12-从仿射F4、B6和E6得到的折叠拟同构异体字
12-fold Quasicrystallography from affine F4, B6, and E6
摘要：
一种可能的方法来获得的准晶格异序结构是高维晶格到2D或3D子空间的投影。在这项工作中，我们介绍了适用于任何高维晶格的一般技术。我们指出，Coxeter数和Coxeter指数的Coxeter Weyl群起着至关重要的作用，在确定平面上的晶格投影以及二面角对称的准晶结构。利用Cartan矩阵的特征向量和特征值来确定nD欧氏空间中的正交向量集，从而选择合适的投影子空间。确定了Coxeter-Weyl群的最大二面角子群，以确定准晶结构的对称性。我们给出了12重对称准晶结构的例子，通过投影由仿射Coxeter-Weyl群Wa（F4），Wa（B6）和Wa（E6）确定的高维晶格获得。这些组共享相同的Coxeter数h=12，具有不同的Coxeter指数。Coxeter群的二面体子群D12可以通过定义两个生成元R1和R2作为Coxeter-Weyl群的生成元的乘积来获得。反射发生器R1和R2在Coxeter平面中工作，其中Coxeter组的Coxeter元件R1 R2表示12阶旋转。晶格的正则投影（条带投影）决定了12重对称的准晶体结构以及4重和6重对称的晶体结构的性质。我们注意到，从点阵Wa（F_4）和Wa（B_6）得到的准晶结构与实验结果是一致的。
英文摘要：
One possible way to obtain the quasicrystallographic structures is the projections of the higher dimensional lattices into 2D or 3D subspaces. In this work we introduce a general technique applicable to any higher dimensional lattice. We point out that the Coxeter number and the Coxeter exponents of a Coxeter-Weyl group play a crucial role in determining the plane onto which the lattice to be projected as well as the dihedral symmetry of the quasicrystal structure. The eigenvectors and eigenvalues of the Cartan matrix are used to determine the set of orthonormal vectors in nD Euclidean space which lead suitable choices for the projection subspaces. The maximal dihedral subgroup of the Coxeter-Weyl group is identified to determine the symmetry of the quasicrystal structure. We give examples for 12-fold symmetric quasicrystal structures obtained by projecting the higher dimensional lattices determined by the affine Coxeter-Weyl groups Wa(F4), Wa(B6) and Wa(E6) . These groups share the same Coxeter number h=12 with different Coxeter exponents. The dihedral subgroup D12 of the Coxeter groups can be obtained by defining two generators R1 and R2 as the products of generators of the Coxeter-Weyl groups. The reflection generators R1 and R2 operate in the Coxeter planes where the Coxeter element R1R2 of the Coxeter group represents the rotation of order 12. The canonical projections (strip projections) of the lattices determine the nature of the quasicrystallographic structures with 12-fold symmetry as well as the crystallographic structures with 4-fold and 6-fold symmetry. We note that the quasicrystal structures obtained from the lattices Wa(F4) and Wa(B6) and are compatible with the experimental results.
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