Quiquandon Marianne, Gratias Denis, Sirindil Abdullah, Portier Richard
Laboratoire de Métallurgie de l'UMR 8247, IRCP Chimie-ParisTech, 11 rue Pierre et Marie Curie, F-75005 Paris, France.
Acta Crystallogr A Found Adv. 2016 Jan;72(Pt 1):55-61. doi: 10.1107/S2053273315018197. Epub 2016 Jan 1.
A twin is defined as being an external operation between two identical crystals that share a fraction of the atomic structure with no discontinuity from one crystal to the other. This includes merohedral twins, twins by reticular merohedry as well as coherent twins by contact where only the habit plane is shared by the two adjacent crystals (epitaxy). Interesting and original cases appear when the invariant substructure is built with positions belonging to the same {\bb Z}-module as, for example, the quinary twin structure first drawn by Albrecht Dürer [(1525). The Painter's Manual: a Manual of Measurement of Lines, Areas and Solids by Means of Compass and Ruler. Facsimile Edition (1977), translated with commentary by W. L. Strauss. New York: Abaris Books]. This paper will show that the Dürer twins, once defined in five-dimensional space, are simple merohedral twins, in the sense of Georges Friedel, leaving the five-dimensional lattice invariant. This analysis will be generalized to some other higher-order {\bb Z}-modules.
孪晶被定义为两个相同晶体之间的一种外部操作，这两个晶体共享一部分原子结构，从一个晶体到另一个晶体没有间断。这包括半面孪晶、网状半面孪晶以及接触形成的共格孪晶，其中相邻的两个晶体仅共享习性面（外延）。当不变子结构由属于同一(\mathbb{Z})-模的位置构成时，就会出现有趣且独特的情况，例如阿尔布雷希特·丢勒首次绘制的五元孪晶结构[（1525年）。《画家手册：用圆规和直尺测量线、面积和立体的手册》。影印版（1977年），由W. L. 施特劳斯翻译并附有注释。纽约：阿巴里斯图书公司]。本文将表明，一旦在五维空间中定义，丢勒孪晶在乔治·弗里德尔的意义上是简单的半面孪晶，使五维晶格保持不变。这种分析将推广到其他一些高阶(\mathbb{Z})-模。
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