved studies and the structure of strain and defects in materials. The journal has two parts, a rapid-publication Advances section and the traditional Foundations section. Articles for the Advances section are of particularly high value and impact. They receive expedited treatment and may be highlighted by an accompanying scientific commentary article and a press release. Further details are given in the November 2013 Editorial. The central themes of the journal are, on the one hand, experimental and theoretical studies of the properties and arrangements of atoms, ions and molecules in condensed matter, periodic, quasiperiodic or amorphous, ideal or real, and, on the other, the theoretical and experimental aspects of the various methods to determine these properties and arrangements.